Quantitative Methods 2 ICMR Workbook

Quantitative Methods

WORKBOOK Volume II

The ICFAI University # 52, Nagarjuna Hills, Hyderabad - 500 082

© 2005 The Icfai University Press. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, used in a spreadsheet, or transmitted in any form or by any means – electronic, mechanical, photocopying or otherwise – without prior permission in writing from The Icfai University Press.

ISBN: 81-7881-964-3 Ref. No. QMWB-II 10200503 For any clarification regarding this book, the students may please write to us giving the above reference number of this book specifying chapter and page number. While every possible care has been taken in type-setting and printing this book, we welcome suggestions from students for improvement in future editions.

Preface The ICFAI University has been upgrading the study material so that it is amenable for self study by the Distance Learning Students. We are delighted to publish a Workbook for the benefit of the students preparing for the examinations. The workbook is divided into three different parts. Brief Summaries of Chapters A brief summary of all the chapters in the textbook are given here for easy recollection of the topics studied. Part I: Questions and Answers on Basic Concepts (with Explanatory Notes) Students are advised to go through the relevant textbook carefully and understand the subject thoroughly before attempting Part I. Under no circumstances the students should attempt Part I without fully grasping the subject material provided in the textbook. Frequently Used Formulae Similarly the formulae used in the various topics have been given here for easy recollection while working out the problems. Part II: Problems and Solutions The students should attempt Part II only after carefully going through all the solved examples in the textbook. A few repetitive problems are provided for the students to have sufficient practice. Part III: Model Question Papers (with Suggested Answers) The Model Question Papers are included in Part III of this workbook. The students should attempt all model question papers under simulated examination environment. They should self score their answers by comparing them with the model answers. Effective from April, 2003, the examinations for all the subjects of DBF/CFA (Level-I) consist of only multiple-choice questions. Each paper consists of Part I and Part II. Part I is intended to test the conceptual understanding of the students. It contains 40 questions carrying one point each. Part II contains problems with an aggregate weightage of 60 points. Please remember that the ICFAI University examinations follow high standards that demand rigorous preparation. Students have to prepare well to meet these standards. There are no shortcuts to success. We hope that the students will find this workbook useful in preparing for the ICFAI University examinations. Work Hard. Work Smart. Work Regularly. You have every chance to succeed. All the best.

Contents

Brief Summaries of Chapters

1

Part I: Questions and Answers on Basic Concepts (with Explanatory Notes)

6

Frequently Used Formulae

130

Part II: Problems and Solutions

139

Part III: Model Question Papers (with Suggested Answers)

359

Detailed Curriculum Paper II

Probability Distribution and Decision Theory: Discrete and Continuous Random Variables, Discrete Uniform, Binomial, Hypergeometric, Continuous Uniform, Normal, Lognormal, t and F Distributions, Decision making under Conditions of Certainty and Uncertainty, Conditional Profit/Loss Table, Expected Value of Perfect Information, Decision Trees. Correlation: The Concept of Covariance and Correlation, Cause and Effect in Correlation, The Scatter Diagram, Correlation analysis, Rank Correlation. Simple Regression: Assumptions, Calculation of Regression Coefficients, Limitations, Testing the Significance of the Model. Multiple Regression: Regression with two Independent Variables, Computers and Multiple Regression, Coefficient of Multiple Correlation, Coefficient of Multiple Determination, Testing the Significance of the Model. Time Series Analysis: Introduction, Variations in Time Series, Trend Analysis, Cyclical Variation, Seasonal Variation, Irregular Variation, Time Series Analysis in Forecasting. Index Numbers: Concept, Types, (Un) Weighted and Weighted Aggregates, Average of Relatives Methods, Quantity and Value Indices. Sampling: Population and Samples, Types of Sampling, Errors in Statistics, Central Limit Theorem, Point Estimates, Interval Estimates, Standard Error, Estimating Population Means. Testing of Hypothesis: Hypothesis testing for Means and Proportions, One Sample Test, Two Sample Test, Runs Test. Quality Control: Control Charts for Mean, Variability, Attributes, Chi-Square Test; Analysis of Variance. Current Developments.

Brief Summaries of Chapters Probability Distribution and Decision Theory Probability distribution gives us an idea about the likely value of a random variable (a variable which assumes different numerical values as a result of random experiments) and the chances of occurrence of the various values. For taking decisions under uncertainty, the concept of expected value of a random variable is used where the expected value is the mean of a probability distribution. •

Given a probability distribution of paired data {x,y}, the covariance of the data measures the variability of a parameter (or data set) in relation to another parameter (or data set). Covariance can be used to find the variance of a sum of two or more random variables.

•

In Discrete Uniform distribution, the random variables assume discrete values. Binomial distribution and Hypergeometric distribution are examples of discrete uniform distribution. In Binomial distribution the Bernoulli experiment (in which a variable has two outcomes: success (1) and failure (0)) is repeated n times with ‘p’ being the probability of success and ‘q’ being the probability of failure. The sampling in this case is done with replacement. If the Bernoulli experiment is repeated without replacement then we get a Hypergeometric distribution.

•

In case of Continuous Uniform distribution, the random variable can take any value between zero and up to but not including 1. The Normal distribution is a continuous distribution in which a large number of observations cluster around the mean value 5 and their frequency drops sharply as we move away from the mean in either direction. The normal distribution is uni-modal and symmetrical and standard normal distribution is a special case of normal distribution with mean = 0 and standard deviation = 1. For a Lognormal distribution, if ln(x) is a normally distributed random variable, then X is said to be a lognormal variable. The t-distribution is symmetrical like the normal distribution but its peak is lower than the normal curve and its tail is a little higher. It is used when sample size is less than or equal to 30. F-distribution is the distribution of the ratio of two random variables.

Decision Theory Decision theory focuses on decision-making environments in which consequences of decision-making are quantifiable. • Decision-making can be of three types: i.

Decision-making under conditions of certainty (where the decision maker is certain about the occurrence of a particular state of nature).

ii.

Decision-making under conditions of uncertainty (where the decision maker has no knowledge about the likelihood of occurrence of various states of nature).

iii.

Decision-making under conditions of risk (where the decision maker has sufficient knowledge about the states of nature to assign probabilities to their likelihood of occurrence).

•

Decision tree is used as a tool to examine all possible outcomes, desirable and undesirable, their chance of occurring, etc.

•

In situations where it is difficult to manage a conditional probability table (used in case of decision-making under risk), the Marginal analysis approach proves to be very useful. Marginal analysis uses the concept of marginal profit and marginal loss, which represent the additional profit (or loss) incurred by increasing the activity level by one unit.

All the above techniques have numerous applications in finance, for example in project evaluation, determination of safety stock level, receivables management, etc. 1

Statistical Inferences •

A population is a collection of all data points being studied and a sample is a part of a population. Sampling is advantageous over a census because (a) population might be too large (b) study of a sample is cheaper than the study of a population (c) chances of errors while processing the data are less. Sampling can be broadly classified into random sampling and judgmental sampling. Random sampling is further divided into simple random sampling, stratified sampling, systematic sampling and cluster sampling. Sometimes wrong conclusions are made either due to non-sampling errors (caused by deficiencies in the collection and editing of data) or due to sampling errors (difference between actual population parameter and the sample statistic). The standard error of a statistic is the standard deviation of the sampling distribution of the statistic. It is used to test whether the difference between the sample statistic and the population parameter is significant or is due to sampling fluctuations. It is also used to find the interval estimate of a population parameter and to test the precision of an estimate.

•

If an estimate of a population parameter is given by a single value, then the estimate is called a point estimate of the parameter. But if an estimate of a population parameter is given by two distinct numbers between which the parameter may be considered to lie, then the estimate is called an interval estimate of the parameter. Interval estimates are usually preferred over point estimates because point estimates provide only a rough guide whereas an interval estimate describes a range of values within which a population parameter is likely to lie.

•

Hypothesis testing is one of the objectives of sampling theory. It begins by making an assumption about the population parameter and then a sample data is collected to determine the sample statistic. To test the validity of the hypothesis, the difference between the hypothesized value and the actual value of the sample statistic is determined and if the difference is large then the hypothesis is rejected. While testing a hypothesis, a critical region has to be selected. A critical region or region of rejection is the area under the sampling distribution in which the test statistic value has to fall for the null hypothesis to be rejected. The various hypothesis tests are large sample test for mean, small sample test for mean, large sample test for proportion, test for equality of two means and test for equality of proportions. While testing a hypothesis, two errors might arise: (a) Type I error: Rejecting a null hypothesis when it is true, (b) Type II error: Accepting the null hypothesis when it is false.

Linear Regression Correlation and Regression analysis help in understanding the relationship between variables. Linear regression constructs linear models while correlation coefficient gives a measure of the relevance of the model. •

Correlation can be defined as the degree of association between two variables. The measure of correlation is called correlation coefficient and it ranges between 0 and 1. Of the several mathematical methods of measuring correlation, the Karl Pearson’s method, popularly known as Pearson’s coefficient of correlation is most widely used in practice.

•

Simple correlation studies the association between two variables whereas Multiple correlation involves measurement of the degree of association between more than two variables, with one variable being dependent and others being independent. If we seek to discover just the influence of one independent variable on the dependent variable while taking into consideration the fact that the other variables are also operative, then we are measuring Partial correlation.

•

In situations where correlation cannot be measured, rank correlation is used to measure the degree of agreement between two sets of ranks.

•

Simple linear regression computes the model that best fits the relationship indicated by coefficient of correlation. Before performing correlation and regression analysis, the cause and effect relationship between the variables should be understood. The assumptions in regression are that the relationship between the distributions X and Y is linear; at each X, distribution of Yx is normal and the variance σx 2 are equal; the Y values are independent of each other. 2

•

The regression line should be drawn on a scatter diagram in such a way that when the squared values of the vertical distance from each plotted point is added, the total amount will be the smallest amount. This criterion is called the method of least squares. Using this principle, we obtain the normal equations which when solved gives us the regression constants. Thus if a causal relationship is established, we can fit a regression line which can be used to predict the value of the dependent variable for a given value of the independent variable.

•

The standard error measures the reliability of the estimate obtained from the regression equation and the coefficient of determination helps us to compute the variation explained by the model. The regression model can also be tested using Analysis of Variances (ANOVA).

The serious limitation in using regression analysis is that past trends are used to estimate future trends. The correlation and regression analysis are applied in the field of finance to find the risk of a portfolio, in Cost-Volume-Profit analysis, in time series and demand forecasting.

Multiple Regression •

In simple regression it is assumed that the dependent variable Y is related to a single variable X. But in practice, a dependent variable may depend on more than one independent variable. In such a situation, a multiple regression equation with more than one independent variable is used. The constants in a multiple regression equation can be computed with the help of the “normal equations”. Computation of the estimators of the standard error of the regression coefficients becomes complex in case of multiple regression and usually computers are used for this purpose. As the degree of correlation between the independent variables increases, the regression coefficients become less reliable, i.e. although the independent variables may together explain the dependent variable, but because of multicollinearity the coefficients of the explanatory variables may be rejected. So in case of a multiple regression equation, the coefficient of multiple correlation should also be computed. Partial correlation coefficients help in finding out the extent to which the variation in the dependent variable is explained by one independent variable if all other independent variables are kept constant.

Index Numbers •

Index numbers are one of the most widely used statistical indicators. Generally, used to indicate the state of the economy, index numbers are aptly called ‘barometers of economic activity’. An index number is a statistical measure designed to show changes in a variable or a group or related variables with respect to time, geographic location or other characteristics such as income, profession, etc. It is calculated as a ratio of the current value to a base value and is expressed as a percentage. The index number for the base year is always 100. Index numbers are very useful in revealing a general trend of the phenomenon under study, in policy making, in determining purchasing power of the rupee and in deflating time series data (i.e. adjusting the original data to reflect reality).

•

There are three types of principle indices:

•

i.

Price index (compares changes in price from one period to another).

ii.

Quantity index (measures change in quantity from one period to another).

iii.

Value index (combines price and quantity changes to present a more spatial comparison).

There are two approaches for constructing an index number, namely the aggregates method and average of relatives method. The index constructed in either of these methods could be an unweighted index or a weighted index. –

An unweighted aggregate index is calculated by totaling the current year/given year’s elements and then dividing them by the sum of the same elements during the base period.

–

In weighted aggregate index, weights are assigned to the various items according to their significance. The important methods of assigning weights to an index are: 1.

Laspeyres method (uses the quantities consumed during the base period in computing the index number). 3

2.

Paasche’s method (uses quantity measures for the current period rather than the base period).

3.

Fixed weight aggregates method (the weights used are neither from the base period nor from the current period but from a representative period).

4.

Fisher’s ideal index number (is the geometric mean of the Laspeyers and Paasche’s indices).

5.

Marshall Edgeworth method (uses both current year as well as base year prices and quantities).

–

In case of unweighted average of relatives method, when a price index is constructed, all price relatives (quantity relatives in case of quantity index) are to be obtained for all items included in the index and then the average of these price relatives is computed.

–

For weighted average of relatives method, the value of each element in the composite is used as the weight. The values could be from the base year, current or any fixed year.

•

In case of Chain index numbers, new relatives are constructed for each year with the previous year as base and these new relatives enable comparison between one period and another period which succeeds it and hence referred to as link relatives.

•

The consistency of index numbers can be tested using the time reversal test, factor reversal test and the circular test.

Time Series Analysis Time series analysis is used to identify and determine the pattern of changes in the data collected over regular intervals of time. The identified patterns are projected into future to get an estimate of the variable under consideration. •

The variations observed in the time series can be broadly classified as (i) the secular trend, (ii) the cyclical fluctuation, (iii) the seasonal variation and (iv) the irregular variation.

•

Secular trend observes the long-term behavior of the variable. Graphically, the secular trend is shown as a straight line with an upward slope, with the actual time series represented as a curve moving towards and away from the trend line. A best-fit trend line can be obtained by using the least squares method. Secular trend is useful in examining whether a policy implemented has yielded the necessary results or not and its future impact; in estimating the variable under study; in studying any other component present in the series; etc.

•

By cyclical variations, we refer to the long-term movement of the variable about the trend line. Generally, the behavior of the variable is considered to be cyclic only if the movements recur after a period of more than one year. The Residual method is used to isolate the component of cyclical variation in a time series and this method can be bifurcated into two measures: percent of trend and relative cyclical residual measures.

•

Under seasonal variation, we observe the similar pattern that the variable under consideration shows during certain months of the successive years. Usually the time period over which this variation is considered can consist of days, weeks, months and at the most one year. The ratio to moving averages method is used to identify the component of seasonal variation in a time series. The use of indices to nullify the seasonal effects is referred to as deseasonalizing the time series.

•

Irregular variation is the movement of the variable in a random fashion without consistency, which makes it highly unpredictable. Since this type of irregularity exists for very short durations, the period under consideration will be of days, weeks and at the most, of months.

Quality Control •

Quality refers to “the product being in a state of readiness for use irrespective of the environmental conditions and the work delivered by it being flawless on a consistent basis”. It is the responsibility of the manufacturer to look into the product’s quality so that it performs the way it was expected to or as claimed by him. This can be achieved if the production process at each stage and the constituent parts which go into the making of the 4

product are carefully monitored. Quality standards in manufacturing process and the service industry can be achieved through the application of statistical methods. Statistical Process Control (SPC) is a technique which not only reduces the variation (present in the manufacturing process) but also helps to identify the causes responsible for such variations. The variations in the manufacturing process are of two types: random and non-random variations. If the process is “out-of-control”, then it indicates the presence of non-random patterns and the management should eliminate such variations in order to bring the process “in-control”. Once this is done, the whole process can be redesigned to improve or reduce the incidence of random or inherent variability. The various control charts, used for monitoring the manufacturing process to identify whether the process has been under control or not, are X charts, R charts and p charts.

Chi-Square Test and Analysis of Variance •

Chi-square test allows us to test whether the difference between the proportions representing more than two samples is significant or not. It is also employed to determine whether the two attributes according to which a population is categorized are independent of each other or not and it also serves as a test for distributional goodness of fit. Analysis of variance (ANOVA) allows us to test whether the differences among more than two sample means are significant or not. This technique overcomes the drawback of the method used in statistical inferences, which allows us to test the significance among the means drawn from two populations only. The technique of ANOVA makes two basic assumptions: (1) the various populations from which the samples are drawn should be normal and have the same variance, (2) ANOVA is based on a comparison of two estimates of the population variance, one estimate is obtained from variance among the sample means and the second estimate is obtained from variation that exists within the samples. ANOVA consists of three basic steps (i) estimating the variance among the sample means, (ii) estimating the variance that exists within the samples and (iii) accepting the null hypothesis if the two estimates are equal or else rejecting the null hypothesis. The decision regarding acceptance or rejection of null hypothesis is based on the value of the F-ratio.

Simulation •

Simulation refers to the creation of conditions that prevail in reality, in order to draw certain conclusions from the trials that are conducted in the artificial conditions. Usually, there are two approaches to decision-making: (a) Analytical approach, (b) Simulation. Simulation is used in situations where the decisions are to be taken under conditions of uncertainty. Simulation as a quantitative method requires the setting up of a mathematical model which would represent the interrelationships between the variables involved in the actual situation in which a decision is to be taken. This is followed by a number of trials or experiments that are conducted to determine the results that can be expected when the variables assume various values. Monte Carlo method is an important technique of simulation. The advantages of simulation are its flexibility and its ability to break down complex systems into sub-systems that can be studied individually or jointly with other sub-systems. There are various drawbacks involved in the simulation technique which are (i) it is very computer-intensive, (ii) it does not produce optimal solutions and is very expensive.

5


Part I: Questions on Basic Concepts Probability Distribution and Decision Theory 1.

The probability of getting exactly 3 heads in four tosses of a fair coin is a. 1/2 b. 1/4 c. 1/8 d. 1/10 e. 1/16.

2.

Find the mean μ and variance σ2 of a binomial distribution with n = 12 and p = 0.45.

3.

4.

5.

6.

a.

μ = 5.40, σ 2 = 2.97

b.

μ = 2.97, σ 2 = 5.4

c.

μ = 5.40, σ 2 = 1.72

d.

μ = 2.43, σ 2 = 2.97

e.

μ = 5.40, σ 2 = 2.43.

The area under the standard normal curve between the points 0 and 2.9 is 0.4981. The area between the points –2.9 and 2.9 is given by a. 0.9962 b. 0.4981 c. 0.3761 d. 0.2491 e. 0.1245. In a semester examination, the mean grade of the students was 72 with a standard deviation of 9. The top 10% of the students are to receive A’s. Assuming that the grades are normally distributed, find the mean grade that a student must get in order to receive an A? a. 83.52. b. 82.53. c. 75.67. d. 63.79. e. 60.01. What is the probability of guessing correctly at least 4 of the 6 answers on a true-false examination? a. 12/32. b. 8/32. c. 10/32. d. 11/32. e. 13/32. When the decision maker possesses information about the probabilities of the possible states of nature, his/her decisions are said to be made under conditions of a. Uncertainty b. Risk c. Certainty d. Probability e. None of the above.

6

Part I

Based on the following information answer questions 7 to 12. A company has under consideration, four new products and one of these will be released in the market. Three demand conditions prevail and these are low, medium and high. We have the following information on annual profits generated by the four products for the various states of nature (demand). Demand Product A

Conditions/Strategy Low Medium High Probability 0.2 0.4 0.4 Profit (in lakh of Rs.) 2.0 6.0 10.0 Product B Probability 0.2 0.4 0.4 Profit (in lakh of Rs.) 5.0 12.0 15.0 Product C Probability 0.2 0.4 0.4 Profit (in lakh of Rs.) 1.0 3.0 6.0 Product D Probability 0.2 0.4 0.4 Profit (in lakh of Rs.) 3.0 8.0 16.0 7. Use the maximax criterion and decide, which product should be released?

8.

9.

a.

A.

b.

B.

c.

C.

d.

D.

e.

B or D.

Use the maximin criterion and decide, which product should be released? a.

A.

b.

B.

c.

C.

d.

D.

e.

B or D.

Use the minimax regret criterion and decide, which product should be released? a.

A.

b.

B.

c.

C.

d.

D.

e.

B or D.

10. The index of optimism α = 0.5. Use Hurwicz criterion and decide, which product should be released in the market? a.

A.

b.

B.

c.

C.

d.

D.

e.

B or D.

11. Use the expected value criterion and decide, which product should be released in the market? a.

A.

b.

B.

c.

C.

d.

D.

e.

B or D.

7


12. What is the expected value of perfect information (in lakh of Rs.)? a. 0.4. b. 0.6. c. 0.8. d. 1. e. 1.2. 13. A decision-maker is using the Hurwicz criterion to take a decision under conditions of uncertainty. In which of the following cases can he be said to have a tendency towards pessimism? a.

Index of optimism α lies between 0.1 and 0.3.

b.

Index of optimism α lies between 0.6 and 0.7.

c. d. e.

Index of optimism α lies between 0.7 and 0.9. Both (a) and (b) above. Both (b) and (c) above.

Based on the following information answer questions 14 to 20. The board of ABC company is considering some major policy changes that will affect the company’s three divisions. The management was keen on getting the employees’ opinion before going ahead with the changes. Meetings were held with groups of employees from the three divisions. The opinions of the employees are summarized in the table given below. Division Finance Garments Strongly oppose 2 2 4 Slightly oppose 2 4 3 Neutral 3 3 5 Slightly support 2 3 2 Strongly support 6 3 1 14. An employee from one of the groups strongly opposes the policy changes. What is the probability, that he is from the Finance division? Opinion

a.

4/45.

b.

1/4.

c.

4/15.

d.

1/2.

e.

3/4.

Textile

15. An employee from one of the groups strongly supports the policy changes. What is the probability, that he is from the garments division? a.

3/45.

b.

1/5.

c.

3/10.

d.

1/2.

e.

2/3.

16. An employee from one of the groups remains neutral to the proposed policy changes. What is the probability, that he is from the textile division? a.

3/45.

b.

11/45.

c.

3/11.

d.

1/2.

e.

3/4.

8

Part I

17. What is the probability that an employee, from the Finance division group strongly opposes the policy changes? a.

4/45.

b.

4/15.

c.

1/3.

d.

1/2.

e.

3/4.

18. What is the probability that an employee, from the garments division group strongly supports the changes? a.

3/45.

b.

1/5.

c.

3/10.

d.

1/3.

e.

1/2.

19. What is the probability, that an employee from the textile division group remains neutral to the proposed policy changes? a.

3/45.

b.

1/5.

c.

3/11.

d.

1/3.

e.

1/2.

20. What is the probability, that an employee from one of the groups strongly supports the policy changes? a.

2/9.

b.

1/3.

c.

1/2.

d.

3/5.

e.

3/4.

Based on the following information answer questions 21 to 23. Assume that a couple prefers the female child and therefore will stop having children only after the birth of the female child. Take the probability of the female child as 0.52. 21. What is the probability that the couple limits the number of their children to 3? a.

0.1406.

b.

0.1298.

c.

0.1198.

d.

0.1106.

e.

0.0575.

22. If the random variable X denotes the number of children that the couple would have, what is the probability distribution of X? a. b. c. d. e.

X P (X) P (X) P (X) P (X) P (X)

1 0.52 0.52 0.52 0.52 0.52

2 0.2496 0.2704 0.2304 0.2496 0.2704

3 0.1198 0.1406 0.1106 0.1298 0.1106

...

k k-1

... ... ... ...

(0.48) (0.52) (0.48)k (0.52) (0.52)k (0.52) (0.48)k (0.52) (0.52)k (0.52)

9


23. Which of the following statements is/are true when decisions are made under conditions of risk? i.

All possible states of nature are not known.

ii.

All possible states of nature are known.

iii.

The decision-maker does not have any knowledge of the probabilities of occurrence of the various states of nature.

iv.

The decision-maker has sufficient knowledge to assign probabilities of occurrence to the various states of nature.

a.

Only (i)

b.

Only (ii)

c.

Both (ii) and (iii)

d.

Both (ii) and (iv)

e.

Only (iii).

24. ABC company has identified a potential offshore oilfield. Studies conducted, indicate that the probability of a dry hole is 0.25. What is the probability, that the company makes a good strike in not more than 3 attempts? a.

0.9844.

b.

0.7523.

c.

0.6321.

d.

0.5137.

e.

0.2513.

Based on the following information answer questions 25 and 26. A newspaper vendor buys newspapers for Rs.0.67 each and sells them for Rs.1.50. The newspapers remaining unsold at the end of the day are sold the next day for Rs.0.50. The daily demand for newspapers is normally distributed with mean 375 and variance 400. The vendor presently buys 375 newspapers daily. 25. Find the Expected Marginal Profit (EMP) and Expected Marginal Loss (EML) in increasing the purchase to 376 newspapers per day. a.

EMP = Rs.0.40, EML = Rs.0.09

b.

EMP = Rs.0.01, EML = Rs.0.32

c.

EMP = Rs.0.50, EML = Rs.0.40

d.

EMP = Rs.0.60, EML = Rs.0.90

e.

EMP = Rs.0.90, EML = Rs.0.19.

26. If the newspaper vendor wants to maximize his profits, how many newspapers, should he purchase daily? a.

380.

b.

384.

c.

394.

d.

410.

e.

415.

10

Part I

27. What is the probability that a value chosen at random from a particular population is larger than the median of the population? a. 0.25. b. 0.5. c. 1.0. d. 0.67. e. 0.48. 28. Following is the probability distribution of rate of return available from security A Rate of return (r%) Probability

7

10

12

15

18

20

0.2

0.1

?

0.2

?

?

The maximum expected rate of return available from security A will be a. 14.5% b. 15.4% c. 15.0% d. 18.0% e. 20.0%. 29. A financial analyst feels that next week the chances are 40% that the price of the shares of Company A would rise, and 10% that the price of the shares of Company B would rise. The two prices are expected to move independent of each other. Then the probabilities that in the next week only one company’s share price would increase is a. 0.68 b. 0.37 c. 0.42 d. 0.47 e. 0.28. Based on the following information answer questions 30 and 31. Consider the following pay-off matrix for the action-event combinations. The probabilities of events are also given. Events B1

B2

B3

Probability

0.2

0.5

0.3

Action 1

75

40

60

Action 2

50

40

70

Action 3

60

80

50

30. Using expected value criterion, the best action which maximizes the expected value is a. A1 b. A2 c. A3 d. All of the above e. Insufficient information. 31. The expected pay-off under perfect information is a. 76 b. 66 c. 78.5 d. 82 e. 80.

11


32. For a normal distribution with mean value of 20 and variance of 16, a Z-score of –1.5 would be used to find the probability that the random variable is: a.

Below 27

b.

Between 17 and 23

c.

Below or above 14

d.

Above 14

e.

None of the above.

33. It is found that a Z-score of –2.5 is associated with an observed value of 38 for a normally distributed random variable. If the variance of the random variable is 16, what is the mode of this distribution? a.

78.

b.

28.

c.

48.

d.

–2.

e.

None of the above.

34. From which of the following can retailers immediately determine how many cases they should stock each day to maximize profits? a.

Expected profit from each stocking action.

b.

Expected loss from each stocking action.

c.

Conditional profit table.

d.

(a), (b) and (c) above.

e.

(a) and (b) but not (c).

35. Consider a conditional-loss table with possible sales levels listed vertically in the first column and possible stock actions listed horizontally in the first row. Any value in a column below a zero indicates a.

An opportunity loss

b.

An obsolescence loss

c.

A profit

d.

Both (a) and (b) above

e.

None of the above.

36. A businessman who is said to be averse to risk a.

Prefers to take large risks to earn large gains

b.

Prefers to act anytime the expected monetary value is positive

c.

Avoids all situations but those with very high expected values

d.

Both (b) and (c) but not (a)

e.

None of the above.

37. Once a decision tree is “solved” the analyst should a.

Immediately go out and make the decision

b.

Specify underlying assumptions to point out, among other things, the limits under which the decision tree is valid

c.

Perform sensitivity analysis to determine how the solution reacts to changes in the inputs

d.

(b) and (c) but not (a)

e.

All of the above.

12

Part I

38. You are the project manager for a construction project. After analyzing the requirements for completing the project, you find that it would take an average of 80 weeks to finish. The completion time is normally distributed with a standard deviation of 9 weeks. The probability of completing the project in 95 weeks or less is: a. 0.4525 b. 0.0475 c. 0.9525 d. Cannot be determined from the information given e. None of the above. 39. A company produces drive shafts for electric motors. The drive shafts are produced from steel rods. The drive shaft of the motor must fit in a groove with a diameter of 57.32 ± 3 cm. The steel rods used for production of drive shafts have a mean diameter of 56 cm with a standard deviation of 2.4 cm. What is the probability of a steel rod fitting in the groove, assuming that the diameter follows a normal distribution? (Choose the value nearest to the true value). a.

0.7221.

b.

0.4641.

c.

0.2580.

d.

0.7031.

e.

0.7580.

40. Assume that for a particular stocking operation, ML = $10 and MP = $30. Then, p* = 0.25. For which of the following situations would you stock the unit in question? a.

The fifth unit when P(requests for 5 or more units) = 0.50.

b.

The third unit when P(requests for fewer than 3 units) = 0.10 and P(requests for exactly 3 units) = 0.09.

c.

The ninth unit when P(requests for more than 9 units) = 0.16 and P(requests for exactly 9 units) = 0.05.

d.

All of the above.

e.


41. If Z is standard normal and P (0 < Z < 2.9) = 0.4981, P (–2.9 ≤ Z ≤ 2.9) a.

0.9961

b.

0.9962

c.

0.9963

d.

1.9963

e. 0. 42. An investment alternative gives the potential of gains/losses to a person of Rs.200, 300, and (–250) with the probabilities of 0.2, 0.3 and 0.5 respectively. The person should a. Undertake the investment because he will certainly make profit b. Avoid such investment because he will book losses c. Undertake the investment because expected gains are positive d. Be indifferent about this opportunity e. Undertake the investment because his funds will be utilized. 43. The daily returns of BSE – Sensex over a long period of time are expected to follow: a. Binomial distribution b. Poisson’s distribution c. Normal distribution d. ‘t’ distribution e. Chi-square distribution.

13


44. The conditional profit table is prepared in decision-making under conditions of a.

Certainty

b.

Risk

c.

Uncertainty

d.

Perfect information

e.

Simulation.

45. Consider the following Probability distribution. X

1

2

3

P(X)

0.3

0.3

0.4

The expected value of X is a.

2.5

b.

2.0

c.

2.3

d.

2.7

e.

2.1.

46. Which of the following is not true for decision-making under conditions of risk? a.

All the possible outcomes are known with certainty.

b.

Regret criterion is one of the approaches of decision-making under risk.

c.

The decision-maker has sufficient knowledge to assign probabilities to all the alternatives.

d.

The probability assigned to each of the alternatives may be subjective or objective.

e.

The determination of conditional profit is one of the examples of decision-making under conditions of risk.

47. Which of the following statements is true for a standard normal distribution curve? a.

It is just like any other normal distribution curve in shape.

b.

It is just like any other normal distribution curve in size.

c.

It is just like ‘t’ distribution in its limits.

d.

It has the probability of 95% that an observation will be in the interval μ ±1.96σ .

e.

It is positively skewed.

48. If the marginal profit and marginal loss for stocking an additional car by a car dealer are Rs.10,000 and Rs.2,000 respectively, then the probability of selling an additional car that will justify the stocking of that car should be at least a.

0.1667

b.

0.2000

c.

0.2500

d.

0.5000

e.

0.6670.

49. Expected Value of Perfect Information (EVPI) is calculated in a.

Decision-making under certainty

b.

Decision-making under risk

c.

Decision-making under uncertainty

d.

Decision tree analysis

e.

Hurwicz criteria.

14

Part I

50. Which of the following is true for a decision tree analysis? a.

Time flows from left to right.

b.

Time flows from right to left.

c.

Time flows from top to bottom.

d.

Time flows from bottom to top.

e.

Time is constant and irrelevant to the problem.

51. Which of the following is not true for normal distribution curve? a.

It is unimodal.

b.

The mean lies at the center of the curve.

c.

The mean, median and mode are same for a normal curve.

d.

The two tails of curve never touch the horizontal axis.

e.

It can be drawn only for discrete observations.

52. The probability distribution of a random variable X is as follows: X 15 Probability 0.3 The expected value of X will be equal to a.

21.05

b.

21.50

c.

21.77

d.

22.50

e.

22.77.

20 0.3

25 0.2

30 0.2

53. The probability distribution of return from a particular investment is as follows: Probability 0.4 0.2 0.2 0.2 Return (%) 10 15 20 25 The probability that the return from above investment will be less than its expected value is a.

0.2

b.

0.4

c.

0.5

d.

0.6

e.

0.8.

54. In decision-making under conditions of uncertainty, the decision-maker a.

Has no idea of possible states of nature

b.

Has sufficient knowledge to assign the probabilities of occurrence to different states of nature

c.

Has insufficient information to assign any probabilities to different states of nature

d.

Has a subjective outlook towards the given selection of any strategy

e.

None of the above.

55. Decision tree is a useful tool in making decisions concerning a.

Investments

b.

Acquisition of property

c.

Profit management

d.

New product strategies

e.

All of the above.

15


56. If the decision-maker is pessimistic then he is likely to use a.

Maximin criteria

b.

Maximum criteria

c.

Regret criteria

d.

Hurwicz criteria

e.

Heuristic tables.

57. A random variable follows a normal distribution with mean 4.6 and standard deviation of 0.2. The probability that the percentage of all observations of the variable will lie between 4.2 and 5.0 is a.

68.2%

b.

75.0%

c.

95%

d.

95.5%

e.

99.2%.

58. Consider the following probability distribution X

1.1

1.2

1.3

1.4

1.5

1.6

1.7

p

0.05 0.10 0.20 0.30 0.20 0.10 0.05

The above probability distribution is a.

Normal probability distribution

b.

Uniform probability distribution

c.

Discrete probability distribution

d.

Poisson’s probability distribution

e.

X2 distribution.

59. Consider the following profit table. Probability

0.2

0.8

A

10

15

B

12

13

Plan

The expected pay-off under perfect information is a.

12.5

b.

13.4

c.

13.5

d.

14.4

e.

15.0.

60. The number of patients seen by a doctor as a function of time spent in clinic is an example of a.

Independent variable

b.

Random sampling

c.

Discrete variable

d.

Continuous variable

e.

None of the above.

16

Part I

61. Which of the following statements is true? a.

Probability distribution can accurately predict the value of a random variable at different time intervals.

b.

The random variable can help in estimating the probability distribution of a given data.

c.

The chances of occurrence of different probability distribution help in decision-making under risk.

d.

The probability distribution can give an idea of the likely values of a random variable.

e.

A random variable can give substantial idea of a probability distribution followed by it in most of the cases.

62. A probability distribution must sum to a. 100 b. 1 c. 0 d. Total number of events e. Total of favorable probabilities. 63. In which of the following conditions is the maximax criterion of decision-making applicable? i.

The decision is made under conditions of certainty.

ii.

The decision is made under conditions of uncertainty.

iii.

The decision-maker is perfectly pessimistic.

iv.

The decision-maker is perfectly optimistic.

v.

The decision-maker is neither perfectly optimistic nor perfectly pessimistic.

a.

Only (i)

b.

Only (ii)

c.

Both (ii) and (iii)

d.

Both (ii) and (iv)

e.

Both (i) and (v).

64. The random variables X and Y are said to be independent if a.

σx = σy

b.

Cov(X, Y) = 0

c.

X=Y

d. E(X) = E(Y) e. Their probability distribution is same. 65. Regret is measured as a. Maximum possible profits – Realized profits b. Maximum of minimum profits – Minimum of minimum profits c. Maximum possible profits – Foregone profits d. Maximum of maximum profits – Maximum of minimum profits e. Maximum possible profits – Minimum of maximum profits. 66. ‘Expected profit’ in a decision problem under risk a. Is not necessarily one of the possible outcomes b. Is always one of the possible outcomes c. Is never one of the possible outcomes d. Gives maximum profit in minimum maximum criteria e. Gives minimum profit in maximum minimum criteria.

17


67. The marginal profit and marginal loss of a product are 10 and 20 respectively. The marginal probability of sale of the product that will justify the stocking of the same is a.

0.100

b.

0.333

c.

0.500

d.

0.667

e.

None of the above.

68. Which of the following frequency distribution(s) is/are not symmetrical with respect to the mean? a.

Poisson’s distribution.

b.

Binomial distribution.

c.

Normal distribution.

d.

ψ2 distribution.

e.

Both (a) and (d) above.

69. If a variable can assume only a limited number of values in a given range at random it is called a.

Independent variable

b.

Dependent variable

c.

Random variable

d.

Continuous random variable

e.

Discrete random variable.

70. Which of the following is true for Bernoulli process? a.

The Bernoulli experiment can always be replicated by an unbiased coin.

b.

The probability of the outcome of any trial remains fixed over time.

c.

The trials are statistically dependent.

d.

The trials have only limited and predetermined outcomes.

e.

The Bernoulli’s process can be applied to a wide range of probability distributions.

71. The probability that an observation from a normal distribution will lie between μ ± 2.33 σ is a.

95%

b.

96%

c.

97%

d.

98%

e.

99%.

72. Which of the following is true for ‘t’ distribution? a.

μ = 1, σ2 = df/(df – 1).

b.

μ = 0, σ2 = df (df – 3).

c.

μ = –1, σ2 = df (df – 2).

d.

μ = –1, σ2 = df (df – 1).

e.

μ = 0, σ2 = df/(df – 2).

73. In decision making under conditions of uncertainty, the decision-maker a.

Has no idea of possible states of nature

b.

Has sufficient knowledge to assign a probability of occurrence to various states of nature

c.

Has insufficient information to assign any probability of occurrence to various states of nature

d.

Has a positive outlook towards occurrence of various states of nature

e.

None of the above.

18

Part I

74. Which of the following statements is false? a.

It is difficult to develop a conditional profit table when there is a large number of scenarios.

b.

The decision tree is considered as a mathematical model of a decision situation.

c.

At every decision node the decision-maker chooses an alternative which gives him the highest probability of success.

d.

The decision tree is a graphic model depicting the entire decision process.

e.

The problem represented by a decision tree is solved from right to left.

75. If the marginal profit and marginal loss of a product are Rs.300 and Rs.200 respectively, the probability of sales that will justify stocking the product is a.

20%

b.

40%

c.

50%

d. 60% e. 66%. 76. The standard normal distribution has a. Mean = 1 and standard deviation = 0 b. Mean = 1 and standard deviation = 1 c. Mean = 0 and standard deviation = 0 d. Mean = 0 and standard deviation = 1 e. None of the above. 77. A security can provide returns as given below: Return (%) Probability

10 0.30

15 0.20

20 0.50

The expected return from the security is a.

12%

b.

15%

c.

16%

d.

18%

e.

19%.

78. The marginal loss of increasing the stock of a particular item by one unit in a departmental store is Rs.50 and the minimum probability of selling the additional unit which justifies increasing the stock of the item by an additional unit is 0.25. The marginal profit for the item is a.

Rs.50

b.

Rs.100

c.

Rs.125

d.

Rs.150

e.

Rs.160.

79. For a binomial distribution, probability of success is 0.60 and number of trials = 8. The approximate probability of getting exactly 5 successes is a.

4%

b.

6%

c.

12%

d.

28%

e.

94%.

19


80. The concept of expected value is used a.

For decision-making under uncertainty

b.

For decision-making under risk

c.

For decision-making under certainty

d.

Under regret criterion

e.

Under Hurwicz criterion.

81. A continuous random variable can assume a.

Only a fixed value

b.

Any value in a given range

c.

Only some discrete values

d.

Only one of two fixed values

e.

None of the above.

82. Which of the following is true? A normal distribution has a.

80% of the observations within μ ±1.28 σ .

b.

80% of the observations within μ ± 1.27σ .

c.

80% of the observations within μ ±1.25σ .

d.

80% of the observations within μ ± 2.33 σ .

e.

80% of the observations within μ ± 1.96σ.

83. Following is the probability distribution of the gains of an investor from an investment: Gain (%)

Probability

50

0.7

30

0.2

20

0.1

The expected gain from the investment is a.

20.00%

b.

30.00%

c.

33.33%

d.

43.00%

e.

None of the above.

84. The probability of getting exactly 3 tails in four tosses of a fair coin is a.

1/2

b.

1/4

c.

1/8

d.

1/10

e.

1/16.

85. The regret criterion for decision-making suggests that a.

The largest regret should be maximized

b.

The smallest regret should be maximized

c.

The largest regret should be minimized

d.

The smallest regret should be minimized

e.

The average regret should be minimized.

20

Part I

86. A discrete random variable can assume a. Only a constant value every time b. Any value in a given range c. Negative values only d. Only some discrete values e. Even numbers only. 87. Which of the following is true? A normal distribution has 95% of the observations within μ ± 1.28 σ . a. b.

μ ± 1.30 σ .

c.

μ ± 1.56 σ .

d.

μ ± 1.82 σ .

e.

μ ± 1.96 σ .

88. When an experiment or a process has only two possible outcomes, the probability of the outcome of any trial remains fixed over time and the outcome of one trial does not affect the outcome of another the appropriate probability distribution to be used is the a. Normal distribution b. t-distribution c. Poisson distribution d. Binomial distribution e. Standard normal distribution. 89. A departmental store is deliberating on the number of units of a perishable product to be stocked. The product costs Rs.12 per unit and can be sold at a price of Rs.15 per unit and has a self life of one day. The minimum required probability of selling an additional unit of the item is a. 0.75 b. 0.80 c. 0.82 d. 0.90 e. 0.92. 90. The regret criterion is used a. To make decisions under uncertain conditions b. To make decisions under certain conditions c. To make decisions under risky conditions d. When the decision-maker is perfectly optimistic e. When the decision-maker is perfectly pessimistic. 91. The maximin criterion for decision-making suggests that the alternative with the a. Minimum of maximum profits must be selected b. Maximum of maximum profits must be selected c. Maximum of minimum profits must be selected d. Minimum of minimum profits must be selected e. Highest of expected profits must be selected. 92. The probability of getting two heads in three tosses of a fair coin is a. 0.285 b. 0.300 c. 0.325 d. 0.345 e. 0.375.

21


93. In a binomial distribution the probability of getting zero or more number of successes is equal to a.

0

b.

The probability of getting zero success

c.

The probability of getting successes in all the trials

d.

1 minus the probability of getting successes in all the trials

e.

1.

94. Which of the following is/are true with regard to the normal distribution? a.

It has a single peak.

b.

It is unimodal.

c.

The mean lies at the center of the distribution.

d.

The two tails extend indefinitely and never touch the horizontal axis.

e.

All of the above.

95. For which of the following distributions, the z-values and the observed values are same? a.

Binomial.

b.

Standard Normal.

c.

Perfectly Normal.

d.

Poisson’s.

e.

Bernoulli’s probability.

96. A random variable follows a normal distribution with mean 5.4 and standard deviation of 0.2. The probability that the percentage of all observations will fall between 5.0 and 5.8 is approximately a.

68.2%

b.

75.0%

c.

95.0%

d.

95.4%

e.

99.2%.

97. Which of the following is not true with regard to the Bernoulli process? a.

Each trial has only two possible outcomes.

b.

The probability of the outcome of any trial varies over time.

c.


d.

The trials are statistically independent.

e.

The probability of ‘r’ successes in ‘n’ trials of the Bernoulli process is ncrprqn–r.

98. Which of the following is a useful tool for making decisions involving many alternatives and different states of nature which can be specified in terms of probabilities? a.

Simple regression analysis.

b.

Multiple regression analysis.

c.

Correlation analysis.

d.

Decision tree analysis.

e.

Variance analysis.

22

Part I

99. If the minimum probability of selling an additional unit of a product which justifies stocking of that unit is 0.60 and the marginal loss on that unit is Rs.15, then the marginal profit on that unit is a.

Rs.5

b.

Rs.6

c.

Rs.10

d.

Rs.15

e.

Rs.18.

100. The 95% confidence interval in a normal distribution is within the range of

101.

102.

103.

104.

a.

± 1.94σ from the mean

b.


c.


d.


± 1.98σ from the mean. e. Which of the following is not true for random variables? a. A discrete random variable can assume only a limited number of values. b. A continuous random variable can assume any value within a given range. c. A random variable follows a predictable sequence. d. A random variable is a variable which assumes different numerical values as a result of random experiments. e. The value of the random variable changes from occurrence to occurrence in no predictable sequence. Which of the following is not true for probability distributions? a. The past frequency distribution may be used as a guide for predicting the future values of the random variable. b. The probability distribution gives the chances of occurrence of the various values of the random variable. c. A subjective probability distribution of the likely future values can be formed on the basis of expert opinion. d. We can never assign probabilities to all likely values of the random variable on the basis of the past data. e. The sum of the probability of occurrence of all possible values must be equal to one. Which of the following is false for expected value of a random variable? a. The concept of expected value of the random variable can be used for making decisions under conditions of uncertainty. b. The expected value is the mean of the probability distribution of the random variable. c. The mean is computed as the weighted average of the values that the random variable can assume. d. The probability assigned to each possible value of the random variable is used as weight. e. The expected value of the random variable predicts the value that the random variable will assume in future. If the covariance of two random variables X and Y is equal to zero then it can be said that a. The random variables are dependent on each other b. The random variables have a positive correlation c. The random variables have a negative correlation d. The random variables are independent of each other e. Covariance of zero does not necessarily mean than the variable an independent.

23


105. Consider a batch of N cricket balls. Each cricket ball may be defective or non-defective. The experiment involves selecting a cricket ball and checking whether it is defective or nondefective. If this experiment is repeated without replacing the balls then the probability distribution function obtained from such an experiment can be appropriately described by a. The binomial distribution b. The hypergeometric distribution c. The normal distribution d. The lognormal distribution e. The t-distribution. 106. Which of the following is not true with regard to the binomial distribution? a. The outcomes are independent of each other. b. Each outcome can be classified as a success or failure. c. The probability of success is constant from trial to trial. d. The random variable of interest is continuous. e. The probability of failure on any trial can be determined on the basis of the probability of success. 107. Which of the following best describes the expected value of a discrete random variable? a.

It is the geometric average of all possible outcomes of the variable.

b.

It is the simple average of all possible outcomes of the variable.

c.

It is the weighted average of all possible outcomes of the variable.

d.

It is the outcome of the variable, which has the highest probability of occurrence.

e.

It is the highest probability of occurrence in probability distribution of the random variable.

108. When sampling without replacement from a finite population such that the probability of success is not constant from trial to trial, the data follow a a.

Binomial distribution

b.

Uniform distribution

c.

Normal distribution

d.

Continuous distribution

e.

Hypergeometric distribution.

109. Which of the following refers to the random (chance) occurrences that can affect the outcome of an individual’s decision? a.

Pay-offs.

b.

States of nature.

c.

Decision alternatives.

d.

Probabilities.

e.

None of the above.

110. The sampling distribution of the mean is a distribution of a.

Means of individual populations

b.

Observations within a population

c.

Observations within a sample

d.

Means of all possible samples of a specific size taken from a population

e.

Means of samples of a specific size taken from different populations.

24

Part I

111. As the sample size increases a.

The variation of the sample mean from the population mean becomes larger

b.

The variation of the sample mean from the population mean becomes smaller

c.

The variance of the sample becomes less than the variance of the population

d.

The standard deviation of the sample becomes less than the standard deviation of the population

e.

The standard deviation of the sample comes close to zero.

112. When estimating the population mean from a sample taken from a normal population with unknown variance, the t-distribution may be used with number of degrees of freedom equal to a.

Sample size

b.

Sample size + 1

c.

Sample size – 1

d.

0

e.

1.

113. The sample mean is an unbiased point estimator of a.

The population variance

b.

The population mean

c.

The population proportion

d.

The standard error of mean

e.

The standard error of proportion.

114. Which of the following criteria is applied for making decisions under conditions of uncertainty if the decision-maker is perfectly optimistic? a.

Maximax criterion.

b.

Maximin criterion.

c.

Hurwicz criterion.

d.

Regret criterion.

e.

Expected value.

Statistical Inferences 115. In random sampling, we can describe mathematically, how objective our estimates are. Why is this? a.

All the samples are of exactly the same size and can be counted.

b.

We always know the chance, that any population element will be included in the sample.

c.

Every sample always has an equal chance of being selected.

d.

Both (b) and (c) above.

e.

None of the above.

116. The central limit theorem a.

Requires some knowledge of the frequency distribution

b.

Relates the shape of a sampling distribution of the mean to the mean of the sample

c.

Permits us to use sample statistics to make inferences about population parameters

d.

Requires the sample to contain fewer than 30 observations

e.

All of (a), (b) and (d) above.

25


Based on the following information answer questions 117 and 118. Assume, that the heights of 3000 male students at a University are normally distributed with mean 173 cm and standard deviation 8 cm. Also, assume that from this population of 3000 all possible samples of size 25 were taken. (Sampling done without replacement). 117. What is the mean of the resulting sampling distribution of means? a. 165. b. 170. c. 173. d. 181. e. 190. 118. What is the standard deviation of the resulting sampling distribution of the means? a. 1.6 cm. b. 8.02 cm. c. 10.3 cm. d. 16.8 cm. e. 24.2 cm. Based on the following information answer questions 119 and 120. The ball bearings in a large consignment of the same, have a mean weight of 142.3 grams and a standard deviation of 8.5 grams. A random sample of 100 ball bearings is taken from the consignment. 119. What is the probability that the sample mean lies between 140.61 and 141.75 grams? a. 0.2345. b. 0.2642. c. 0.2778. d. 0.4864. e. 0.05139. 120. What is the probability, that the sample mean is more than 144.60 grams? a. 0.0035. b. 0.3936. c. 0.4987. d. 0.5313. e. 0.5987. 121. The following sample of eight observations is from an infinite population with a normal distribution. 75.3 76.4 83.2 91.0 80.1 77.5 84.8 81.0 Estimate the population standard deviation. a. 1.82 b. 2.71 c. 3.90 d. 4.82 e. 5.15. 122. Which of the following are true? a. The probability of Type I error is equal to significance level of the test. b. Rejecting a null hypothesis, when it is true is called a Type I error. c. Accepting a null hypothesis, when it is false is called a Type I error. d. Both (a) and (b) above. e. Both (a) and (c) above.

26

Part I

123. Suppose that a hypothesis test is being performed for a process in which a Type I error will be very costly, but a Type II error will be relatively inexpensive and unimportant. Which of the following would be the best choice for α in this test? a. 0.01. b. 0.10. c. 0.25. d. 0.50. e. None of the above. 124. The normal distribution is the appropriate distribution, to use in testing hypothesis about: n

p H 0 > 5 and n q H 0 > 5

a.

A proportion, when

b.

A mean, when σ is known and the population is normal

c. d. e.

A mean, when σ is unknown but n is large Both (a) and (b) above All of (a), (b) and (c) above.

Based on the following information answer questions 125 to 127. Two groups, A and B, consist of 100 people each who have a disease. A serum is given to group A but not to group B (which is called the control); otherwise the two groups are treated identically. It is found that in groups A and B, 75 and 65 people, respectively, recover from the disease. Let p1 and p2 denote the population proportions cured by (1) using the serum and (2) not using the serum, respectively. 125. The null and alternative hypothesis for testing, whether the serum helps cure the disease are a.

H0: p1 = p2, H1: p1 > p2

b.

H0: p1 > p2, H1: p1 < p2

c.

H0: p1 = p2, H1: p1 ≠ p2

d.

Both (a) and (c) above

e.

None of the above.

126. The estimated standard error of the difference between the two proportions σ p1 − p 2 is a.

0.0648

b.

0.0558

c.

0.0523

d.

0.0414

e.

0.0323.

127. We test the null hypothesis(a), chosen in question 125 at various levels of significance. When do we conclude that the serum helps cure the disease? a.

α = 0.01.

b.

α = 0.05.

c.

α = 0.10.

d.

α = 0.20.

e.

None of the above.

Based on the following information answer questions 128 and 129. An examination was given to two classes consisting of 40 and 50 students, respectively. In the first class the mean grade was 74 with a standard deviation of 8, while in the second class the mean grade was 78 with a standard deviation of 7. Suppose that the two classes come from two populations having the respective means μ1 and μ2.

27


128. We need to test if there is any significant difference in the performance of the two classes. The null hypothesis and alternative hypothesis for this purpose are a. H0: μ1 = μ 2 , H1: μ 1 ≠ μ 2 b.

H0: μ1 = μ 2 , H1: μ 1 > μ 2

c.

H0: μ1 = μ 2 , H1: μ 1 < μ 2

d. Both (b) and (c) above e. None of the above. 129. The estimated standard error of the difference between the two means σˆ x1 − x 2 is

130.

131.

132.

133.

a. 1.406 b. 1.606 c. 1.906 d. 2.306 e. 2.706. The Central Limit Theorem claims that the sampling distribution of the mean a. Is always normal b. Is always normal for large sample sizes c. Approaches normality as sample size increases d. Is a t-distribution if N is less than 30 e. Approaches normality if N is greater than or equal to 30. Which of the following is a method of selecting samples from a population? a. Judgment sampling. b. Random sampling. c. Probability sampling. d. All of the above. e. (a) and (b) but not (c). Suppose you are performing stratified sampling on a particular population and have divided it into strata of different sizes. How can you now make your sample selection? a. Select at random an equal number of elements from each stratum. b. Draw equal numbers of elements from each stratum and weigh the results. c. Draw numbers of elements from each stratum proportional to their weights in the population. d. Both (a) and (b) above. e. Both (b) and (c) above. The dispersion among sample means is less than the dispersion among the sampled items themselves because a. Each sample is smaller than the population from which it is drawn b. Very large values are averaged down, and very small values are averaged up c. The sampled items are all drawn from the same population d. None of the above e. (b) and (c) but not (a).

134. In which of the following situations would σ x = σ / n be the correct formula to use? a. Sampling is from an infinite population. b. Sampling is from a finite population with replacement. c. Sampling is from a finite population without replacement. d. Both (a) and (b) above. e. Both (b) and (c) above.

28

Part I

135. In testing of hypothesis, the decision to use a single-tailed or two-tailed test depends upon a. The size of the sample b. The alternative hypothesis c. The values of the sample statistic d. The null hypothesis e. None of the above. 136. When a null hypothesis is accepted, it is possible that a. A correct decision has been made b. A Type I error has been made c. Both (a) and (b) have occurred d. Neither (a) nor (b) has occurred e. None of the above. 137. With a lower significance level, the probability of rejecting a null hypothesis that is actually true a. Decreases b. Remains the same c. Increases d. Slightly decreases e. All of the above. 138. Decision-makers make decisions on the appropriate significance level by examining the cost of a.

Performing the test

b.

A Type I error

c.

A Type II error

d.


e.


139. The net sales of a randomly selected manufacturing company is assumed to follow a normal distribution. (2.00 crore, 4.20 crore) is a 95% confidence interval for μ = average net sales of all the manufacturing companies, computed on the basis of a random sample of 50 manufacturing companies. Then a.

P(μ > 4.20) = 0.025

b.

95% of all possible samples of size 50 will have average net sales between 2.00 crore and 4.20 crore

c.

95% of all manufacturing companies will have net sales between Rs.2.00 crore and 4.20 crore

d.

(a), (b) and (c) above

e.

None of the above.

140. Suppose that 200 members of a group were asked whether or not they like a particular product. Fifty said yes; 150 said no. Assuming “yes” means a success, which of the following is correct? a.

p = 0.33.

b.

p = 0.25.

c. d. e.

p = 0.33. p = 0.25. Both (b) and (d) above.

29


141. Assume that you take a sample and calculate x as 100. You then calculate the upper limit of a 90 percent confidence interval for μ; its value is 112. What is the lower limit of this confidence interval? a. 88. b. 89. c. 92. d. 100. e. It cannot be determined from the information given. 142. If we say that α = 0.10 for a particular hypothesis test, then we are saying that

a. Ten percent is our minimum standard for acceptable probability b. Ten percent is the risk we take of rejecting a hypothesis that is true c. Ten percent is the risk we take of accepting a hypothesis that is false d. Both (a) and (b) above e. Both (a) and (c) above. 143. A two-tailed test means a. A test where a second sample is taken based on the first b. A test where null hypothesis is accepted if test statistic exceeds a given value c. A test where both upper and lower limits of confidence interval are used to take a decision of accepting null hypothesis d. A test used only when the sample size is small e. A test used only to measure the difference between two proportions. 144. Which of the following is used to reduce the sampling error while studying a heterogeneous population? a. Cluster sampling. b. Systematic sampling. c. Survey sampling. d. Stratified sampling. e. Simple random sampling. 145. For listing H0 : μ = 5, a t-distribution is required when a.

Sample size is small

b. S is known but not σ c. The population is very large d. Both (a) and (b) but not (c) e. (a), (b) and (c) of the above. 146. Which of the following is true? a.

P (Type I error) = α.

b.

P (Rejecting H0/H0 is true) = α.

c. P (Accepting H0/H0 is true) = α. d. Both (a) and (b) above. e. Both (a) and (c) above. 147. The finite population multiplier need not be used when the sampling fraction is a.

< 0.50

b.

> 0.50

c.

< 0.05

d.

> 0.05

e.

> 0.99.

30

Part I

148. Which of the following is an example of Type I error? The null hypothesis is a. True and is accepted. b. False and is rejected. c. True and is rejected. d. False and is accepted. e. Rejected whether true or false. 149. Which of the following statements is true? As the degree of freedom increases a. The normal distribution approaches ‘t’ distribution. b. The ‘t’ distribution approaches binomial distribution. c. Binomial distribution approaches ‘F’ distribution. d. ‘F’ distribution approaches ‘t’ distribution. e. ‘t’ distribution approaches normal distribution. 150. In hypothesis testing α indicates a. The probability of not committing Type I error b. The probability of not committing Type II error c. The probability of committing Type I error d. The probability of committing Type II error e. The validity of the test. 151. In a sampling technique, the first element is randomly selected and thereafter all other elements are selected based on a predetermined plan. This sampling technique can be described as a. Simple random sampling b. Stratified sampling c. Systematic sampling d. Cluster sampling e. Judgmental sampling. 152. Which of the following is not a type of random sampling? a. Simple random sampling. b. Stratified sampling. c. Systematic sampling. d. Judgmental sampling. e. Cluster sampling. 153. The value which defines the boundaries of acceptance and rejection region in hypothesis testing is called a. Observed value b. Critical value c. Acceptance value d. Rejection value e. Appropriate value. 154. Which of the following statements is false about the standard error of the statistic? a. It measures the variability arising from sampling error due to chance. b. It follows a normal distribution only when sample mean is equal to population mean. c. It is used as a tool in tests of hypothesis of significance. d. It helps in determining the limits within which the parameters are expected to lie. e. It gives an idea about the reliability and precision of a sample.

31


155. Which of the following is an example of Type II error? a. The null hypothesis is true and rejected. b. The null hypothesis is true and accepted. c. The null hypothesis is false and accepted. d. The null hypothesis is false and rejected. e. The alternative hypothesis is false and accepted. 156. For a heterogeneous population consisting of relatively homogeneous groups, which of the following sampling techniques is most suitable? a. Simple random sampling. b. Stratified sampling. c. Judgmental sampling. d. Systematic sampling. e. Cluster sampling. 157. A confidence interval is also known as a. Interval estimate b. Confidence limits c. Central estimate d. Dual point estimate e. Confidence level. 158. The managers use sample statistics to estimate a. Sampling distribution b. Population parameters c. Sample characteristics d. Population size e. Accuracy of the distribution. 159. Which of the following statements is true when the sample size is large?

a.

σ x will be larger than σ.

b.

x will be normally distributed.

c.

μ will be normally distributed.

d.

x will have standard deviation equal to σ2/n.

e.

σ μ will be equal to σ / n.

160. In hypothesis testing, the power of the test is equal to α a.

b.

β

c.

1–α

d.

1–β

e. β – 1. 161. Which of the following sampling method is most susceptible to subjectivity in selection? a. Simple random sampling. b. Stratified sampling. c. Systematic sampling. d. Judgmental sampling. e. Cluster sampling.

32

Part I

162. If the population variance and the sample size are 26 and 5 respectively, then the standard error of the sample mean will be a. 0.192 b. 0.438 c. 0.980 d. 2.280 e. 11.62. 163. Which of the following statements is false?

The probability of Type I error is denoted by α. Each trial has only two possible outcomes in Bernoulli’s process. Census refers to the measurement of every element in the population. Within a population, groups that are essentially similar to each other are known as clusters. e. None of the above. A normal population whose 99.78 percent elements are between 10 and 19 has a standard deviation of a. 1.5 b. 3 c. 9 d. 10 e. 29. Cluster sampling is used when the population is divided into groups and a. The individual groups do not represent the population as a whole b. There is no variation within each group and the groups are not similar to each other c. There is a small variation within each group but there is a wide variation among the groups d. There is considerable variation within each group but the groups are essentially similar to each other e. None of the above. The significance of the Central Limit Theorem is that a. It allows us to study the entire population only when the frequency distribution of that population is symmetrical b. It does not allow us to make inferences about the population parameters without knowing anything about the frequency distribution of the population c. It allows us to use sample statistics to make inferences about the population parameters without knowing anything about the frequency distribution of the population d. It allows us to use the process of sample study for making inferences about finite populations only e. It does not allow the use of sample study for making inferences about finite populations. In hypothesis testing, the critical region is that region of the sampling distribution a. In which the value of sample statistic must fall for the null hypothesis to be accepted b. In which the value of sample statistic must fall for the decision-maker to remain indifferent to the acceptance or rejection of the null hypothesis c. In which the value of sample statistic must fall for the null hypothesis to be rejected and the alternative hypothesis to be accepted d. Which always lies in the left tail of the sampling distribution e. Which always lies in the right tail of the sampling distribution. a. b. c. d.

164.

165.

166.

167.

33


168. When a null hypothesis is accepted on the basis of sample information, it means that

a.

The null hypothesis is true

b.

The null hypothesis is false

c.

The alternative hypothesis is false

d.

The alternative hypothesis is true

e.

There is no statistical evidence to reject the null hypothesis.

169. The standard error of sample mean is

a.

The standard deviation of the sample

b.

The standard deviation of the population

c.

The standard deviation of the distribution of sample means

d.

The error in computing the sample statistic

e.

The error in computing the standard deviation of the sample means.

170. The standard error of sample mean is equal to

a. b. c. d. e.

σ2 n σ n σ n σ n2

σ2 . n

171. The t-distribution is used for estimating the population parameters when

a.

Sample size is equal to 30 and the population standard deviation is known

b.

Sample size is less than 30 and the population standard deviation is known

c.

Sample size is more than 30 and the population standard deviation is not known

d.

Sample size is 30 or less, the population standard deviation is not known and the population is assumed to be normal

e.

Sample size is 30 or less and the population standard deviation is known and the population is assumed to be normal or approximately normal.

172. Type I error is caused when the null hypothesis is

a.

True and accepted

b.

False and rejected

c.

True and rejected

d.

False and accepted

e.

Incorrectly formulated.

173. In a left tailed test of hypothesis with α = 10% the cut-off Z-value for accepting the null hypothesis is

a.

–1.64

b.

–1.28

c.

–1.00

d.

0.00

e.

1.28.

34

Part I

174. Which of the following statements is not true? a. Cluster sampling is used when the population is divided into groups which are not representative of the population as a whole. b. Systematic sampling requires less time and lower costs than simple random sampling. c. In systematic sampling each element does not have an equal chance of being selected. d. The chances of errors are less while dealing with census. e. Judgmental sampling is free from subjectivity. 175. If x1 and x2 are two random variables independent of each other then, the variance of x1 + x2 is equal to the a. Standard deviation of x1 plus standard deviation of x2 b. Standard deviation of x1 minus standard deviation of x2 c. Variance of x1 plus variance of x2 d. Variance of x1 minus variance of x2 e. Covariance of x1 and x2. 176. If the standard error of mean for a sample of size 9 is 6, then the population standard deviation will be equal to a. 2 b. 6 c. 9 d. 12 e. 18. 177. The variance of the standard normal distribution is

a.

Equal to its mean

b.

Less than its mean

c.

Equal to 0

d.

Equal to its standard deviation

e.

Greater than its standard deviation.

178. A student doing a market research will waste resources by taking larger samples, if the estimator is not a. Unbiased b. Efficient c. Consistent d. Sufficient e. Inconsistent. 179. A statistic which comes very close to the value of the population parameter as the sample size increases is said to be a. An unbiased estimator b. An efficient estimator c. A consistent estimator d. A sufficient estimator e. An accurate estimator. 180. If the null hypothesis is rejected in spite of it being true then it results in a a. Type I error b. Type II error c. Type III error d. Type IV error e. Type V error.

35


181. If the significance level is increased the probability of accepting a null hypothesis that is actually false

a.

Increases

b.

Decreases

c.

Remains unchanged

d.

Increases for large samples

e.

Increases for small samples.

182. The variance of a population is known to be 5.76. If a sample of 36 items is drawn from it the standard error of mean will be

a.

0.25

b.

0.40

c.

1.60

d.

1.80

e.

2.20.

183. According to the central limit theorem which of the following is true?

a.

The mean of the sampling distribution of mean will be equal to the population mean for large samples only.

b.

The mean of the sampling distribution of mean will be equal to the population mean for small samples only.

c.

The mean of the sampling distribution of mean will be equal to the population mean for normal populations only.

d.

The mean of the sampling distribution of mean will be equal to the population mean for non-normal populations only.

e.

The mean of the sampling distribution of mean will be equal to the population mean regardless of the sample size even if the population is not normal.

184. When the sample size is less than 30, σ is not known and the population is assumed to be normal the appropriate distribution for making statistical inferences is

a.

Normal distribution

b.

Binomial distribution

c.

Symmetrical distribution

d.

Non-symmetrical distribution

e.

t-distribution.

185. Which of the following is/are example(s) of probability sampling?

a.

Simple random sampling.

b.

Stratified sampling.

c.

Systematic sampling.

d.

Cluster sampling.

e.

All of the above.

186. In making estimates of population parameters using sample statistics, efficiency refers to the

a.

Tendency of the estimator to take values that are different from the population parameter

b.

Size of the standard error of the sample statistic

c.

Possibility that the sample statistic gets closer to the population parameter as the sample size increases

d.

Usage of sample information by the statistic

e.

Degree of representativeness of the sample information.

36

Part I

187. Other things remaining the same, the standard error of mean a. Increases when the sample size increases b. Decreases when the sample size increases c. Remains same irrespective of any change in the sample size d. Increases when the standard deviation of the population decreases e. Remains same irrespective of any change in the population standard deviation. 188. From a population of size 200 a sample of 20 items is taken. The finite population correction factor will be a. 0.901 b. 0.921 c. 0.951 d. 0.961 e. 0.981. 189. The following information is given: σ = 18; n = 36; and x = 25. The lower limit of 95% confidence interval is a. 18.15 b. 19.12 c. 19.80 d. 20.25 e. 21.50. 190. If a random variable X has a lognormal distribution then which of the following statements is true? a. X is normally distributed. b. ln(X) is normally distributed. c. The expected value of X is zero. d. The expected value of X is 1. e. None of the above. 191. The distortion of the representativeness of the data caused by the procedure of data collection is known as a. Procedural bias b. Sampling error c. Biased observation d. Cluster error e. Non-response bias. 192. Which of the following is/are true? a. The probability of Type I error is equal to significance level of the test. b. Rejecting a null hypothesis, when it is true is called a Type I error. c. Accepting a null hypothesis, when it is false is called a Type I error. d. Both (a) and (b) above. e. Both (a) and (c) above. 193. How will the standard error of mean be affected if the sample size is reduced? a. It will decrease. b. It will remain unchanged. c. It will increase. d. It will change unpredictably. e. It will be equal to the population variance.

37


194. If a sample of size 48 is collected from a population of size 800 then the finite population correction factor to be used for calculation of standard error is equal to a. 0.941 b. 0.965 c. 0.970 d. 0.978 e. 0.999. 195. Probability of Committing Type I error in testing a hypothesis = α and probability of Committing Type II error in testing the same hypothesis = β. If β is higher than the acceptance level, then one should a. Decrease the sample size b. Increase the sample size c. Increase α d. Both (a) and (c) above e. Both (b) and (c) above. 196. In many situations managers resort to sampling to draw some conclusions about a population. Which of the following is not an advantage of sampling over a census? a. The population may be too large to be studied in full, hence sampling is the only feasible means. b. A study of sample is usually cheaper than a census. c. Sampling usually provides information quicker than a census. d. In destructive testing sampling is the only available course. e. The conclusions obtained from sampling are more accurate than census. 197. Rahul automobiles, wants to study the consumer preference in terms of the features that vehicle owners of the city want to have in their vehicle. It divided the city in fifty sectors, each representing the characteristics of the whole city. Now they randomly selected five sectors and interviewed each household of these sectors for their preferences. They plan to forward the result of this survey to the vehicle producers to make them aware about the consumer preferences. This method of sampling is termed as a. Cluster sampling b. Stratified sampling c. Systematic sampling d. Judgmental sampling e. Simple random sampling. 198. Which of the following is not true for estimation? a. The probability that we associate with an interval estimate is called the confidence interval. b. If the population standard deviation is high, we can go for large confidence interval. c. The number of degrees of freedom used in a t-distribution is equal to the sample size. d. The t–distribution need not be used in estimating if you know the standard deviation of the population. e. The confidence interval is the range of the estimate we are making. 199. If in a test of statistical hypothesis, α be the level of significance and β be the probability of committing Type II error then which of the following is not true for the hypothesis test? a. The probability that the null hypothesis is true but rejected is α.

b. c. d. e.

The probability that the null hypothesis is false but accepted is β. (1 – β) is known as the power of test. A high (1 – β) value implies that the test is poor. There is a trade-off between the Type I and Type II error.

38

Part I

200. A company manufacturing a brand of detergent powder, Sunshine, has found the proportion of people using their brand of detergent powder in three different geographic regions by choosing suitable samples from each region. The company wishes to test whether the regions have significantly different proportions. Assuming p1 , p2 and p3 to be the true proportions for the three regions, which of the following represents the alternative hypothesis for this test?

a.

H 0 : p1 = p 2 = p3 .

b.

H1 : p1 = p 2 = p3 .

c.

H 0 : p1 ≠ p 2 ≠ p3 .

d.

H1 : p1 ≠ p 2 ≠ p3 .

e.

H1 : p1 , p 2 and p3 are not all equal .

201. Which of the following is not a step used in Analysis Of Variance (ANOVA)?

a.

We determine an estimate of the population variance from the variance that exists among the sample means.

b.

We determine an estimate of the population variance from the variance that exists within the samples.

c.

We prepare a table of observed and estimated frequencies.

d.

We accept the null hypothesis if the two estimates of the population variance are not significantly different.

e.

We reject the null hypothesis if the F ratio is too high and falls in the critical region.

202. If F(n, d, α) is the right tail value in a F distribution, then the corresponding left tail value will be

a.

F (n, d, 1 – α)

b.

F (d, n, 1 – α)

c.

1/ F (n, d, 1 – α)

d.

1/ F (d, n, 1 – α)

e.

F (d, n, α).

203. If each of the values of a random variable is transformed into a Z-value, the distribution of the resulting values will have a standard deviation equal to

a.

Zero

b.

One

c.

The mean of the original distribution

d.

The standard deviation of the original distribution

e.

A variable, depending upon the shape and spread of the original distribution.

204. Which type of sampling is appropriate when the population consists of well-defined groups such that the elements within each group are heterogeneous and each group is a representative of the population as a whole?

a.

Simple random sampling.

b.

Cluster sampling.

c.

Stratified sampling .

d.

Systematic sampling.

e.

Judgmental sampling.

39


205. In which of the following conditions a Type I error is said to have occurred, in testing a hypothesis? a. The sample statistic is incorrectly calculated. b. The sample variance is incorrectly calculated. c. Null hypothesis is rejected though it is true. d. Null hypothesis is accepted though it is false. e. The sample size is small. 206. If a hypothesis is tested at a significance level of 10% then it means that a. There is a 10% probability that the null hypothesis will be rejected though it is true b. There is a 10% probability that the null hypothesis will be false c. There is a 10% probability that the null hypothesis will be true d. There is a 90% probability that the alternative hypothesis will be false e. There is a 90% probability that the alternative hypothesis will be true. 207. The property of efficiency with regard to an estimator refers to the a. Sample size b. Size of the standard error of the sample statistic c. Size of the standard deviation of the sample d. Size of the sample variance e. Size of the sample mean.

Linear Regression Based on the following information answer questions 208 to 212.

Given below are the returns on a stock and the market index for a five year period. Year

Return on the stock

Return on the market index

1

0.29

(0.10)

2

0.31

0.24

3

0.10

0.11

4

0.06

(0.08)

5 (0.07) 0.03 208. What is the beta of the stock? a. 0.0586. b. 0.2421. c. 0.2772. d. 0.3133. e. 0.4312. 209. What is the coefficient of determination? a. 0.0044. b. 0.0587. c. 0.1061. d. 0.2421. e. 0.3671. 210. Using the coefficient of determination, the systematic risk of the stock will be a. 0.0015 b. 0.0021 c. 0.0051 d. 0.0063 e. 0.0079.

40

Part I

211. The total risk of the stock is a. 0.0014 b. 0.0259 c. 0.0362 d. 0.0731 e. 0.0831. 212. What is the unsystematic risk for the stock? a. 0.0155. b. 0.0244. c. 0.0311. d. 0.0353. e. 0.0423. 213. Suppose, the fraction of variation in Y that is unexplained by the independent variable X is 1/4. Then, the coefficient of determination r2 is a. 1/16 b. 1/4 c. 9/16 d. 11/16 e. 3/4. Based on the following information answer questions 214 to 217. Consider the following data on the variables X and Y.

X 214.

215.

216.

217.

1

3

4

6

8

9

11

14

Y 1 2 4 4 5 7 8 9 Find the slope of the regression line for the two variables. a. 0.5313 b. 0.5452 c. 0.6364 d. 0.6921 e. 0.7900. What fraction of the variation in Y is explained by the variable X? a. 0.9770. b. 0.957. c. 0.7538. d. 0.6316. e. 0.5892. Find the estimate of Y when X = 16, by using the regression line a. 10.7278 b. 12.0608 c. 12.3374 d. 13.3791 e. 13.9792. Compute the standard error of the estimate given by the regression line a. 0.3317 b. 0.4309 c. 0.4421 d. 0.6499 e. 0.7612.

41


218. For the estimating equation to be a perfect estimator of the dependent variables, which of these would have to be true? a. The standard error of the estimate is zero. b. All the data points are on the regression line. c. The coefficient of determination is –1. d. Both (a) and (b) above. e. All of (a), (b) and (c) above. Based on the following information answer questions 219 to 222. Consider the following data about two securities A and B. The returns from the two securities are perfectly negatively correlated.

Return (%)

Security A

Security B

7 or 11

13 or 5

Probability 0.5 each return 0.5 each return 219. Calculate the standard deviation of the returns from each of the securities. What are they? a.

σ A = 2, σ B = 3 .

b.

σ A = 2, σ B = 4 .

c.

σ A = 4, σ B = 2 .

d.

σ A = 3, σ B = 2 .

e.

σ A = 4, σ B = 3 .

220. We construct a portfolio of the two securities with proportions of A and B being 0.75 and 0.25 respectively. What is the portfolio risk as measured by its standard deviation? a. 0.35. b. 0.50. c. 0.75. d. 0.80. e. 0.95. 221. Consider the above problem. What would be the portfolio risk (as measured by the standard deviation) when the proportions of the two securities are interchanged? a. 0.5. b. 1.0. c. 2.5. d. 3.5. e. 3.75. 222. If the securities constitute equal portions of the portfolio, what is the portfolio risk (as measured by standard deviation)? a. 0.5. b. 1.0. c. 2.5. d. 3.1. e. 3.5. ˆ = 5 + 4X. Then 223. Two variables X and Y have the regressional relationship Y

a. b. c. d. e.

For a given value of Y = 21, the estimated value of X is 4 For a given value of Y = 25, the estimated value of X is 5 The slope of the line is 4 The intercept of the above line is 5 All of the above.

42

Part I

224. X1 is a normal variable with mean 25 and standard deviation 3. X2 is another normal variable with mean 50 and standard deviation 4. If X1 and X2 are independent P(X2 – X1 ≥ 30) would equal a. P(Z ≥ 10/25) b. P(Z ≥ 10/5) c. P(Z ≥ 1) d. P(Z ≥ 7) e. P(Z ≥ 10). 225. For a two-asset portfolio the returns on the two assets exhibit perfect negative correlation with each other. The standard deviations of returns on the assets A and B are σ A = 5%, and σ B = 15%. The standard deviation of the portfolio will be zero if

226.

227.

228.

229.

a. 75% of the funds are invested in asset A b. 50% of the funds are invested in asset A c. 33% of the funds are invested in asset B d. 67% of the funds are invested in asset B e. 75% of the funds are invested in asset B. Based on daily prices for the past one year, the stockbroker obtained the following sample regression equation E(Y|X = x) = 13 + 0.1 × X where X denotes the daily returns on the Sensex and Y denotes the daily returns on stock Y. She was then interested in the relationship between Y and another market index. A research analyst informed her that Z, the daily returns on the new index, equals (0.5 × X – 50), and therefore, there is no need to run the regression again. The regression equation is a. E(Y|Z = z) = 13 + 0.1 x Z b. E(Y|Z = z) = 8 + 0.05 x Z c. E(Y|Z = z) = 23 + 0.2 x Z d. E(Y|Z = z) = 113 + 0.2 x Z e. The regression must run again. If Y = a + bX, the sample regression line, and Y = A + BX, the true unknown population regression equation, are equivalent, then the following must be true a. The estimating equation is a perfect estimator of the dependent variable b. All the data points are on the regression line c. r2 = 1 d. Both (a) and (b) above e. None of the above. X and Y are two independent random variables, following normal distributions with means 20 and 30 and standard deviations 4 and 6 respectively. If Y = 20, then what is the probability that X is greater than Y? a. 0.90. b. 0.95. c. 0.75. d. 0.50. e. None of the above. If the regression coefficient of the independent variable in a simple regression equation is negative then, which of the following statements is correct? a. The coefficient of correlation between the variables is zero. b. The coefficient of correlation between the variables is the positive square root of the coefficient of determination. c. The coefficient of correlation between the variables is the negative square root of the coefficient of determination. d. The coefficient of correlation between the variables is 1. e. None of the above.

43

Quantitative Methods ∧

230. Suppose that the regression equation calculated for a set of data turned out to be y = 4 – 3x. Which of the following statement(s) is/are true for this situation?

a.

The y-intercept of the line is 3.

b.

The slope of the line is negative.

c.

The line represents an inverse relationship.

d.

All of (a), (b) and (c) above.

e.


231. Two variables are said to be perfectly negatively correlated if

a.

Covariance between them is negative

b.

Covariance between them is –1

c.

Covariance between them is equal to −σ 1 σ2 (σ1 and σ2 are standard deviations of the variables)

d.

Correlation coefficient of these variables is –1

e.

Both (c) and (d) above.

232. Markowitz has proposed the Portfolio Theory to evolve strategies to reduce risk. For a portfolio with investments in two securities with equal weightages, the risk is zero when

a.

The mean returns of both securities is same

b.

The standard deviation of both securities is same

c.

The covariance between these two securities is –1

d.

The standard deviation of either of them is zero

e.

None of the above.

233. We know that the standard error is the same at all points on a regression line because we assume that

i.

Observed values are normally distributed around each estimated value of y.

ii.

The variance of the distributions around each possible value of y is the same.

iii.

All available data points were taken into account when the regression line was calculated.

a.

Only (i) above

b.

Only (ii) above

c.

Only (iii) above

d.

Only (i) and (ii) above

e.

All of (i), (ii) and (iii) above.

234. A statistician is trying to explore the relationship between percentage fat content (Y) and price (X), of various dairy products. He finds that the coefficient of determination is 0.4624 and that the estimated regression line is

Y = 0.5 – 0.02X. The coefficient of correlation would be a.

0.68

b.

– 0.68

c.

0.68 and – 0.68

d.

– 0.02

e.

None of the above.

44

Part I

235. X and Y are two random variables and we are interested in estimating the regression equation,

Y = a + bX. For a pilot study, data on four samples are collected and these values are given below: (X, Y): (3, 5) (4, 10), (6, 9), (7, 12) The value of the slope ‘b’ of regression line would be: (choose the value nearest to the true value) a.

1.3

b.

– 12.4

c.

0.0433

d.

1.74

e.

– 1.74.

236. The correlation between money supply and inflation is

a.

Positive

b.

Zero

c.

Negative

d.

Cannot say

e.

Oscillatory.

237. A study has been carried out to find the degree of association between the monthly income, X and net savings, Y. If the standard deviation of X and Y series has been estimated as 3.71 and 4.05 respectively and the covariance between the monthly income and net savings is 12.3, Pearson’s correlation coefficient is

a.

0.819

b.

0.916

c.

1, as there is perfect correlation

d.

Cannot be determined, more information required

e.

None of the above.

238. Mrs. C. Falguni Anand has some surplus funds which she wants to invest in securities with the least risk. As a student of portfolio management she knows that the risk in an investment is normally measured by the variance in the returns from the mean or expected return and that higher the value of dispersion (standard deviation), the higher is the risk. Her research enabled her to identify two securities A and B with the following features.

Rate of Return on Share A

Prob. of Occurrence

Rate of Return on Share B

Prob. of Occurrence

25%

0.36

26%

0.50

20%

0.40

18%

0.30

15%

0.24

11%

0.20

Assuming that Mrs. C. F. Anand invests only in one of the two securities (with the least risk), she should invest in a.

Security A

b.

Security B

c.

Security A or Security B since both of them are equally risky

d.

Cannot be determined, more information required

e.

None of the above.

45


239. If the correlation coefficient is 0.9 the coefficient of determination is

a.

0.3

b.

± 0.3

c.

0.81

d.

± 0.81

e.

0.9.

240. If the function of variation in Y unexplained by the Explanatory variable X is ¼, then R2 is

a.

1/16

b.

3/16

c.

4/16

d.

9/16

e.

3/4.

241. The correlation coefficient of a variable with itself is

a.

1

b.

–1

c.

0

d.

(a) or (b) above

e.

None of the above.

242. In case of a simple regression Y = A + BX

a.

Y is a dependent variable

b.

X is a dependent variable

c.

A is a dependent variable

d.

B is a dependent variable

e.

X and Y are independent variables.

243. Two variables are perfectly positively correlated when the covariance between them is

a.

0

b.

1

c.

–1

d.

σ1σ2 (where σ 1 and σ 2 are standard deviations of the two variables)

e.

Having the sign opposite to their correlation coefficient.

244. Which of the following can be used to predict the value of a dependent variable?

a.

Regression analysis.

b.

Correlation coefficient.

c.

Coefficient of variation.

d.

Frequency distribution.

e.


245. The correlation between X and Y does not depend on

a.

Standard Deviation of X

b.

Standard Deviation of Y

c.

Means of X and Y respectively

d.

Covariance between X and Y

e.

Ranges of X and Y respectively.

46

Part I

246. The random variables X and Y are said to be independent if σX = σY a.

247.

248.

249.

250.

251.

252.

b.

Cov(X,Y) = 0

c.

X=Y

d. E(X) = E(Y) e. Their probability distribution is the same. Many questions in finance and economics are studied by constructing a. Mathematical models b. Physical models c. Real models d. Practical models e. Relative models. The value of a variable X and its proability p is shown below X 5.0 10.0 18.0 20.0 p 0.2 0.5 0.1 0.2 In a Monte Carlo simulation method, if a random number 37 is generated between the interval 1 – 100, the corresponding value of variable X will be a. 5 b. 10 c. 15 d. 18 e. 9. Given the equation of the regression line on a scatter diagram which of the following can be found? a. Intersection of the line on Y-axis. b. Intersection of the line on X-axis. c. Slope of the line. d. Expected value of the dependent variable when independent variable is known. e. All of the above. If the intercept on Y-axis and the slope of a regression line are 4 and 3 respectively, then the equation of the regression line can be written as a. Y = 3X + 4 b. Y = 4X + 3 c. X = 3Y + 4 d. X = 4Y + 3 e. 4X + 3Y = 0. If the correlation coefficient between two variables is –1, then all the pairs of observations when plotted on a scatter diagram will lie a. On the regression line b. Above the regression line c. Below the regression line d. Close to the regression line e. Away from the regression line. Beta of a share is used to measure its a. Systematic risk b. Unsystematic risk c. Total risk d. Company’s business risk e. Company’s security risk.

47


253. If points (x1, y1) and (x2, y2) lie on a regression line Y = a + bX, then b will be equal to

a.

y2/y1

b.

x2/x1

c.

(y2 – y1)/(x2 – x1)

d.

(x2 – x1)/(y2 – y1)

e.

(y2 – x2)/(y1 – x1).

254. When stock market is expected to be bullish in the near to medium term, it makes sense in investing in shares having a beta which is

a.

Less than one

b.

Zero

c.

Equal to one

d.

More than one

e.

Equal to market beta.

255. The variance of a stock’s return and the market return are 33.7 and 22.8 respectively. If beta of the stock is 1.1, then its unsystematic risk will be equal to

a.

6.11

b.

25.08

c.

27.58

d.

37.07

e.

40.78.

256. If the Total Sum of Squares (TSS) is 25 and the Error Sum of Squares (ESS) is 9, then the coefficient of determination will be equal to

a.

16/25

b.

9/25

c.

9/16

d.

4/5

e.

3/5.

257. Which of the following statements is true?

a.

r2 is always less than or equal to the correlation coefficient (r) in magnitude.

b.

r is always less than or equal to r2 in magnitude.

c.

r2 > 0 implies r > 0.

d.

r2 is always less than or equal to r.

e.

r is always less than or equal to r2.

258. On a scatter diagram a regression line is drawn in such a way that the sum of the squares of the vertical distances of each point to the regression line is

a.

Maximum

b.

Minimum

c.

Zero

d.

Positive

e.

Negative.

48

Part I

259. The standard error of the estimate of a regression line is given by ∧

a.

∑ (Y − Y) (n − 2)

b.

∑ (Y − Y) 2 (n − 2)

c.

∑ (Y − Y) 2 (n −1)

d.

∑ (Y − Y) 2 (n − 2)

e.

∑ (Y − Y) 2 . (n −1)

∧

∧

∧

∧

260. The standard deviations of two variables and covariance are 0.1, 0.2 and 0.02 respectively. The two variables

a.

Have a low correlation coefficient

b.

Are not correlated to each other

c.

Are perfectly positively correlated

d.

Are negatively correlated

e.

Are perfectly negatively correlated.

261. If two variables when plotted on a scatter diagram indicate a relationship, then the coefficient of correlation between them

a.

Will be positive

b.

Will be negative

c.

Will be unity

d.

Will be zero

e.

Cannot be determined by the available information.

262. As the degree of correlation between independent variable and dependent variable increases, the regression coefficients will be

a.

More reliable

b.

Less reliable

c.

As reliable as earlier

d.

More predictive

e.

Less predictive.

263. If B in a regression line Y = A + BX is equal to zero, then the regression line will

a.

Pass through the origin

b.

Be parallel to the x-axis

c.

Be parallel to the y-axis

d.

Have the same intersection point on x-axis as well as y-axis.

e.

Never intersect x or y-axis.

49


264. If the proportion of variation in a dependent variable that is explained by the independent variable is 75%, then the coefficient of determination (R2) will be equal to a. 1/16 b. 1/4 c. 9/16 d. 3/4 e. 0.866. 265. If the systematic risk of a security is 13.6 and the variance of market return is 12.8, then the beta of the security is equal to a. 0.94 b. 0.97 c. 1.03 d. 1.06 e. 1.13. 266. Which of the following statements is true? When two variables are associated with each other it means that. a. The dependent variable is the cause of variation in independent variable. b. The independent variable is the cause of variation in dependent variable. c. The dependent variable may be the cause of variation in independent variable. d. The independent variable may be the cause of variation in dependent variable. e. There is no cause and effect relationship between the two variables. 267. Rank correlation is used when the variables a. Have a high variation b. Involved are too many c. Represent data in ordinal form d. Represent data in matrix form e. Represent data in the discrete form. 268. The fixed and variable components of costs can be segregated with the help of a. Cost-volume-profit analysis b. Regression analysis c. Correlation analysis d. Simulation e. Probability theory. ∧

269. A regression line is represented as Y = 3.5X + 4.3. The value of Y at point (0, 4) is equal to a. 0 b. 3.5 c. 4.3 d. 14.0 e. 18.3. 270. If the Error Sum of Squares (ESS) in 207 and the coefficient of determination (R2) is 0.81, then Total Sum of Squares (TSS) will be equal to a. 0.19 b. 39.33 c. 167.67 d. 544.73 e. 1089.47.

50

Part I

271. If the correlation coefficient between the movement of share prices of two securities is 0, then the coefficient of determination between them is equal to

a.

100

b.

1

c.

0

d.

–1

e.

–100.

272. If the variance of X and Y is 3.4 and 5.1 and the covariance between them is 1.7, then the coefficient of correlation between X and Y is

a.

0.098

b.

0.167

c.

0.408

d.

2.449

e.

10.200.

273. If the error sum of squares = 100 and the total sum of squares = 169, what percentage of changes in the dependent variable is explained by the changes in the independent variable?

a.

35.0%.

b.

40.8%.

c.

42.8%.

d.

48.2%.

e.

49.0%.

274. When the slope of the regression equation is negative

a.

There is no correlation between dependent and independent variables

b.

There is a negative correlation between dependent and independent variables

c.

The regression line is parallel to the horizontal axis

d.

The regression line is perpendicular to the horizontal axis

e.

The regression line slopes upwards as one moves along the positive side of the horizontal axis.

275. For a given coefficient of correlation, the coefficient of determination

a.

Is always negative

b.

Is always positive

c.

Has the same sign as the coefficient of correlation

d.

Is always higher in magnitude than the coefficient of correlation

e.

Is sometimes higher and sometimes lower in magnitude.

276. Which of the following is true about the coefficient of determination?

a.

The existence of high coefficient of determination indicates the existence of the cause and effect relationship between the two variables.

b.

It indicates the fraction of total variation of Y that is not explained by the regression line.

c.

If it is equal to one, the dependent and the independent variables are perfectly positively correlated.

d.

If it is equal to one then the error sum of squares is equal to the total sum of squares.

e.

If it is equal to one then the error sum of squares is equal to zero.

51


277. For two variables X and Y the covariance between X and Y is 550 and the variance of the independent variable, X is 250. The slope of the regression line is

a.

0.40

b.

0.45

c.

2.20

d.

2.50

e.

2.80.

278. In the expression for standard error of estimate the denominator is n – 2 because

a.

Two degrees of freedom are gained in computing the coefficients, ‘a’ and ‘b’

b.

Two degrees of freedom are lost in computing the coefficients, ‘a’ and ‘b’

c.

Two variables are used in formulating the regression equation

d.

Two variables (Y and Y ) are used in the numerator

e.

It is customary to reduce the number of data points by 2.

∧

279. If two variables X and Y are perfectly positively correlated and their standard deviations are 5 and 10 respectively, then their covariance is

a.

0.40

b.

0.50

c.

2.00

d.

4.00

e.

50.00.

280. If the coefficient of correlation between variables Y and Z is –0.90 and Y is the independent variable and Z is the dependent variable, then what percentage of the variations in Z is explained by the variations in independent variable Y?

a.

8%.

b.

10%.

c.

19%.

d.

81%.

e.

90%.

281. The return from a share having a beta of –1.5

a.

Increases by 1.5% when the market return increases by 1%

b.

Decreases by 1.5% when the market return decreases by 1%

c.

Is not affected at all by any change in the market return

d.

Increases by 1.5% when the market return decreases by 1%

e.

Is always less than the market return by 1.5%.

282. In a regression model Error Sum of Squares is 30 and Total Sum of Squares is 90. The coefficient of determination is equal to

a.

1/3

b.

2/3

c.

10

d.

60

e.

70.

52

Part I

283. The slope and the intercept on y-axis for a regression line are 2 and 3 respectively. The equation of the line is

a.

y = 2x + 3

b.

y = 3x + 2

c.

y=x+6

d.

3y = 2x

e.

2y = 3x.

284. For the estimating equation to be a perfect estimator of the dependent variables, which of these would have to be true?

a.

The standard error of estimate is zero.

b.

All the data points are on the regression line.

c.

The coefficient of determination is 1.

d.

Error sum of squares = 0.

e.

All of the above.

285. Suppose that the regression equation calculated for a set of data turned out to be Y = 4 – 3X. Which of the following statement(s) is/are true for this situation?

a.

The y-intercept of the line is –3.

b.

The slope of the line is 4.

c.

There is an inverse correlation between X and Y.

d.

Y increases as X increases.

e.

None of the above.

286. An expert is trying to explore the relationship between per capita income (Y) and disposable income (X) for a particular community. He finds that the coefficient of determination is 0.4624 and that the estimated regression line is Y = 0.5 – 0.02X. The coefficient of correlation would be a. 0.68 b. – 0.68 c. 0.68 and –0.68 d. –0.02 e. None of the above. 287. The regression relationship between the returns from a share (Rx) and returns from the market portfolio (Rm) is Rx = 5.5 + 1.25 Rm If the return on the market portfolio is 12% the return from the share will be a. 15.5% b. 17.5% c. 20.5% d. 22.5% e. 25.0%. 288. If the coefficient of correlation between two data sets is –0.85 and the variances of the two sets are 16 and 9 respectively, then the covariance between the data sets is a. –10.20 b. – 4.20 c. 3.20 d. 7.20 e. 10.20.

53


289. If total sum of squares = 625 and regression sum of squares = 550, then the coefficient of determination is a. 0.84 b. 0.86 c. 0.88 d. 0.90 e. 0.92. 290. If the regression equation is not a perfect estimator of the dependent variable then which of the following will be true? a. The standard error of estimate is not zero. b. The data points do not lie on the regression line. c. The coefficient of determination is not one. d. Error sum of squares is not zero. e. All of the above. 291. The following information is given with regard to the regression relationship between two variables X and Y (X is independent variable and Y is dependent variable): X = 6.8; Y = 12.2; b = 1.5. The regression equation is ∧

a.

Y = 1.5 + 2.0X

b.

Y = 2.0 + 1.5X

c.

Y = 2.5 + X

d.

Y = 1.0 + 2.5X

∧

∧

∧

∧

Y = 1.5 + 2.5X . e. 292. Alpha company has identified a potential market for innovative products. Studies conducted indicate that the probability of a successful product launch in one attempt is 0.25. The probability that the company has a successful product launch in 3 attempts is

a.

0.9844

b.

0.7523

c.

0.6321

d.

0.5137

e.

0.4219.

293. If covariance between the returns from a stock and market returns = 225 and variance of market returns = 180, the beta coefficient for the stock is

a.

1.25

b.

1.50

c.

1.75

d.

2.00

e.

2.25.

294. The regression line of Y on X is given by Y = 35.82 + 0.476X and that of X on Y is given by X = –3.38 + 1.036Y. Then, the value of r2 is

a.

0.369

b.

0.459

c.

0.493

d.

0.536

e.

0.662.

54

Part I

295. Which of the following is not required for finding out the standard deviation of returns of a portfolio of 2 securities X and Y?

a.

Proportions of the securities X and Y in the portfolio.

b.

Standard deviation of returns of X.

c.

Standard deviation of returns of Y.

d.

Coefficient of correlation between X and Y.

e.

Mean returns of individual securities X and Y. ∧

296. If for a simple linear regression equation has ∑(Y − Y) 2 = 158.76 and there are 11 pairs of observations then the standard error of estimate is

a.

3.80

b.

4.20

c.

12.60

d.

17.64

e.

18.00.

297. Which of the following is not true with regard to simple linear regression?

a.

There is one independent variable and one dependent variable.

b.

The relationship between the independent and dependent variables is necessarily linear.

c.

The relationship between the independent and dependent variables is a relationship of association.

d.

The relationship between the independent and dependent variables necessarily implies cause and effect.

e.

There may be other factors which cause changes in both the dependent and independent variables.

298. Which of the following is not true with regard to coefficient of determination?

a.

It indicates the percentage of variations in the dependent variable that is explained by the regression relationship.

b.

0 ≤ coefficient of determination ≤ 1.

c.

The magnitude of coefficient of determination is less than the magnitude of coefficient of correlation.

d.

It is different for positive and negative correlation of the same magnitude.

e.

None of the above.

299. Which of the following is false with regard to simple regression and correlation analysis?

a.

If the coefficient of correlation between two variables is close to one, then there is a strong correlation between the two variables.

b.

If the slope of a regression equation is positive then the coefficient of correlation between the variables involved is positive.

c.

A high correlation always indicates a cause-effect relationship between the variables involved.

d.

If the slope of a regression equation is negative then the coefficient of correlation between the variables involved is negative.

e.

If the coefficient of correlation between two variables is equal to zero, then there is no correlation between the two variables.

55


300. According to the ‘method of least squares’ criterion, the regression line should be drawn on the scatter diagram in such a way that

a.

The sum of the squared values of the vertical distances from each plotted point to the line is maximum

b.

The sum of the squared values of the vertical distances from each plotted point to the line is minimum

c.

The sum of the squared values of the horizontal distances from each plotted point to the line is maximum

d.

The sum of the squared values of the horizontal distances from each plotted point to the line is minimum

e.

The sum of the squared values of the vertical distances from each plotted point to the line is equal to zero.

301. The standard error of estimate of a simple regression line is

a.

Equal to the slope of the regression line

b.

Equal to the square root of the coefficient of determination

c.

Equal to the Y-intercept of the regression line

d.

The measure of the variability of the observed values around the regression line

e.

Equal to the coefficient of correlation between two variables.

302. If r is the coefficient of correlation between two variables X and Y, then the variation in the observed Y values which is not explained by the regression line is equal to

a.

r2

b.

1− r2

c.

r

d.

1–r

e.

1 + r2.

303. If RSS be the Regression Sum of Squares and ESS be the Error Sum of the Squares in a linear regression analysis between two variables with n observations, then the ratio RSS /1 follows ESS /(n − 2)

a.

A normal distribution

b.

A t-distribution

c.

An F-distribution

d.

A chi-square distribution

e.

A binomial distribution.

304. Which of the following is true with regard to covariance?

a.

It is the average of all the values that may be assumed by two random variables, considered together.

b.

It is the variance of all the values that may be assumed by two random variables, considered together.

c.

It is the standard deviation of all the values that may be assumed by two random variables, considered together.

d.

It is the range of all the values that may be assumed by two random variables, considered together.

e.

It is a single number that captures the extent to which two random variables move together.

56

Part I

305. If two random variables are independent of each other then their covariance is equal to a. 0 b. 1 c. A positive value d. A negative value e. An infinite value. 306. The Y-intercept in the simple linear regression equation (X is the independent variable and Y is the dependent variable) represents the a. True value of Y when X = 0 b. Change in average value of Y per unit change in X c. Expected value of Y when X = 0 d. Standard deviation of the values of X e. Mean of the values of X. 307. The slope of the simple linear regression equation (X is the independent variable and Y is the dependent variable) represents the a. Mean value of Y when X = 0 b. Change in mean value of Y per unit change in X c. True value of Y for a fixed value of X d. Variance of the values of X e. Variance of the values of Y for a fixed value of X. 308. Which of the following is true with regard to simple regression and correlation analyses? a. Correlation analysis provides a measure, which indicates how well an equation for estimating one variable from another, explains the changes in the estimated variable. b. If the coefficient of correlation between two variables is close to 1, then there is no correlation between the two variables. c. If the slope of a regression equation is positive, then the coefficient of correlation between the variables involved is negative. d. If the slope of a regression equation is negative, then the dependent variable decreases as the independent variable decreases. e. Correlation analysis necessarily indicates the presence of cause-effect relationship between the variables involved. 309. Which of the following is indicated if the standard error of estimate for a regression equation is zero? a. The standard deviation of the observed values of the dependent variable is zero. b. The standard deviation of the independent variable is zero. c. The observed value of the dependent variable will always be equal to its estimated value, for a given value of the independent variable. d. There is no correlation between the variables involved. e. The correlation coefficient between the variables is greater than zero but less than 1. 310. Which of the following is true when the slope of a regression line is negative? a. The correlation coefficient between the dependent and independent variables is 1. b. The correlation coefficient between the dependent and independent variables, lies between 0 and 1. c. There is a negative correlation between the dependent and independent variables. d. The regression line is parallel to the horizontal axis. e. The regression line passes through the intersection of the horizontal and vertical axes.

57


311. Which of the following is true with regard to a given coefficient of correlation and its corresponding coefficient of determination?

a.

The coefficient of determination is always greater than or equal to zero, and less than or equal to 1.

b.

The coefficient of determination is always less than zero.

c.

The coefficient of determination is always equal to 1.

d.

The coefficient of determination always has the same sign as the coefficient of correlation.

e.

The magnitude of coefficient of determination is always higher than the magnitude of coefficient of correlation.

312. Which of the following is the graphical plot of the values of the dependent and independent variables, in the context of regression analysis?

a.

Scatter diagram.

b.

Frequency polygon.

c.

Histogram.

d.

p chart.

e.

Ogive.

Multiple Regression Based on the following information answer questions 313 to 315.

The following data pertains to 3 variables – X1, X2 and X3. r12 = 0.8196; r13 = 0.7698; r23 = 0.7984. Note: X1 is the dependent variable. 313. Find the coefficients of linear partial correlation r12.3

a.

0.2391

b.

0.3346

c.

0.3931

d.

0.4580

e.

0.5334.

314. The multiple correlation coefficient between X1 and both X2 and X3 is

a.

0.3346

b.

0.4127

c.

0.6580

d.

0.7334

e.

0.8417.

315. What percentage of the variation in X1 can be explained by the multiple regression equation in terms of X2 and X3?

a.

70.85.

b. c. d. e.

65.37. 61.21. 54.75. 50.20.

58

Part I

Based on the following information answer questions 316 to 320. The management consultant employed by a firm to examine the efficiency of the plants it operates considers that the profitability of each plant is positively related to the total sales of each plant (X1) and capital expenditure (X2) but inversely related to a cost efficiency index (total costs/sales), denoted by X3, and the number of employees (X4). The computer output obtained by regressing profits on the four explanatory variables, using data from a random sample of plants is given below.

The regression equation is Profits = –27.4 + 0.0464 X1 + 8.21 X2 + 12.3 X3 + 0.109 X4 Analysis of Variance

Source Regression Error Total Root M.S.E = (3) Variable

DF 4 1 5

Sum of Squares (1) 1.3 22239.4

DF Parameter Estimate Intercept –27.383 X1 1 0.046369 X2 1 8.2096 X3 1 12.288 X4 1 0.10928 316. The value appearing in blank (1) is a.

21238.1

b.

22238.1

c.

22239.1

d.

24239.1

e.

25239.1.

Mean Square 5559.5 1.3

Std. Error 3.188 0.002742 0.2466 3.064 0.02139

F Value

Prob > F

(2)

0.011

R-Square = (4) T for H0: Parameter = 0 –8.59 (5) 33.29 4.01 5.11

Prob > |t| 0.074 0.038 0.019 0.156 0.123

317. The value appearing in blank (2) is

a.

4276.54

b.

4288.09

c.

4388.71

d.

4399.09

e.

4791.72.


a.

1.11

b.

1.12

c.

1.14

d.

1.15

e.

1.19.


a.

90%

b.

92%

c.

93%

d.

95%

e.

100%.

59



a.

13.79

b.

14.86

c.

15.91

d.

16.91

e.

17.91.

321. Which of the following is true with regard to multicollinearity in multiple regression analysis?

a.

It increases the influence of the individual variables in the model.

b.

It reduces the predictive power of the model.

c.

It increases the reliability of the regression coefficients.

d.

It reduces the effectiveness of sensitivity analysis of the model.

e.

None of the above.

322. The standard error of the estimate can be decreased for a regression equation by

a.

Increasing the number of observations

b.

Using ‘t’ distribution

c.

Using two tails of a ‘t’ distribution

d.

Choosing an observation point close to the estimate required

e.

Increasing the number of independent variables.

323. Which of the following statements is not true with regard to the coefficient of multiple correlation R1.23 between the dependent variable, Y, and the independent variables, X1 and X2?

a.

It is a non-negative real number.

b.

It can take values between 0 and 1.

c.

It is dependent on the coefficient of correlation between Y and X1.

d.


e.

It is independent of the coefficient of correlation between X1 and X2.

324. If it is given that the correlation coefficients between X1 and X2 is 0.36, between X2 and X3 is 0.64, between X3 and X1 is 0.49, the multiple correlation coefficient between X1 and both X2 and X3 will be given by

a.

0.36

b.

0.49

c.

0.64

d.

0.73

e.

None of the above.

325. A normal distribution may be used to approximate the ‘t’ distribution for multiple regression whenever the degrees of freedom (n minus the number of estimated regression coefficients) are

a.

Less than 40

b.

More than 10

c.

Equal to 5

d.

More than 30

e.

None of the above.

60

Part I

326. We have said that the standard error of estimate has n – k – 1 degrees of freedom. What does the k stand for in this expression?

a.

Number of elements in the sample.

b.

Number of independent variables in the multiple regression.

c.

Mean of the sample values of the dependent variable.

d.

Number of samples.

e.

None of the above.

327. Suppose that you have run a multiple regression and have found that the value of b1 is 1.66. Historical data, however, indicate that the value of B1 should be 1.34. You wish to test, at a 0.05 level of significance, the null hypothesis that B1 is still 1.34. Assuming that you have access to any tables you may need, what other information is required for you to perform your test?

a.

Degrees of freedom.

b.

Sb .

c.

Se.

d.


e.

(a) and (c) but not (b).

1

The next 3 questions deal with a director of personnel who is trying to determine a predicting equation for longevity in his plant. He has used a computer package to regress months employed for several employees on their education levels (years of schooling), age when hired, score on the company’s psychological maturity test, and number of dependents (including the employee). Here, are his results: Dep Variable: Longev length of Employment (Months) Analysis of Variance

Source

DF

Model

4

7325.325

1831.331

Error

5

898.275

179.655

C Total

9

8223.600

Root MSE Variable

Sum of Squares Mean Square

13.403541

R-Square

F Value Prob > F 10.194

0.0127

0.8908

DF

Parameter Estimate

Standard Error

T for H0: Prob > parameter = 0

Intercep

1

82.237454

81.737817

1.006

0.3605

School

1

–1.552644

4.362058

–0.356

0.7364

Age

1

–1.685367

1.252534

–1.346

0.2362

Score

1

0.110216

0.290813

0.379

0.7203

Dependen

1

6.875539

7.657836

0.898

0.4104

T

328. How much of the variation in length of employment is explained by the regression? a. 94%. b. 82%. c. 89% d. 13%. e. None of the above.

61


329. Suppose you wish to test whether years of school is a significant explanatory variable for longevity. The degrees of freedom you would use would be a. 4 b. 10 c. 6 d. 5 e. None of the above. 330. What is the value of Sb3 ?

a. 13.4. b. 0.29. c. 0.38. d. 0.11. e. None of the above. 331. Signs of the possible presence of multicollinearity in a multiple regression are a. Significant t values for the coefficients b. Low standard errors for the coefficients c. A sharp increase in a t value for the coefficient of an explanatory variable when another variable is removed from the model d. Both (a) and (b) above e. None of the above. 332. A general multiple regression equation which has n-independent variables, will have

a.

n constants

b.

n + 1 constants

c.

n – 1 constants

d.

n dependent variables

e.

No constants.

333. Multicollinearity is the

a.

Existence of correlation among independent variables

b.

Absence of correlation among independent variables

c.

Existence of correlation among dependent variables

d.

Absence of correlation among dependent variables

e.

Predictive power of the constants.

334. Multiple regression analysis is used to

a.

Relate changes in more than one dependent variable to one independent variable

b.

Relate changes in more than one dependent variable to more than one independent variable

c.

Relate changes in one dependent variable to more than one independent variable

d.

Relate changes in one dependent variable to one independent variable

e.

Establish a cause-effect relationship between multiple pairs of dependent and independent variables considered at the same time.

335. A general multiple regression equation will

a.

Have many dependent variables

b.

Have only one dependent variable

c.

Have only one independent variable

d.

Not have any independent variables

e.

Not have any constants.

62

Part I

336. Which of the following statements regarding multicollinearity is false?

a.

As degree of correlation between the independent variables increases, the regression coefficients become less reliable.

b.

It reduces the accuracy of the model.

c.

It hurts any sensitivity analysis.

d.

It can happen that the model may be accepted through the F test and ANOVA but the individual coefficients may be rejected through t test.

e.

None of the above.

337. Which of the following is false for the multiple correlation coefficient between dependent variable Y and two independent variables X1 and X2?

a.

It depends on the correlation coefficient between Y and X1.

b.

It depends on the correlation coefficient between X1 and X2.

c.

It depends on the correlation coefficient between Y and X2.

d.

It will take values between 0 and 1.

e.

It will take values between –1 and +1.

338. In a multiple regression analysis involving two independent variables, X and Y, we find that there is no correlation between dependent variable Z and independent variable Y. We also observe that there is no correlation between two independent variables X and Y. The partial coefficient of correlation between Z and X when Y is kept constant is equal to a. Simple coefficient of correlation between Z and X b. Simple coefficient of correlation between Z and Y c. Simple coefficient of correlation between X and Y d. Partial coefficient of correlation between Z and Y when X is kept constant e. Partial coefficient of determination between Z and X when Y is kept constant. 339. In a multiple regression with n observations and k independent variables, the standard error of estimate is calculated using the number of degrees of freedom as a. n – 2 b. n – k c. n – k –1 d. n – k +1 e. n + k –1. 340. In multiple-regression analysis, the regression coefficients often become less reliable as the degree of correlation between the independent variables increases. This problem is referred by statisticians as a. Random variations b. Non-random variations c. Irregular variation d. Multicollinearity e. Cyclical variation. 341. A multiple regression relationship is given by : Y = a + b1X1 + b2X2 Which of the following conditions indicates the presence of multicollinearity? a. The regression coefficients ‘a’, ‘b1’ and ‘b2’ are all negative. b. Only the regression coefficients ‘b1’ and ‘b2’ are negative. c. There is a strong correlation between the variables X1 and Y. d. There is a strong correlation between the variables X2 and Y. e. There is a strong correlation between the variables X1 and X2.

63


342. Which of the following is false with regard to the coefficient of multiple correlation between the dependent variable, Y, and the independent variables, X1 and X2?

a.

It can take values in the range of 0 to 1.

b.

It is the positive square root of coefficient of multiple determination.

c.

It does not depend upon the coefficient of correlation between X1 and X2.

d.

It depends upon the coefficient of correlation between Y and X1.

e.

It depends upon the coefficient of correlation between Y and X2.

343. Which of the following is true with regard to a multiple regression equation?

a.

It has only one dependent variable.

b.

It has multiple dependent variables.

c.

It has only one independent variable.

d.

All the regression coefficients are equal to 1.

e.

For given values of the independent variables the estimated value of the dependent variable is always equal to its observed value.

Index Numbers Based on the following information answer questions 344 to 348.

The following data pertains to the consumption of material by a bakery. Inputs

Units

Prices (in Rs.)

Quantities Used

P0

P1

Q0

Q1

1993

1996

1993

1996

Flour

Kilo

15

30

500

700

Eggs

Dozen

8

14

100

70

Milk

Liter

6

17

200

120

Sugar

Kilo

10

16

50

70

344. Laspeyres Price Index for the given data is

a.

184.5

b.

190.0

c.

201.4

d.

206.0

e.

250.5.

345. Laspeyres Quantity Index for the given data is

a.

123.4

b.

124.8

c.

198.2

d.

206.0

e.

210.1.

346. Paasche’s Quantity Index for the given data is

a.

122.04

b.

124.80

c.

197.81

d.

201.44

e.

205.31.

64

Part I

347. The unweighted average of relatives price index for the given data is

a.

123.41

b.

199.22

c.

201.44

d.

204.58

e.

307.31.

348. When the base year values are used as weights, the weighted average of relatives price index is the same as

a.

The Laspeyres price index

b.

The Paasche’s price index

c.

The unweighted average of relatives price index

d.

Fisher’s ideal index

e.

Marshall-Edgeworth index.

349. Which of the following is/are true?

a.

Laspeyres Index tends to overestimate the rise in prices or has an upward bias.

b.

As opposed to the Laspeyres Index the Paasche Index has a downward bias.

c.

Fisher’s Ideal Index is free from any bias.

d.

Both (a) and (b) above.

e.


350. What can be concluded if the weighted aggregates price index for a set of prices was calculated as 106 index using the Laspeyres method and 112 using the Paasche method?

a.

The Paasche index is incorrect.

b.

There is a trend towards less expensive goods.

c.

There is a trend towards more expensive goods.

d.

The difference can be attributed to a poor estimation of consumer attitudes.

e.


351. Using 1976 as the base year, the consumer price index was 197 in 1989. If a person’s annual salary had increased from Rs.7,000 in 1976 to Rs.11,000 in 1989, his adjusted salary has

a.

Decreased by 97%

b.

Decreased by 20%

c.

Increased by 97%

d.

Increased by 20%

e.

None of the above.

352. To measure changes in monetary worth, one should calculate

a.

Price index

b.

A value index

c.

A quantity index

d.

No index can be useful

e.

None of the above.

65


353. If the weighted aggregates price index for a set of prices was calculated as 117 using the Paasche method and as 121 using the Laspeyres method, what can be concluded from this?

a.

The Paasche index is incorrect.

b.

There is a trend toward less expensive goods.

c.

There is a trend toward more expensive goods.

d.

The Laspeyres index is incorrect.

e.

The difference between the two indices can be attributed to a poor estimation of consumer attitudes.

354. Distortion of index numbers is due to the following

a.

Availability of limited data

b.

Inappropriate weighting of factors

c.

Selection of an improper base

d.

Incomparability of indices

e.

All of the above.

355. Which of the following is an example of unweighted index?

a.

∑ P1 x100. ∑ P0

b.

∑ P1 Q0 x100. ∑ P0 Q0

c.

∑ Q1 P0 x100. ∑ Q0 P0

d.

Paasche’s Index.

e.

Fisher’s Ideal Index.

356. The Link relatives are used in

a.

Average of relatives method

b.

Value index numbers

c.

Chain index numbers

d.

Factors reversal test

e.

Circular test.

357. Which of the following is an example of the simplest form of index numbers?

a.

Unweighted Index.

b.

Weighted Index.

c.

Price Index.

d.

Laspeyre’s Index.

e.

Fisher’s Index.

358. Which of the following is not a source of Index Numbers?

a.

RBI.

b.

CSO.

c.

Labor Bureau.

d.

FAO.

e.

FEDAI.

66

Part I

359. Seasonal variations have minimal impact on a. Price Index b. Quantity Index c. Weighted Index d. Unweighted Index e. Chain Index. 360. Which of the following is not true for index numbers? a. They are one of the most widely used statistical indicators. b. They are extensively used in indicating the price levels in the share markets. c. They are called barometers of economic activity. d. Their role in trade and industry is not very significant. e. There are various types of index numbers in use today. 361. Which of the following indices satisfy Factor Reversal Test? a. Laspeyre’s index. b. Paasche’s index. c. Fisher’s ideal index. d. Marshall-Edgeworth index. e. Value index. 362. The composite price index of a basket of goods was 310 in 1997 and 334 in 1998. If both the indices were computed from a base of 100 in 1980, how much did the general price level rise between 1997 and 1998? a. 7.18%. b. 7.56%. c. 7.74%. d. 17.86%. e. 19.52%. 363. The test in which the resulting index is expected to be the reciprocal of original index, when time periods are reversed is known as a. Time reversal test b. Factor reversal test c. Circular test d. Chain test e. Reciprocal test. 364. Which of the following statements is false? a. An unweighted aggregates price index does not attach any importance to the consumption or usage levels of the items covered. b. Laspeyres price index is an unweighted aggregates price index. c. Paasches price index is a weighted aggregates price index. d. A price index reflects the overall change in price for the basket of goods considered. e. None of the above. 365. An index has increased from 141 to 163 in a period of 2 years from 1997 to 1999. The percent relative for 1999 taking the base as 1997 will be a. 107.51 b. 115.60 c. 141.00 d. 150.67 e. 163.00.

67


366. If the Laspeyre’s and Paasche’s indices are 104 and 106 respectively, then the Fisher’s Ideal Index will be equal to

a.

104

b.

105

c.

106

d.

110

e.

None of the above.

367. The units of index numbers are a. Same as that of underlying variables b. Index number does not have a unit of measurement c. Close to 100 d. Always positive e. Multiple. 368. If the Laspeyre’s and Paasche’s Index are 102.41 and 103.2 respectively, then Fisher’s Ideal Index is equal to a. 99.23 b. 100.77 c. 102.41 d. 102.80 e. 103.20. 369. If the consumer price index increases from 100 in year X to 250 in year Y, the real purchasing power of the rupee in year Y is a. 30 paise b. 35 paise c. 40 paise d. 45 paise e. 50 paise. 370. If, for a basket of commodities, Laspeyres price index is 135 and Paasche’s price index is 120 then Fisher’s ideal price index will be a. 127.0 b. 127.3 c. 127.5 d. 127.8 e. 130.0. 371. Unweighted aggregates price index is equal to

a.

P1 x 100 P0

b.

∑

c.

∑ P1 x 100 ∑ P0

d.

∑ P1 x 100 P0

e.

P1 x100. ∑ P0

P1 x 100 P0

68

Part I

372. If the Laspeyres and Paasche’s price indices are 108 and 112.4 respectively, then Fisher’s Ideal price index will be equal to a. 109 b. 110.2 c. 113 d. 115 e. None of the above. 373. Which of the following indices is a geometric mean of the Laspeyres Index and Paasche’s Index?

a.

Weighted aggregates index.

b.

Unweighted aggregates index.

c.

Fisher’s ideal index.

d.


e.

Chain index.

374. Given: ∑ P1 Q 1 = 365 ∑ P0 Q 1 = 245 ∑ P1Q 0 = 308 ∑ P0 Q0 = 205 P0 and P1 are base year and current year prices respectively; Q0 and Q1 are base year and current year quantities respectively.

The price index by the Paasche’s method is a.

143.15

b.

144.25

c.

146.75

d.

148.98

e.

150.18.

∑ P1 Q 1 x 100, where P1 and Q1 are the price and quantity in year 1, and P0 and ∑ P0 Q 0 Q0 are the price and quantity in year 0 respectively.

375. Price index =

Which of the following types of indices use the above expression? a.

Laspeyres price index.

b.

Paasche’s quantity index.

c.

Value index.

d.

Fisher’s ideal quantity index.

e.

None of the above.

376. Which of the following is/are used for testing the consistency of Index numbers?

i.

Runs test

ii.

Time reverval test

iii.

Serial correlation test

iv.

Circular test.

a.

Both (i) and (ii) above.

b.

Both (i) and (iii) above.

c.

Both (ii) and (iii) above.

d.

Both (ii) and (iv) above.

e.

All of the above.

69


377. The index of industrial production is an example of

a.

Price index

b.

Value index

c.

Quantity index

d.

Unweighted index

e.

None of the above.

378. In a certain year the nominal wage is Rs.945 and the consumer price index is 108. The real wage for that year is

a.

Rs.850

b.

Rs.875

c.

Rs.900

d.

Rs.920

e.

Rs.945.

379. If, for a set of commodities, the Laspeyre’s price index = 130 and Paasche’s price index = 127, then Fisher’s ideal price index is equal to

a.

127.75

b.

128.00

c.

128.49

d.

129.54

e.

130.00.

380. Which of the following is false about index numbers?

a.

Index numbers are expressed as a percentage.

b.

Index number for the base year is always 100.

c.

Index number has a unit of the variable that is being compared.

d.

Index number is calculated as a ratio of the current value to a base value.

e.

None of the above.

381. Which of the following is not true about the use of index numbers?

a.

Index numbers when analyzed reveal a general trend of the phenomenon under study.

b.

Index numbers help in taking policy decisions, like dearness allowance paid to employees.

c.

Index numbers help in determining the purchasing power of money.

d.

Index numbers can be used to deflate time series data to reflect real values.

e.

The value index is the most popular index as it can distinguish the effects of price and quantity separately.

382. Which of the following price indices has a downward bias?

a.

Marshall-Edgeworth price index.

b.

Fisher’s price index.

c.


d.

Paasche’s price index.

e.

None of the above.

70

Part I

383. According to the factor reversal test if we interchange the price and quantity terms in an index number then the product of the original index number and the index number obtained by reversing the factors should be equal to

a.

One

b.

One hundred

c.

The value index

d.

The Fisher’s ideal index

e.

The chain index number.

384. Which of the following price indices uses only current year quantities as weights?

a.

Fisher’s ideal price index.

b.

Laspeyre’s price index.

c.


d.


e.

Fixed weight aggregates price index.

385. Which of the following is true with regard to index numbers?

a.

The fixed weight aggregates price index does not allow the flexibility to select the base period for comparison of prices.

b.

Paasche’s price index tends to underestimate the rise in prices.

c.

Unweighted aggregates price index numbers link the price changes to the usage or consumption levels.

d.

Laspeyre’s price index has a downward bias.

e.

The weighted average of relatives price index uses base year prices as weights.

386. Which of the following is true with regard to Fisher’s ideal price index?

a.

It does not consider the base year prices.

b.

It does not consider the base year quantities.

c.

It does not consider the current year prices.

d.

It does not consider the current quantities.

e.

It is the geometric mean of the Laspeyre’s and Paasche’s price indices.

Time Series Analysis 387. A hotelier found that the total number of bookings during 1998 was 10,000. The seasonal index during the second quarter was found to be 155. Then the number of rooms that will be booked during second quarter in 1999 will be

a.

8735

b.

7853

c.

5837

d.

3875

e.

None of the above.

388. In the third quarter of 1998 it was found that the amount of rice exported by a firm was Rs.3 crore. The centered moving average for that period was found to be 2.76. The value of the seasonal component, if expressed in terms of percentage, will be

a.

108.69

b.

102.67

c.

95.84

d.

91.23

e.

87.65.

71


389. The values during the fourth quarter of four consecutive years were as follows: 88.3, 105.5, 109.8 and 94.5. The unadjusted index for the fourth quarter is, therefore,

a.

100.00

b.

110.00

c.

116.35

d.

118.50

e.

120.00.

390. The unadjusted and adjusted seasonal indices were 112.36 and 113.45 respectively. The adjusting factor is

a.

0.95

b.

1.00

c.

1.01

d.

1.03

e.

1.05.

391. Consider the following data.

X Y

–2 7

–1 8

0 5

1 6

2 8

The value of “b” is a.

0.00

b.

1.00

c.

1.50

d.

2.00

e. 2.25. 392. The adjusting factor for a particular year, when four quarters were considered was 0.9234. The sum of modified means in this case is a. 433.00 b. 433.18 c. 433.21 d. 433.25 e. 433.30. 393. Which of the following is/are true? a. The various components of variation in a time series can be isolated even if the data is collected at irregular intervals of time. b. Only after the isolation of the cyclical trend, we are able to conduct analysis for seasonal variation. c. The cancelation of the highest and the lowest of the seasonal values reduces the extreme cyclical and irregular variations. d. The relative cyclical residual can be obtained by subtracting the percent of trend value from 100. e. All of the above. 394. The estimating equation regarding the sale of refrigerators for three years 1996, 1997 and ∧

1998 was found to be Y = 2.65 + 1.10X. Then the estimate of sales for 1999 is _________. a. 4.85 b. 10.75 c. 11.00 d. 11.25 e. 11.60.

72

Part I

395. Which of the following class of variations can be identified with the help of time series analysis? a. Secular trend. b. Cyclical fluctuation. c. Seasonal variation. d. Irregular variation. e. All of the above. 396. Which of the following series indicates the moving average of two data points of the observations 4, 5, 7, 8, 5, 2? a. 4.5, 6, 7.5, 6.5, 3.5. b. 9, 12, 15, 13, 7. c. 1, 2, 1, 3, 3. d. 2, 2.5, 3.5, 4, 2.5. e. 3, 4, 5, 4, 2. 397. Percent of trend measure is used to a. Study secular trend b. De-seasonalizing a time series c. Relative cyclical residual d. Observe the variations from a trend line e. Study irregular variation. 398. The long-term moving average of a variable indicates its a. Secular trend b. Cyclical trend c. Seasonal trend d. Irregular trend e. None of the above. 399. Relative Cyclical Residual Measure is expressed as

a.

Y−Y Y ∧

b.

Y−Y ∧

Y ∧

c.

Y−Y Y ∧

d.

Y−Y Y ∧

e.

Y−Y

. Y 400. The variation due to events like floods, strikes and war etc., in a time series data can be classified as a. Secular trend b. Cyclical variation c. Seasonal variation d. Irregular variation e. None of the above.

73


401. Which of the following statements is/are true? a. Secular trend allows us to describe a historical pattern in the data. b. Secular trend represents the short-to medium-term variation in time series data. c. The component of a time series that oscillate above and below the trend line for a period smaller than one year is known as irregular variation. d. Seasonal variations are periodic and repetitive in nature. e. Both (a) and (d) above. 402. If the upper control limit and the lower control limit of a x chart are 10.967 and 10.367 respectively, then the standard deviation of the corresponding sampling distribution is a. 0.100 b. 0.300 c. 0.600 d. 10.667 e. None of the above. 403. Patterns of change in a variable occurring within a year and which tend to be repeated from year to year are known as a. Secular trend b. Cyclical fluctuation c. Seasonal variation d. Irregular variation e. None of the above. 404. The business cycle is an example of a. Secular trend b. Cyclical fluctuation c. Seasonal variation d. Irregular variation e. Regular variation. 405. Repetitive and predictable movement around the trend line in one year or less is known as a. Secular trend b. Cyclical fluctuation c. Seasonal variation d. Irregular variation e. Temporary variation. 406. In time series analysis a trend which reveals the long-term behavior of a variable is known as a. Cyclical fluctuation b. Seasonal variation c. Secular trend d. Regular variation e. Irregular variation. 407. The estimated value of a variable according to the trend equation is 84.5 in a particular year. The actual value of the variable is 86.4. The relative cyclical residual is a. 0.84 b. 1.16 c. 1.54 d. 1.96 e. 2.25.

74

Part I

408. In a time series the actual value of the variable plotted against time is 54 for a particular year. For this year, the estimated value of the variable is 50. The relative cyclical residual is equal to

a.

0.08

b.

1.08

c.

8.00

d.

80.00

e.

108.00.

409. Which of the following methods is used to isolate the cyclical variation component from the time series?

a.

Residual method.

b.

Linear regression analysis.

c.

Non-linear regression analysis.

d.

Ratio to Moving average method.

e.


410. An ice cream parlor has been witnessing similar fluctuations in the demand of ice creams in the same quarters each year over the last five years. The owner wants to study the movement of the demand to streamline his procurement plans. Which of the following components of the time series should be studied to get an accurate behavior of the demand?

a.

Secular trend.

b.

Cyclical variation.

c.

Seasonal variation.

d.

Irregular variation.

e.

Regular variation.

411. The movement of the dependent variable above and below the secular trend line over periods longer than one year is known as

a.

Seasonal variation

b.

Secular variation

c.

Irregular variation

d.

Cyclical variation

e.

Regular variation.

412. Which of the following is not true about the ratio to moving average method?

a.

This method is used to identify the seasonal variations in a time series.

b.

An index is constructed with a base of 100.

c.

The magnitude of seasonal variations is measured by the individual deviations from the base of 100.

d.

The index is used to eliminate the seasonal effects in the time series.

e.

We multiply the original data points with the relevant seasonal index to deseasonalize the time series.

413. Which of the following components of time series can be separated using the percent of trend measure?

a.


b.

Secular trend.

c.

Seasonal variation.

d.

Cyclical variation.

e.

None of the above.

75


414. Which of the following components of time series shows the long-term behavior of the variable?

a.

Seasonal variation.

b.

Cyclical variation.

c.

Secular trend.

d.


e.

None of the above.

Quality Control 415. Which of the following is/are true?

a.

Luxury is necessarily one of the facets of quality.

b.

Generally high quality items are costly compared to items of lower quality.

c.

If the variability in the production process is within the accepted limits, then the process is said to be maintaining quality.

d.

It is sufficient that the quality aspect is maintained for some crucial aspects of the production process.

e.

It is cheaper to check the finished product rather than checking each of the individual parts.

416. In x charts, a process is said to be in control, when the points plotted

a.

Are on either side of the center line with small deviations

b.

Follow a random path and are between the control limits

c.

Lie beyond the control limits

d.

Lie near the control limits

e.


417. The LCL in the R control charts when n

a.

1

b.

0.5

c.

0

d.

Insufficient data

e.

None of the above.

≤

6 is

418. Which of the following example has the characteristics of an attribute?

a.

The result of the candidate who has contested for the position of general secretary of a labor union.

b.

A person being a postgraduate or not.

c.

Nuclear status of a country (in case of arms).

d.


e.


419. The lower control limit in a R chart if n = 6, R = 5.3, d2 = 2.534 and d3 = 0.848 is

a.

–10.62

b.

0

c.

10.62

d.

10.76

e.

10.94.

76

Part I

420. The grand mean is calculated as

a.

∑ x2 ∑x

b.

∑ k x2 ∑k x

c.

∑ x2 ∑k x

d.

∑x k

e.

∑ x −2 . ∑kx

421. The upper control limit for n = 48 and p = 0.23 in p-chart will be

a.

0.05

b.

0.06

c.

0.18

d.

0.23

e.

0.41.

422. Which of the following is/are the example(s) of control chart(s)?

a.

x charts.

b.

R charts.

c.

p charts.

d.


e.


423. The upper control limit of x chart for a product having a mean dia of 1.20 cm, population standard deviation of 0.10 cm and sample size of 5 will be

a.

1.066

b.

1.245

c.

1.289

d.

1.334

e.

1.500.

424. In case of p charts, if n = 48 and p = 0.23, the value of center line will be given by

a.

0.033

b.

0.115

c.

0.230

d.

0.400

e.

11.040.

425. The mean and standard deviation of a sampling distribution are 10.00 and 0.02 respectively. The Upper Control Limit (UCL) for the corresponding x chart will be

a. b. c. d. e.

9.94 10.00 10.02 10.04 10.06.

77


426. Which of the following charts are used to study the characteristics of attributes? a. p charts. x chart. b. c. R charts. d. Both (a) and (b) above. e. All of (a), (b) and (c) above. 427. Which of the following is plotted using sample range? a. p chart. x chart. b. c. R chart. d. Ogive. e. Frequency polygon. 428. Which of the following charts deal(s) with qualitative factors possessed by the data? x charts i. ii. p charts iii. R charts. a. Only (i) above. b. Only (ii) above. c. Only (iii) above. d. Both (i) and (ii) above. e. Both (ii) and (iii) above. 429. Which of the following control charts is/are used to monitor the variability in a process?

a. x chart. b. R chart. c. p chart . d. Scatter diagram. e. Both (b) and (c) above. 430. Which of the following is not true about the statistical process control? a. Quality standards in manufacturing process and the service industry can be achieved through the application of statistical methods. b. The variability present in the manufacturing process can be minimized to the extent possible. c. The variation in the specifications of the product due to wearing of the machine is a systematic or non-random variation. d. The variation arising out of measurement errors is a random variation. e. If the process is out of control, then random variation will be observed. 431. The quality manager of the bottling plant of a cola major wants to ensure that the quantity of soft drinks in the bottles is as printed on the label of the bottle. He takes the samples of ten bottles every day and plots a control chart for the mean quantity of beverage in the bottle. Towards the end of the week he observes that the observations are within the control limits but there are jumps in the level around which the observations vary. Which of the following inferences is most appropriate? a. The variations have been reduced to a great extent. b. The means of two different populations are being observed. c. The process mean has been shifting. d. The process is out of control. e. None of the above.

78

Part I

432. In a control chart there are some observations that lie outside the upper and lower control limits. These points are referred to as the

a.

Outliers

b.

Attribute

c.

Inherent variation

d.

Assignable variation

e.

Common variation.

433. Which of the following patterns of x chart indicates that the variations have been reduced significantly?

a.

Hugging the control limits.

b.

Hugging the center line.

c.

Cycles.

d.

Decreasing trend.

e.

Increasing trend.

434. Which of the following is true with regard to the p chart in quality control?

a.

The center line is drawn on the basis of overall sample mean.

b.

The lower control limit can only be greater than or equal to zero.

c.

The upper control limit is equal to the overall sample proportion.

d.

The lower control limit is equal to the overall sample proportion.

e.

The upper and the lower control limits are based on the overall sample mean.

Chi-Square Test and Analysis of Variance 435. Which of the following is/are true?

a.

In an analysis of variance, two estimates of the population variance are compared to test the equality of two means.

b.

When the populations have same variance, the between-column variance tends to be smaller than the within-column variance.

c.

The area under any F distribution curve is 1.

d.

For different degrees of freedom in the numerator and the denominator, the skewness of the F distribution shifts to the right.

e.


436. Which of the following is/are false?

a.

The specific shape of an F distribution depends on the number of degrees of freedom of the numerator and the denominator.

b.

The value of F statistic can be lower than 1.

c.

Chi-square distribution is used to test the hypothesis about proportions, variances and as a test of distributional goodness of fit.

d.


e.


79


437. Consider the data given below.

Attitude Towards Social Legislation Occupation Support Neutral Oppose Blue collar worker 25 21 30 White collar worker 26 30 28 Professionals 29 25 30 The expected frequency for the cell (white collar, neutral) is a.

26.16

b.

28.54

c.

30.21

d.

31.24

e.

32.64.

438. Which of the following is/are true?

a.

While calculating between-column variance, it is necessary that the samples should have the same size.

b.

The number of degrees of freedom in chi-square distribution is dependent upon number of rows and columns.

c.

Expected frequency of a cell in a contingency table can be found if the totals of row and the column containing that particular cell are known.

d.

Chi-square distribution is a probability distribution like Normal or t-distribution.

e.

Both (b) and (d) above.

439. Consider the data given below.

i.

Between-column variance = 6, within-column variance = 5

ii.


iii.


iv.


v.

Between-column variance = 11, within-column variance = 9.

This data pertains to different groups of samples and both the estimates of population variance have equal number of degrees of freedom. Given this data, for which sample the null hypothesis of equal means is more likely to be accepted? a.

Only (i).

b.

Only (ii).

c.

Only (iii).

d.

Only (iv).

e.

Only (v).

440. To find the value of F (n, d, 0.95), we consider

a.

The value of F (n, d, 0.05)

b.

The inverse value of F (n, d, 0.95)

c.

The inverse value of F (n, d, 0.05)

d.

Both (a) and (c) above

e.

None of the above.

80

Part I

441. A contingency table prepared for chi-square test contains two rows and two columns. The degrees of freedom in this case will be

a.

0

b.

1

c.

2

d.

3

e.

4.

442. The chi-square statistic is given by

a.

(f − f )2 ∑ o e fo

b.

∑

c.

(f − f )2 ∑ o e fe

d.

[∑ (f o − f e )]2 fe

e.

[∑ (f o − f e )]2 . fo

(f o − f e )1/2 fe

443. The analysis of variance allows us to test whether the difference(s) among __________ sample means is/are significant or not.

a.

More than one

b.

More than two

c.

Less than three

d.

Less than four

e.

Any number of.

444. Consider the following data

fo

5

2

fe

3

4

The ψ 2 statistic in the above case is a.

3 7

b.

6 7

c.

7 6

d.

7 3

e.

7.

81


445. Consider the following contingency table.

Annual Sales of Product Q State of Economy

High

Medium

Low

Boom

245

200

175

Moderate

195

170

155

Recession 165 140 The number of degrees of freedom in the above data is

125

a.

3

b.

4

c.

6

d.

8

e.

9.

446. Consider the following data regarding the sales projections of a new product.

Rs. crore Sales

State of Economy Low Promotion 29

Boom Moderate 25 Growth Slow 22 Growth Recession 15 The degrees of freedom in the above data is a.

3

b.

4

c.

5

d.

6

e.

None of the above.

Moderate Promotion 33

High Promotion 48

28

35

26

29

20

22

447. For conducting a chi-square test for testing a hypothesis the data have been collected and tabulated in 5 rows and 4 columns. The relevant degree of freedom is

a.

19

b.

18

c.

16

d.

15

e.

12.

448. Which of the following is/are false?

a.

Chi-square value is negative if the expected value is greater than the actual value.

b.

The mean calculated when all the individual observations are taken into account is called grand mean.

c.

Chi-square distribution consists of a family of distribution.

d.

F distribution is unimodal in nature.

e.

None of the above.

82

Part I

449. Which of the following is false about Chi-square distribution?

a.

There will be a different chi square distribution for each degrees of freedom.

b.

If the number of degrees of freedom is small, the curve will be skewed to the right.

c.

As the number of degrees of freedom increases the curve tends to become symmetrical.

d.

The area under the probability distribution curve increases as the number of degrees of freedom increase.

e.

None of the above.

450. The psychology department of an institute has developed an IQ test. It has compiled the result of the test conducted on a number of students and faculty members of the institute. After analyzing the test results they have arrived at this conclusion that the distribution of results can be said to follow a binomial distribution with the probability of scoring above average IQ level to be 60%. They want to test whether this particular result really follows a binomial distribution. Which of the following statistical techniques should be used to verify these findings?

a.

Analysis of variance.

b.


c.


d.

Chi-square test.

e.

Marginal analysis.

451. When the chi-square test is used as a test of independence, the number of degrees of freedom is determined by

a.

The sample size

b.

The ratio of sample size and the population

c.

Number of rows in the contingency table, only

d.

Number of columns in the contingency table, only

e.

Both the number of rows and the number of columns in the contingency table.

452. Which of the following techniques is suitable for testing whether there is a significant difference between more than two sample means?

a.

Chi-square test.

b.


c.

Simple regression analysis.

d.


e.


453. The value of chi-square statistic cannot be negative, because

a.

The observed frequencies are always greater than the expected frequencies

b.

The differences between the observed and expected frequencies are squared

c.

The chi-square statistic is based upon the absolute value of the differences between the observed frequencies and expected frequencies

d.

The chi-square statistic is the summation of squares of observed frequencies

e.

The chi-square statistic is the summation of squares of expected frequencies.

454. In a test involving ANOVA the F statistic is calculated on the basis of a. The mean of the largest sample b. The mean of the smallest sample c. The standard deviation of the smallest sample d. The estimated population variance based on the variance among the sample means and the estimated population variance based on the variance within the samples e. The variance of the largest sample.

83


Simulation Based on the following information answer questions 455 and 456.

The following table given the profitability distribution of the daily demand of a particular item. Demand (’00 units) Probability

2

3

5

30%

60%

10%

The opening inventory was 1000 units and the daily demand is to be simulated. 455. In one run, the random number generated was 45. The closing inventory will be

a.

1000

b.

800

c.

700

d.

500

e.

0.

456. If instead 90 was generated the closing inventory will be

a.

1000

b.

800

c.

700

d.

500

e.

0.

457. Simulation is used under

a.

Conditions of certainty

b.

Conditions of uncertainty

c.

Conditions of risk

d.

Conditions of known probabilities

e.

Decision tree approach.

458. Mathematical modeling requires the setting up of mathematical relationships which would represent the

a.

Problem

b.

System

c.

Criterion function

d.

Linear programming problem

e.

Decision variables.

459. Consider the following probability distribution of a variable X

X

5.0 10.00 15.00 20.0 25.0 30.0 35.0

p

0.1

0.05 0.05

0.1

0.3

0.2

0.2

In a Monte Carlo simulation process the number 80 was generated in a particular run from a uniform distribution of 0 – 99. The value of X in this run will be a.

5

b.

10

c.

25

d.

30

e.

35.

84

Part I: Answers on Basic Concepts (with Explanatory Notes) Probability Distribution and Decision Theory 1.

(b) P =

n! (p)r (q) n − r r!(n − r)!

= 4!/ [3! (4 – 3)!] (0.5)3 (0.5)4-3 = 0.25 or 1/4. 2.

(a) μ = np = 12(0.45) = 5.40; σ2 = npq = 12(0.45) (0.55) = 2.97.

3.

(a) The area under the standard normal curve between the points 0 and 2.9 is 0.4981 and thus, the area under the standard normal curve between the points –2.9 to 0 is also equal to 0.4981 by symmetricity. Hence, the area between the points –2.9 and 2.9 is equal to 0.4981 + 0.4981 = 0.9962.

4.

(a) 72 + 1.28(9) = 72 + 11.52 = 83.52 [Hint: see Z-value for 0.4000 (i.e. 0.5000 – 0.1000) from area tables of normal distribution].

5.

(d) 6!/[(6 – 4)! 4!] (0.5)4 (0.5)6-2 + 6!/[(6 – 5)! 5!] (0.5)5 (0.5)6-5 + 6!/[(6 – 6)! 6!] (0.5)6 (0.5)0 = 0.34375.

6.

(b) When the decision maker possesses information about the probabilities of possible states of nature, his/her decisions are said to be made under the conditions of risk.

7.

(d) Strategy Products

Low P(L) = (0.2)

Medium P(M) = (0.4)

High P(H) = (0.4)

A

2.0

6.0

10.0

B

5.0

12.0

15.0

C

1.0

3.0

6.0

D

3.0

8.0

16.0

According to Maximax Criterion, the decision maker identifies the maximum profit associated with the selection of each alternative and the maximum of these is chosen as the decision. In this case, strategy high is associated with the maximum profit and among products A, B, C and D, product D has the maximum profit. 8.

(b) Strategy Low Medium High Products P(L) = (0.2) P(M) = (0.4) P(H) = (0.4) A 2.0 6.0 10.0 B 5.0 12.0 15.0 C 1.0 3.0 6.0 D 3.0 8.0 16.0 According to Maximin Criterion, the decision maker first identifies the lowest profit associated with each decision alternative and he chooses that alternative which is the maximum of the above minimum profits. Thus, product B will be selected if maximin criterion is used.

9.

(b)

Regret Table Strategy

Low

Medium

High

Products A B C D

P(L) = (0.2) 5.0 - 2.0 5.0 - 5.0 5.0 - 1.0 5.0 - 3.0

P(M) = (0.4) 12.0 - 6.0 12.0 - 12.0 12.0 - 3.0 12.0 - 8.0

P(H) = (0.4) 16.0 - 10.0 16.0 - 15.0 16.0 - 6.0 16.0 - 16.0


Strategy Low Medium High Max. Regret Products P(L) = (0.2) P(M) = (0.4) P(H) = (0.4) A 3.0 6.0 6.0 6.0 B 0.0 0.0 1.0 1.0 C 4.0 9.0 10.0 10.0 D 2.0 4.0 0.0 4.0 Under this criterion, the decision maker strives to minimize the largest regret and hence selects product B. 10. (b) Under Hurwicz Criterion, weighted profits are calculated as [α (maximum profit for alternative) + (1 – α) (minimum profit for alternative)]. Strategy Weighted Gains Amount (in rupees) A 0.5 (10) + 0.5 (2) 6.0 B 0.5 (15) + 0.5 (5) 10.0 C 0.5 (6) + 0.5 (1) 3.5 D 0.5 (16) + 0.5 (3) 9.5 11. (b) Expected Profits: Product A = 0.2(2) + 0.4(6) + 0.4(10) = 6.8 Expected Profits: Product B = 0.2(5) + 0.4(12) + 0.4(15) = 11.8 Expected Profits: Product C = 0.2(1) + 0.4(3) + 0.4(6) = 3.8 Expected Profits: Product D = 0.2(3) + 0.4(8) + 0.4(16) = 10.2 As the expected profit is highest in case of B, the product B should be released in the market. 12. (a) The expected value of perfect information is the difference between the expected profit under certainty (sum of the products of maximum demand in each condition and its probability) and the best expected monthly profit without any predictions of the future (Expected profit of the product to be released in the market) as calculated before. The expected value under perfect information = 5(0.2) + 12(0.4) + 16(0.4) – 11.8 (refer Q.No.11) = 12.2 – 11.8 = 0.4. 13. (a) Index of pessimism is (1 – α) and it lies between 0.1 and 0.3. 14. (d) P(F/STO) = P(F and STO)/P(STO). P(F and STO) = P(F) . P(STO/F) = 15/45 . 4/15 P(STO) = 8/45 Thus, P(F/STO) = 15/45 x 4/15 x 45/8 = 1/2. 15. (c) P(G/SS) = P(G and SS)/P(SS) P(G and SS) = P(G).P(SS/G) = 15/45 . 3/15 P(SS) = 10/45 Thus, P(G/SS) = 15/45 x 3/15 x 45/10 = 3/10. 16. (c) P(T/N) = P(T and N)/P(N) P(T and N) = P(T) . P(N/T) = 15/45 . 3/15 P(N) = 11/45 Thus, P(T/N) = 15/45 x 3/15 x 45/11 = 3/11. 17. (b) Number of employees in Finance department strongly opposes the policy changes (4) divided by total number of employees in Finance department (15). Thus, P(F and STO) = 4/15. 18. (b) Number of employees in Garments department strongly supports the policy changes (3) divided by total number of employees in Garments department (15). Thus, P(G and SS) = 3/15 = 1/5. 19. (b) Let n = Number of employees is textile division that remains neutral to the proposed policy changes. S= Total number of employers in textile division. n 3 1 P= = = . s 15 5 86

Part I

20. (a) Number of employees strongly supports the policy (6 + 3 + 1 = 10) divided by total number of employees (15 + 15 + 15 = 45). Thus, P(SS) = 10/45 = 2/9. 21. (c) P(MMF) = 0.48 x 0.48 x 0.52 = 0.1198. (Stopped after having 3 children, means that first male, second male and third female.) 22. (a) When the couple had a female child, they will immediately stop having children. This implies that all the children except the last one (which is a female) of the couple are male. Thus, the probability of having a male child is equal to (0.48)k–1 where k is the number of expected children and the probability of having a female child is 0.52. Thus, the probability distribution of X is equal to (0.48)k–1 (0.52), where X denotes the number of children that the couple would have. 23. (d) When decisions are made under conditions of risk all possible states of nature are known to the decision maker and the decision-maker has sufficient knowledge to assign probability of occurrence to the various states of nature. 24. (a) Probability of making a good strike in not more than three attempts = P(making a good strike in first attempt) + P(making a good strike in second attempt) + P(making a good strike in third attempt) = 0.75 + 0.25 x 0.75 + 0.25 x 0.25 x 0.75 = 0.984375. [P(dry) = 0.25 and P(good) = 1 – 0.25 = 0.75]. 25. (a) P(D > 376) = 0.5 – [(376 – 375) / 400 ] = 0.5 – 0.05 = 0.45 and P(D < 376) = 1 – 0.45 = 0.55. Thus, the Expected Marginal Profit (EMP) when the purchase is equal to 376 = 0.45 x (1.50 – 0.67) = 0.37 or 0.40 approximately and the expected marginal loss EMI = 0.55(0.67 – 0.50) = 0.09 approximately. 26. (c) P* = ML/(ML + MP) = 0.17/(0.17 + 0.83) = 0.17. If we see the z table for the corresponding value for probability 0.5 – 0.17 = 0.33, the value of z = 0.96. If we multiply 0.96 with the standard deviation, 400 = 20, we get 19.2. Thus, the vendor should buy 375 + 19.2 = 394.2 newspapers to maximize profits. 27. (b) If a value is chosen at random from a particular population, it has equal chances of being smaller than and larger than the median of the population. Hence, the probability that a value chosen at random from a population is larger than the medium of the population is 0.5. 28. (b) Security A will give the maximum return when the remaining 0.5% probability is attached to the rate of return of 20%. Then the expected return = 0.2 x 7 + 0.1 x 10 + 0.0 x 12 + 0.20 x 15 + 0.0 x 18 + 0.5 x 20 = 15.4%. 29. (c) The probability that the price of the shares of Company A would rise and that of Company B would fall is equal to (0.4) (1 – 0.1) = 0.36. The probability that the price of the shares of Company A falls and that of Company B rises is equal to (1 – 0.4) (0.1) = 0.6. Thus, the probability that in the next week only one company’s share price would increase is 0.36 + 0.06 = 0.42. 30. (c) Events Expected Value B1 B2 B3 Action 1 0.2 (75) 0.5 (40) 0.3 (60) 53 Action 2 0.2 (50) 0.5 (40) 0.3 (70) 51 Action 3 0.2 (60) 0.5 (80) 0.3 (50) 67 31. (a) The expected pay-off under perfect information = 75(0.2) + 80(0.5) + 70(0.3) = 76. 32. (c) z = (x – μ)/σ – 1.5 = (x − 20)/ 16 – 6 = x – 20 x = 20 – 6 = 14. 33. (c) z = (x – μ)/σ – 2.5 = (38 − μ)/ 16 μ

= 48. (Since Mean = Median = Mode = μ ). 87


34. (e) Conditional profit table shows the profit at different levels of demand, but it does not show expected profit or loss due to occurrence of that event, which is required to determine the profit maximization stock level. 35. (a) In a conditional-loss table with possible sales levels listed vertically in the first column and possible stock actions listed horizontally in the first row. Any value in a column below a zero indicates an opportunity loss. 36. (c) A businessman who is averse to risk, avoids all situations but those with very high expected values (expected value = probability x amount) as the events have high probability of occurrence. In other words, the events that have high expected values will have minimum risk. 37. (d)Mere construction of decision tree does not help the analyst to take an immediate decision. The analyst, for the construction of decision tree, should specify underlying assumptions to point out, among other things, the limits under which the decision tree is valid and should also perform sensitivity analysis to determine how the solution reacts to changes in the inputs. 38. (c) z = (x – μ)/σ z = (95 – 80)/9 = 1.67; the corresponding value for z-score 1.67 is equal to 0.4525 (see z table). Since this is a right tailed test, in addition to the above area we have to consider the total area of left hand side, which is equal to 0.5. Thus, P (0 < z < 95) = 0.4525 + 0.5 = 0.9525. 39. (a) z = (x1 – μ)/σ, where x1 = 57.32 + 3 = 60.32 z = (60.32 – 56)/2.4 = 1.8 The P (56 < x < 60.32) = 0.4641 (see the corresponding value for z-score, 1.8) z = (x2 – μ)/σ, where x2 = 57.32 – 3 = 54.32 z = (54.32 – 56)/2.4 = – 0.7 The P (54.32 < x < 56) = 0.2580 (see the corresponding value for z-score, 0.7). Thus, the probability of a steel rod fitting in the groove is equal to 0.4641 + 0.2580 = 0.7221. 40. (e) A particular unit would be included in the stock, if the probability of stocking that unit is greater than P*. Here according to option (a), P(requests for 5 or more than 5 units) = 0.50, which is greater than P*. Hence, we can include the fifth unit in the stock. According to option (b), P(requests for fewer than 3 units) = 0.10 and P(requests for exactly three units) = 0.09. Therefore, P(requests for 3 or more units) = 1 – (0.09 + 0.10) = 0.81, which is greater than P*. Hence, we can include the third unit in the stock. P

According to option (c), P(request for 9 or more units) = 0.05 + 0.16 = 0.21, which is less than P*, so we cannot include the ninth unit in the stock. Hence (e) is the required answer. 41. (b) If P(0 < Z < 2.9) is equal to 0.4981, then P(–2.9 < Z < 0) would be equal to 0.4981. Thus, P(–2.9 < Z < 2.9) is equal to 0.4981 + 0.4981 or 0.9962. 42. (c) Expected Gain/Loss = 200(0.2) + 300(0.3) + (–250)(0.5) = 130 – 125 = Rs.5. As the expected gains are positive the person should undertake the investment. 43. (c) The normal distribution reflects the various values taken by many real life variables like the heights and weights of people or the marks of students in a large class. In all these cases a large number of observations are found to be clustered around the mean value μ and their frequency drops sharply as we move away from the mean in either direction. Stock return on BSE-Sensex in the long run is almost like a normal distribution, but in short-run in may not be normal distribution. 44. (b) Conditional profit table is prepared in decision-making under the conditions of risk. 45. (e) Expected value = [1(0.3) + 2(0.3) + 3(0.4)] = 0.3 + 0.6 + 1.2 = 2.1. 46. (b) Regret criterion is used only under the conditions of uncertainty. 47. (a) The Standard Normal Distribution is a normal distribution with a mean μ = 0 and a standard deviation σ = 1. The observation values in a standard normal distribution are denoted by the letter Z. 88

Part I

48. (a) P* = ML/(ML + MP) = 2000/(10000 + 2000) = 2000/12000 = 1/6 = 0.1667. 49. (b) The expected value criterion to evaluate the decisions under the conditions of risk. But, as long as uncertainty exists, there is the possibility that the expected value criterion may lead to the wrong course of action. Thus, the decision maker may prefer to calculate Expected Value of Perfect Information (EVPI) under the conditions of risk. The expected value of perfect information is the difference between the expected profit under certainty and the bestexpected profit without any predictions of the future. 50. (a) For a decision tree analysis, the time always flows from left to right. 51. (e) The characteristics of normal probability distribution with reference to the figure are •

The curve has a single peak; thus it is unimodal.

•

The mean of a normally distributed population lies at the center of its normal curve.

•

Because of the symmetry of the normal probability distribution, the median and the mode of the distribution are also at the center.

•

The two tails of the normal probability distribution extend indefinitely and never touch the horizontal axis.

52. (b) The expected value of X is equal to [15(0.3) + 20(0.3) + 25(0.2) + 30(0.2)] = 4.5 + 6 + 5 + 6 = 21.5. 53. (d) The expected value of the investment = [0.4 x 10 + 0.2 x 15 + 0.2 x 20 + 0.2 x 25] = 16. Thus, the probability that the return from the investment will be less than its expected value is equal to 0.4 (i.e. Return 10%) + 0.2 (i.e. Return 15%) = 0.6. 54. (c) With decision-making under uncertainty, the decision maker is aware of different possible states of nature, but has insufficient information to assign any probabilities of occurrence to them. 55. (e) Decision tree is considered as a mathematical model of the decision situation and can be used for whether investment, acquisition of property, Profit management or new product strategies. 56. (a) If the decision maker is a pessimist, he would first identify the lowest profit associated with each decision alternative and then will choose that alternative which is the maximum of that minimum profit. This criteria is called Maximin Criterion. 57. (d) z = (x1 – μ)/σ, where x1 = 4.2 z = (4.2 – 4.6)/0.2 = 2 P (4.2 < x < 4.6) = 0.4772 (see the corresponding value for z-score, 2) z = (x2 – μ)/σ, where x2 = 5.0 z = (5.0 – 4.6)/0.2 = 2 The P (4.6 < x < 5.0) = 0.4772 (see the corresponding value for z-score, 2) Thus, P (4.2 < x < 5.0) is equal to 0.4772 + 0.4772 = 0.9544 or 95.5% (approximately). 58. (c) Discrete Probability Distribution is a probability distribution in which the variable is allowed to take on only a limited number of values. While Poision, Binomial are Discrete Probability distribution. Normal Probability is continuous probability distribution. While Chi-Square is non-parametric test. 59. (d) The expected pay-off under perfect information = 12(0.2) + 15(0.8) = 14.4. (The sum products of probability and the maximum profit in each column). 60. (c) If the random variable can assume only a limited number of values, it is called a discrete random variable. The number of patients seen by a doctor as a function of time spent in clinic are examples of discrete variable as these values could only be whole numbers. You cannot have 18.7 patients visiting a doctor. 61. (d) A probability distribution provides an idea of the likely values of a random variable. 62. (b) A probability distribution must sum to 1. 63. (d)The maximax criterion of decision-making is applicable when the conditions are uncertain and the decision maker is perfectly optimistic. It is not applicable when the conditions are certain, or the decision maker is perfectly pessimistic, or the decision maker cannot be strictly classified to be perfectly optimistic or perfectly pessimistic. 89


64. (b) A measure of the variability of one variable (or data set) in relation to another variable (or data set), is the covariance. Hence for two independent variables, the covariance would be zero. 65. (a) Regret is a measure of magnitude of loss/cost (opportunity loss or cost) incurred by not selecting the best alternative. The regret we realize is measured by the difference between the maximum profits we would have realized had we known the state of nature and the profit we realize. 66. (a) Expected profit in a decision problem under risk is not necessarily one of the possible outcomes. The expected daily profits for each stock decision can be determined by weighing each conditional profit by its likelihood of occurrence (which is the probability of the corresponding level of demand) and then conditional table or probability tree can draw for calculating optimum profit. 67. (d) P* = ML/(ML + MP), where P*represents the minimum required probability of selling at least one additional unit to justify the stocking of that additional unit. P* = 20/(10 + 20) = 0.667. 68. (e) Poisson’s distribution and ψ2 distribution are non symmetrical in nature, But Binomial distribution and Normal distributions are symmetrical. 69. (e) If the random variable can assume only a limited number of values, it is called a discrete random variable. The number of patients seen by a doctor as a function of time spent in clinic are examples of discrete variables as these values could only be whole numbers. You cannot have 18.7 patients visiting a doctor. 70. (b) Bernoulli process can be used only when the following conditions are fulfilled. i. Each trial has only two possible outcomes. ii. The probability of the outcome of any trial remains fixed over time. iii. The trials are statistically independent. 71. (d) See the corresponding value for z-score +2.33 in z table. 72. (e) The ‘t’ distribution is a theoretical probability distribution. The ‘t’ distribution is symmetrical, bell-shaped, and to some extent similar to the standard normal curve. So for t-distribution μ = 0, and variance is defined as σ2 = df/(df – 2). 73. (c)In decision-making under conditions of uncertainity, the decision-maker is aware of different possible states of nature, but has insufficient information to assign any probabilities of occurrence to them. 74. (c) Decision tree is considered as a mathematical model of the decision situation. All decision trees contain decision points or nodes and state of nature points or nodes. At every decision node the decision-maker chooses an alternative which gives him the highest profitability. 75. (b) P* = ML/(ML + MP), where P* represents the minimum required probability of selling at least one additional unit to justify the stocking of that additional unit. P* = 200/(200 + 300) = 2/5 = 40%. 76. (d) The standard normal probability distribution is a normal probability distribution with mean μ = 0 and standard deviation σ = 1. 77. (c) The expected return for the security = [10(0.3) + 15(0.2) + 20(0.5)] = 3 + 3 + 10 = 16%. 78. (d) P* = ML/(ML + MP), where P* represents minimum required probability of selling at least one additional unit to justify the stocking of that additional unit. 0.25 = 50/(50 + MP); MP = 200 – 50 = Rs.150. 79. (d) Probability of r successes in n trials = n!/[r!(n – r)!] (p)r (q)n-r = 8!/[5! (8 – 5)!] (0.6)5 (0.4)8-5 = (56) (0.005) = 0.28 or 28%. 80. (b) The expected value criterion is used under the conditions of risk. P

90

P

Part I

81. (b) A random variable which can assume any value including fractions within a given range is called a continuous random variable. 82. (a) If σ is the standard deviation of the normal distribution, 80% of the observation will be in the interval μ –1.28σ to μ + 1.28σ.95% of the observations will be in the interval μ – 1.96σ to μ + 1.96σ.98% of the observations will lie in the interval μ – 2.33 σ to μ + 2.33 σ. 83. (d) The expected gain from the investment = [50(0.7) + 30(0.2) + 20(0.1)] = 35 + 6 + 2 = 43%. 84. (b) Probability of r successes in n trials = n!/[r!(n – r)!] (p)r (q)n–r = 4!/[3! (4 – 3)!] (0.5)3 (0.5)4–3 = 0.25 or 1/4. 85. (c) Regret criteria is based upon the regret one might have from making a particular decision. The regret is referred to as the opportunity loss or opportunity cost – a measure of magnitude of loss incurred by not selecting the best alternative. The regret is measured by the difference between the maximum profit that we would have realized in case of known state of nature and the profit we realize. This criterion suggests that the decision maker should strive to minimize the largest regret. 86. (d) A discrete random variable can assume only a countable number of values. The number of persons traveling from Delhi to Mumbai everyday and the number of patients seen by a doctor each day are examples of discrete random variables as these values could only be whole numbers. You cannot have 353.5 persons traveling or 18.7 patients visiting the doctor. 87. (e) If σ is the standard deviation of the normal distribution, 80% of the observation will be in the interval μ –1.28σ to μ + 1.28σ.95% of the observations will be in the interval μ – 1.96σ to μ + 1.96σ.98% of the observations will lie in the interval μ – 2.33 σ to μ + 2.33 σ. 88. (d) Bernoulli Process can be sum up as follows: i. Each trial has only two possible outcomes. ii. The probability of the outcome of any trial remains fixed over time. In our example, the probability of the bulb being defective or non-defective remains fixed throughout. iii. The trials are statistically independent. In our example, the outcome of the bulb being defective or non-defective does not affect the outcome of any other bulb being so. 89. (b) (12/15) x 1 = 0.8. 90. (a) With decision-making under uncertainty, the decision maker is aware of different possible states of nature, but has insufficient information to assign any probabilities of occurrence to them. Variety of criteria have been proposed, each based upon certain attitude of the decision maker. Some of the important criteria are i. Maximax Criterion ii. Maximin Criterion iii. Hurwicz Criterion iv. Regret Criterion. 91. (c) Under Maximin Criterion, the decision maker first identifies the lowest profit associated with each decision alternative and then chooses that alternative which is the maximum of the above minimum profits. 92. (e) Probability of r successes in n trials = n!/[r!(n – r)!] (p)r (q)n–r = 3!/[2! (3 – 2)!] (0.5)2 (0.5)3–2 = 0.375.

91


93. (e)In a binomial distribution the probability of getting zero or more number of successes = P(x = 0) + P(x = 1) + … P(x = n) = 1.00. 94. (e) All the given statements that are made with respect to normal distribution are correct. 95. (b) The Standard Normal Distribution is a normal distribution with a mean μ = 0 and a standard deviation σ z = 1. The observation value in a standard normal distribution is denoted by the letter Z and is the z-values and the observed values are same. 96. (d) z = (x1 – μ)/σ, where x1 = 5.0 z = (5.0 – 5.4)/0.2 = 2 The P (5.0 < x < 5.4) = 0.4772 (see the corresponding value for z-score, 2) z = (x2 – μ)/σ, where x2 = 5.8 z = (5.8 – 5.4) / 0.2 = 2 The P(5.4 < x < 5.8) = 0.4772 (see the corresponding value for z-score, 2). Thus, the probability that the observations will fall between 5.0 and 5.8 is 0.4772 + 0.4772 = 0.9544 or 95.4% (approximately). 97. (b) Bernoulli process can be used only when the following conditions are fulfilled. i.

Each trial has only two possible outcomes.

ii.


iii.

The trials are statistically independent.

Thus, if the probability of the outcome of any trial varies over time, Bernoulli process cannot be used. 98. (d) With decision-making under conditions of risk all possible states of nature are known and the decision maker has sufficient knowledge to assign probabilities to their likelihoods of occurrence. These probabilities may range from subjective assignments based upon the decision maker’s feelings and experience to objective assignments based on the collection and analysis of numerous data related to the states of nature. 99. (c) P* = ML/(ML + MP), where P* represents the minimum required probability of selling at least one additional unit to justify the stocking of that additional unit. P

0.6 = 15/(15 + MP) MP = 25 – 15 = Rs.10. 100. (c) If σ is the standard deviation of the normal distribution, 80% of the observation will be in the interval μ –1.28σ to μ + 1.28σ.95% of the observations will be in the interval μ – 1.96σ to μ + 1.96σ.98% of the observations will lie in the interval μ – 2.33 σ to μ + 2.33 σ. 101. (c) (a) This is a true statement. A discrete random variable can assume only a limited number of values. If the number of students attending a particular class ranges from 50 to 60. It is an example of discrete random variable, as this variable can assume numerical values which are positive integers lying between 50 and 60. It cannot assume any fractional value. (b) This is a true statement. A continuous random variable can assume any value within a given range. The maximum temperature of a city for a given day is an example of continuous random variable as it can assume any value between a given range i.e., it can have values like 29.35 oC, 29.48 oC, 29.57 oC etc. between 29 oC to 30 oC. (c) This is not a true statement for random variables. A random variable is defined as a value or magnitude that changes from occurrence to occurrence in no predictable sequence. (d) This is a true statement for random variables. A random variable is a variable, which assumes different numerical values as a result of random experiments. (e) This is a true statement for random variables. The value of the random variable changes from occurrence to occurrence in no predictable sequence. 92

Part I

102. (d) (a)

This statement is true for probability distribution. Usually, the past behavior of the variable is studied and the frequency distribution of the data is formed. If the past behavior can be taken as a representative pattern for the future also, then the past frequency distribution can be used as a guide for predicting the future values of the random variable.

(b) This is a true statement for probability distribution. The probability distribution gives an idea of the likely values of a random variable and the chances of occurrence of the various values. (c)

This is a true statement for probability distribution. If the past behavior does not help in forming a probability distribution then, a subjective probability distribution of the likely future values is formed on the basis of expert opinion.

(d) This is not true for probability distribution. We can assign probabilities to all likely values of the random variable on the basis of the past data. (e)

This is a true statement for probability distribution. The probability distribution for a random variable provides a probability for each possible value and that these probabilities must sum to 1.

103. (e) (a)

This is true for expected value of the random variable. The concept of expected value of the random variable is used for making decisions under conditions of uncertainty.

(b) This is true for expected value of the random variable. The expected value is the mean of the probability distribution. (c)

This is true for expected value of the random variable. The mean is computed as the weighted average of the values that the random variable can assume.

(d) This is true for expected value of the random variable. The probabilities assigned to each random variables are used as weights. Thus, expected value of the random variable is computed by summing up the random variables multiplied by their respective probabilities of occurrence. E [X] =∑ X P (X). (e)

This is not true for expected value of the random variable. The expected value of the random variable does not predict the value that the random variable will assume in future. Although, there are many different possible values that the random variable can take, the expected value is a single figure. We say that the random variable is most likely to assume this value, it does not predict the value as such.

104. (e) (a)

This is a wrong answer. If the covariance of two random variables X and Y is equal to zero then it can be said that the random variables are independent of each other. If the variables are dependent on each other then the covariance will assume any non-zero value.

(b) This is a wrong answer. If the coefficient of correlation is positive then we say that there is a positive correlation between two variables. (c)

This is a wrong answer. If the coefficient of correlation is negative then we say that there is a negative correlation between two variables.

(d) This is the wrong answer. If the covariance of two random variables X and Y is equal to zero then it is not necessary mean that the random variables are independent of each other. (e)

This is a correct answer. If X and Y are independent, then covariance is zero but is covariance is zero, then the variables may be or may not be independent.

93


105. (b) (a)

This is a wrong answer. If the balls are replaced after inspection then it will follow the Bernoulli process and can be described by the binomial distribution.

(b) This is the right answer. If the Bernoulli experiment is repeated without replacement, it follows hypergeometric distribution. Therefore the random variables in the given case are best described by hypergeometric distribution. (c)

This is the wrong answer. Any variable studied for a large population is assumed to follow a normal distribution. If we draw samples of size more than 30 from any population the sample means are approximately normally distributed. But in our case the distribution of the random variable can be best described by hypergeometric distribution.

(d) This is a wrong answer. If ln (X) is a normally distributed random variable, the X is said to be a lognormal variable. The random variables in given case are not described by lognormal distribution. (e)

This is a wrong answer. t-distribution is used for testing of hypothesis when the sample size is 30 or less than 30 and the population standard deviation is not known. The random variables in the given case are not described well with the help of t-distribution.

106. (d) i.

In a binomial distribution the outcomes are independent of each other.

ii.

In a binomial distribution each outcome can be classified as a success or failure.

iii.

In a binomial distribution the probability of success is constant from trial to trial.

iv.

In a binomial distribution the random variable of interest is continuous

v.

In a binomial distribution the probability of failure can be determined from the probability of success.

107. (c) i.

The expected value of a discrete random variable is not a geometric average of the outcomes of the variable.

ii.

The expected value of a discrete random variable is not a simple average of the outcomes of the variable.

iii.

The expected value of a discrete random variable is a weighted average of the outcomes of the variable.

iv.

The expected value of a discrete random variable is not the outcome, which has the highest frequency.

v.

The expected value of a discrete random variable is not the highest probability of occurrence in the distribution of the random variable

108. (e)

94

i.

The data may follow a binomial distribution if the probability of success is constant from trial to trial.

ii.

The data may follow a uniform distribution if the probabilities of the outcomes of the trial are equal.

iii.

The data may follow a normal distribution if the probability of success is constant from trial to trial.

iv.

The data may follow a continuous distribution if the outcome is any value within a given range of values.

v.

When sampling is done without replacement from an finite population such that the probability of success is not constant from trial to trial, the data follow a hypergeometric distribution.

Part I

109. (b) i. The pay-offs represent the outcomes of certain decisions. ii. The states of nature refer to the chance occurrences that can affect the outcome of an individual’s decision. iii. The decision alternative courses of action that are available to the decision-maker. iv. The probabilities refer to the likelihood of the chance occurrences or the states of nature. 110. (d) i. The sampling distribution of mean is not a distribution of means of individual populations. ii. The sampling distribution of mean is not a distribution of observations within a population. iii. The sampling distribution of mean is not a distribution of observations within a sample. iv. The sampling distribution of mean is a distribution of means of all possible samples of a specific size taken from a population. v. The sampling distribution of mean is not a distribution of means of samples of a specific size taken from different population. 111. (b) i. As the sample size increases the variation of the sample mean from the population mean becomes smaller. ii. From above we can see that (b) is incorrect. iii. It cannot be said with certainty that as the sample size increases the variance of the sample becomes less than the variance of the population. iv. It cannot be said with certainty that as the sample size increases the standard deviation of the sample becomes less than the standard deviation of the population. v. It cannot be said with certainty that as the sample size increases the standard deviation of the sample comes close to zero. 112. (c) (c) When estimating population mean from a sample taken from a normal population with unknown variance the t-distribution may be used with number of degrees of freedom equal to sample size – 1. (a), (b), (d) & (e) represent incorrect degrees of freedom for using the t-distribution. 113. (b) i. The sample mean is not an estimator of the population variance. ii. The sample mean is an unbiased estimator of the population mean. iii. The sample mean is not an estimator of the population proportion. iv. The sample mean is not an estimator of the standard error of mean. v. The sample mean is not an estimator of the standard error of proportion. 114. (a) i.

For decision-making under conditions of uncertainty the maximax criterion is applied if the decision maker is perfectly optimistic.

ii.

For decision-making under conditions of uncertainty the maximin criterion is applied if the decision maker is perfectly pessimistic.

iii.

For decision-making under conditions of uncertainty the Hurwicz criterion is applied if the decision maker cannot be classified as perfectly optimistic or perfectly pessimistic.

iv.

For decision-making under conditions of uncertainty the regret criterion is applied if the decision maker wants to take into account the opportunity cost his decisions.

v.

The expected value criterion is applied under conditions of risk.

95


Statistical Inferences 115. (d) In random sampling, every sample has an equal chance of being selected. We always know the chance, that any population element will be included in the sample. 116. (c) The central limit theorem permits us to use sample statistics to make inferences about population parameters. 117. (c) When the population is normally distributed, the sampling distribution of mean has a mean equal to the population mean. Symbolically, μ = mean of x = X = μ = E( x ). 118. (a) When the population is normally distributed, the sampling distribution has a standard deviation (standard error) equal to the population standard deviation divided by the square root of the sample size. Symbolically, σ = σ/ n = 8/ 25 = 8/5 = 1.6. 119. (a) z = (x − μ)/σ x , Where σ x =

σ n

=

8.5 100

= 0.85

z = (140.61 – 142.3)/0.85 = –1.99. (The corresponding value for a z-score of –1.99 = 0.4767.) Thus, P(140.61 < z < 142.3) = 0.4767. z = (141.75 – 142.3)/0.85 = –0.65. (The corresponding value for a z-score of –0.65 = 0.2422.) Thus, P(141.75 < z < 142.3) = 0.2422. Hence, P(140.61 < z < 141.75) = 0.4767 – 0.2422 = 0.2345. 120. (a) Z = (x − μ)/(σ/ n ), σ / n = 0.85.For x = x (sample mean) x=μ+z

σ

n P (x > 1 4 4 .6 ) = P (μ + z

σ

> 1 4 4 .6 ) n = P (z > 2 .7 0 5 ) = P (0 < z < ∞ ) − P (0 < z < 2 .7 0 5 ) = 0 .5 − 0 .4 6 5 = 0 .0 0 3 5

For x = 144.60, z = (144.60 – 142.3)/0.85 = 2.705. The z table gives us an area of 0.4965 corresponding to a z-value of 2.705. If we subtract 0.4965 from the total area on right hand side, i.e. 0.5, we get 0.0035 (i.e. 0.5 – 0.465) as the total probability that the sample mean will be more than 144.60. 121. (e) Population variance, S2 = [∑x2 – n x 2]/(n – 1) ∑x2 = 52884.9; x 2 = (81.1625)2 = 6587.35; nx 2 = 52698.811; [∑x2 – n x 2] = 52884.59 – 52698.811 = 185.778; S2 = [∑x2 – n x 2]/(n – 1) = 185.778/7 = 26.539; hence S = S 2 = 26.539 = 5.15. 122. (d) Rejecting (accepting) a null hypothesis, when it is true (false) is called Type I error (Type II error) and its probability (is also the significance level of the test) is symbolized α. 123. (a) Since the cost of rejecting a null hypothesis is more, decision-makers will set very low levels of significance in its testing. 124. (e) In the entire given situation the normal distribution is the appropriate distribution. 96

Part I

125. (a) In order to test whether the serum helps to cure the disease or not, we set a null hypothesis that there is no significant effect after using medicine and an alternative hypothesis that there is significant effect after using medicine. When there is no significant effect after using medicine, the proportion of people recovered from the disease in group A would be equal to the proportion of people recovered from the disease in group B, where medicine is not used. On the contrary, when there is significant effect after using medicine, the proportion of people recovered from the disease in group A would be more than the proportion of people recovered from the disease in group B, where medicine is not used. Thus, H0: p1 = p2; H1: p1 > p2. 126. (a) σ p1 − p 2 =

(pq/n1 ) + (pq/n 2 ) ,

Where p = (n1 p1 + p2 n2)/(n1 + n2) = [100(0.75) + 100(0.65)]/100 + 100 = 140/200 = 0.7. Thus, σ p1 − p 2 = (0.7) (0.3)/100 + [(0.7) (0.3)/100] =

0.0042 = 0.0648.

127. (c) The hypothesized difference between the two population proportions is zero; the difference between the sample proportions, p1 – p2 = 0.75 – 0.65 = 0.10. Z=

p1 − p 2 0.10 = = 1.543 σp1 − p2 0.0648

For α = 0.10, we have Z = 1.543 greater than 0.102384, then we reject Ho, and conclude that serum helps to cure the disease. From the table, it is clear that we accept the null hypothesis only when α or the significance level is 0.10. α 0.01 0.05 0.10

Acceptance level 0 + 2.58 (0.0648) = 0 + 0.167184 0 + 1.96 (0.0648) = 0 + 0.127008 0 + 1.58 (0.0648) = 0 + 0.102384

128. (a) H0: μ1 = μ2: null hypothesis: there is no difference. H1: μ1 ≠ μ2: alternative hypothesis: a difference exists. 129. (b) The estimated standard error of the difference between the means σˆ x1 − x 2 = (8 2 ) / 40 + (7) 2 / 50 = (64 / 40) + (49 / 50)

= 2.58 = 1.606. 130. (c) The Central Limit Theorem claims that the sampling distribution of the mean approaches normality as sample size increases. 131. (d) If a statistical hypothesis is tested, is the null hypothesis is true, but it is rejected; the error made is called Type I error. The error committed in (d) is called Type II error. In either case a wrong decision is taken. P(Committing a Type I Error) = P (The Null Hypothesis is true but is rejected) = P (The Null Hypothesis is true but sample statistic falls in the rejection region) = α , the level of significance. 132. (e) Stratified sampling is generally used when the population is heterogeneous. In this case, the population is first subdivided into several parts (or small groups) called strata according to some relevant characteristics so that each stratum is more or less homogeneous. Each stratum is called a sub-population. Then a small sample (called sub-sample) is selected from each stratum at random. All the sub-samples combined together form the stratified sample. The sample selection is made by drawing equal numbers of elements from each stratum and weight the results or by drawing number of elements from each stratum proportional to their weights in the population. 97


133. (b) The dispersion among sample means is less than the dispersion among the sampled items themselves because very large values are averaged down, and very small values are averaged up . 134. (d) σ x = σ / n is used for when Sampling is from an infinite population or from a finite population with replacement. 135. (b) In testing of hypothesis, the decision to use a single-tailed or two-tailed test depends upon the alternative hypothesis. 136. (a) When a null hypothesis is accepted, there is a possibility that we have made a correct decision or have committed Type II error. 137. (a) When we choose 5% level of significance in a test procedure, there are about 5 cases in 100 that we would reject the hypothesis when it should be accepted, that is, we are about 95% confident that we have made the right decision. Similarly, if we choose 1% level of significance in testing a hypothesis, then there is only 1 case in 100 that we would reject the hypothesis when it should be accepted. So with a lower significance level, the probability of rejecting a null hypothesis that is actually true will decreases and the significance level is increased the probability of accepting a null hypothesis that is actually false Decreases. 138. (e) Decision makers make decisions on the appropriate significance level by examining the cost of Type I and Type II errors. If the cost of rejecting a null hypothesis is more (Type I error), decision makers set a lower level of significance in its testing. By contrast, if the cost of accepting a null hypothesis, when it is false (Type II error) is more, decision makers set a higher significance level in its testing. 139. (d) P(2.00 < μ < 4.2) = 0.95 (given). Hence, the probability that the normal variable will take a value in the interval 2.00 to 4.2 is 95 percent and P(μ > 4.2) = 0.05/2 = 0.025. 140. (d) P =

50 = 0.25,P =1 − P =1 − 0.25 = 0.75. 200

141. (a) If upper limit is more than sample mean by 12, then lower limit will be less than sample mean by 12. Hence, the lower limit of confidence interval is equal to 100 – 12 = 88. 142. (b) α, level of significance, indicates the chances or probability of committing Type I error, that is, rejecting a null hypothesis when it is true. Thus, α = 0.1 indicates that ten percent is the risk we take in rejecting a hypothesis when it is true. 143. (c) A two-tailed test is a test where both upper and lower limits of confidence interval are used to take a decision of accepting null hypothesis. 144. (d) Stratified sampling is generally used when the population is heterogeneous. In this case, the population is first subdivided into several parts (or small groups) called strata according to some relevant characteristics so that each stratum is more or less homogeneous. Each stratum is called a sub-population. Then a small sample (called sub-sample) is selected from each stratum at random. All the sub-samples combined together form the stratified sample. 145. (d) t-distribution or student’s t-distribution is used when the sample size is 30 or less and the population standard deviation is not known. 146. (d) Rejecting a null hypothesis, when it is true is called Type I error and its probability (is also the significance level of the test) is symbolized α. 147. (c) Finite population multiplier,

(N − n)/(N − 1) is used for a finite population of size N,

when a sample is drawn without replacement. If the value of the sampling fraction n/N is less than 0.05, then finite population correction factor

(N − n)/(N − 1) need not be used.

148. (c) Rejecting a null hypothesis, when it is true is called Type I error and its probability (is also the significance level of the test) is symbolized α. 149. (e) As the degree of freedom increases, ‘t’ distribution approaches normal distribution. 150. (c) Rejecting a null hypothesis, when it is true is called Type I error and its probability (is also the significance level of the test) is symbolized α. 98

Part I

151. (c) In systematic sampling each element among first k elements (k < n) has an equal chance of being selected, but each sample does not have the same chance of being selected. Here, the first element of the population is randomly selected. But thereafter the elements are selected according to a systematic plan. 152. (d) Judgmental sampling is not a type of random sampling. In judgmental sampling the sample is selected according to the judgment of the investigators or experts. Hence, there is a degree of subjectivity in the selection. 153. (b) Suppose we are test a hypothesis at 95%, then the set of z-scores outside the range –1.96 and 1.96 constitutes the critical region or region of rejection of the hypothesis or the region of significance. Thus, critical region is the area under the sampling distribution in which the test statistic value has to fall for the null hypothesis to be rejected. On the other hand, the set of z-scores inside the range –1.96 to 1.96 is called the region of acceptance of the hypothesis. The values –1.96 and 1.96 are called critical values at 5% level of significance. 154. (b) The standard error of a statistic θ (standard error of sample mean, or sample standard deviation or sample proportion of defectives, etc.) is the standard deviation of the sampling distribution of the statistic. Standard error measures the variability arising from sampling error due to chance. 155. (c) Accepting a null hypothesis, when it is false is called Type II error and its probability is symbolized as β. 156. (b) Stratified sampling is generally used when the population is heterogeneous. In this case, the population is first subdivided into several parts (or small groups) called strata according to some relevant characteristics so that each stratum is more or less homogeneous. Each stratum is called a sub-population. Then a small sample (called sub-sample) is selected from each stratum at random. All the sub-samples combined together form the stratified sample. 157. (a) Confidence interval is also termed as interval estimate. 158. (b) Sample statistics is used by the managers to estimate population parameters, that are useful for decision-making. 159. (b) For a sample of large size x will be almost normally distributed. 160. (d) Since rejecting a null hypothesis when it is false is exactly what a good test ought to do, the value of 1–β reflects the power of the test. A high value of 1 – β means the test is working quite well and a low value of 1 – β means that the test is working very poorly. 161. (d) Judgmental sampling is not a type of random sampling. In judgmental sampling the sample is selected according to the judgment of the investigators or experts. Hence, there is a degree of subjectivity in the selection. 162. (d) σ2 = 26 and n = 5, σ =

26 and standard error σ = σ/ n =

26/5 =

5.2 = 2.28.

163. (e) All of the above statement in the given question is correct, so the alternative (e) is correct. 164. (a) If P(10 < z < 19) = 99.78%, then the z value should be equal to 6 (i.e. 3 + 3). Z = (x–μ)/σ. Thus, the standard deviation, σ = (x – μ)/z = (19 – 10)/6 = 1.5. 165. (d) In cluster sampling, the population is divided into clusters or groups and then random sampling is done for each cluster. Cluster sampling differs from stratified sampling. In the case of stratified sampling, the elements of each stratum are homogenous. In cluster sampling, the elements of each cluster are not homogenous. Each cluster is representative of the population. 166. (c) Self-explanatory. 167. (c) In hypothesis testing, the critical region is that region of the sampling distribution in which the value of sample statistic must fall for the null hypothesis to be rejected and the alternative hypothesis to be accepted. Suppose we are test a hypothesis at 95%, then the set of z-scores outside the range –1.96 and 1.96 constitutes the critical region or region of rejection of the hypothesis or the region of significance. 99


168. (e) When a null hypothesis is accepted on the basis of sample information, it means that there is no statiscal evidence to reject the null hypothesis. 169. (c) The standard error of sample mean is the standard deviation of the distribution of sample means. 170. (b) The standard error of sample mean = σ/ n . 171. (d) t-distribution or student’s t-distribution is used when the sample size is 30 or less and the population standard deviation is not known. Furthermore, in using the t-distribution, we assume that the population is normal or approximately normal. 172. (c) Rejecting a null hypothesis, when it is true is called Type I error and its probability (is also the significance level of the test) is symbolized α. 173. (b) If σ is the standard deviation of the normal distribution, 80% of the observation will be in the interval μ –1.28σ to μ + 1.28σ. 174. (d) All of the above statement is true except alternative (d) as the sampling involves less work as compared with a census. Hence, the chances of errors while processing the data are less. 175. (c) Var (x1 + x2) = Var (x1) + Var(x2) + 2Cov (x1, x2) If x1 and x2 are independent then, Cov (x1, x2) = 0 ∴ Var (x1 + x2) = Var (x1) + Var (x2). 176. (e) The standard error of sample mean = σ/ n ; 6 = σ/ 9 . Thus, σ = 3 x 6 = 18. 177. (d) For a standard normal distribution σ = 1 ∴ Variance = σ2 = 12 = 1 = standard deviation. 178. (c) If the estimators are not consistent then any research will waste resource by taking large sample. 179. (c) If a statistic comes very close to the value of the population parameter as the sample size increases, it is said to be a consistent estimator. 180. (a) Rejecting a null hypothesis, when it is true is called Type I error and its probability (is also the significance level of the test) is symbolized α. 181. (b) When we choose 5% level of significance in a test procedure, there are about 5 cases in 100 that we would reject the hypothesis when it should be accepted, that is, we are about 95% confident that we have made the right decision. Similarly, if we choose 1% level of significance in testing a hypothesis, then there is only 1 case in 100 that we would reject the hypothesis when it should be accepted. So, if the significance level is increased the probability of accepting a null hypothesis that is actually false. Decreases and with a lower significance level, the probability of rejecting a null hypothesis that is actually true will decreases. 182. (b) σ2 = 5.76 and n = 36. σ = 5.76 and standard error = σ/ n =

5.76/36 = 2.4/6 = 0.4.

183. (e) According to the Central Limit Theorem: i.

If X be the mean of a random sample of size n drawn from a population having mean μ and standard deviation (σ ) , then the sampling distribution of the sample mean X is approximately a normal distribution with mean μ and Standard Deviation equal to Standard Error of X provided the sample size n is sufficiently large.

ii.

If p be the proportion of defectives in a random sample of size n drawn from a population having the proportion of defectives p, then the sampling distribution of the sample proportion of defectives p is approximately a normal distribution with mean p and standard deviation is equal to Standard Error of p , provided the sample size n is sufficiently large.

Usually, a sample of size 30 or more is considered as a ‘large sample’. However, the larger the value of n, the better is the approximation. 100

Part I

184. (e) t-distribution or student’s t-distribution is used when the sample size is 30 or less and the population standard deviation is not known. 185. (e) All of the above alternatives are the example of probability sampling. 186. (b) In making estimates of population parameters using sample statistics, efficiency refers to the size of the standard error of the sample statistic. 187. (b) Ceteris paribus, the standard error of mean decreases when the sample size increases. 188. (c) Finite population multiplier,

(N − n)/(N − 1) is used for a finite population of size N, when

a sample is drawn without replacement. If the value of the sampling fraction n/N is less than 0.05, then finite population correction factor (N − n)/(N − 1) need not be used.

(N − n)/(N − 1) =

[(200 − 20) /( 200 − 1) = 180 / 199 =

0.905 = 0.951.

189. (b) Lower limit of 95% confidence interval = x – 1.96 (σ); σ x = σ/ n = 18/6 = 3. Thus, x – 1.96(σ) = 25 – 1.96 (3) = 19.12. 190. (b) If a random variable x has a lognormal distribution, then ln(x) will be normally distributed. The expected value of the variable need not necessarily be 0 or 1. 191. (a) Procedural bias is the distortion of the representativeness of the data due to the procedure adopted in collecting the data. For instance in our retailers example, Procedural bias may creep in, if the retailer excludes all customers making purchases under Rs.2,000. In effect she will then study only high value customers, not all customers. 192. (d) Rejecting a null hypothesis, when it is true is called Type I error and its probability (is also the significance level of the test) is symbolized α. 193. (c) The standard error of mean will increase if sample size is reduced.

σx = 194. (c)

σ n

, If ‘n’ is reduced then σ x will increase.

(N − n)/(N − 1) = 752/ 799 = 0.941 = 0.97.

195. (e) Self-explanatory. 196. (e) (a)

This is an advantage of sampling. The population may be too large to be studied in full. In such situations, it is always better to use sampling compared to the study of whole population. (b) This is an advantage of sampling. A study of sample is usually cheaper than a study of the population. As the number of observations required is less the cost incurred in obtaining and processing these information is also less for sampling. (c) This is an advantage of sampling. Sampling usually provides information quicker than a census. Since the time required for studying the whole population is large, it is advisable to go for sampling to reduce the time spent in the studying the whole population. (d) This is an advantage of sampling. In destructive testing, where the product becomes unusable after the performance of test, we cannot test the whole population. In such cases, sampling is the only method of testing. (e) This is not an advantage of sampling. In a sampling exercise we use a small number of samples to study the whole population the degree of accuracy is not very high. For accurate information census is always the preferable method compared to sampling. 101


197. (a) (a) This is the right answer. In cluster sampling, we divide the population into groups, or clusters, and then select a random sample of these clusters. We assume that these individual clusters are representative of the population as a whole. Then we choose a certain number of clusters randomly. Every household of these clusters is interviewed. (b) This is a wrong answer. In stratified sampling we divide the population into relatively homogeneous groups, called strata. Then we use one of two approaches. Either we select at random from each stratum a specified number of elements corresponding to the proportion of that stratum in the population as a whole or we draw an equal number of elements from each stratum and give weight to the results according to the stratum’s proportion of total population. (c) This is a wrong answer. In systematic sampling, elements are selected from the population at a uniform interval that is measured in time, order, or space. In this method of sampling each element has an equal chance of being selected but each sample does not have an equal chance of being selected. (d) This is a wrong answer. In the judgmental sampling the sample is selected according to the judgment of the investigators or experts. (e) This is a wrong answer. In simple random sampling we do not divide the population in groups. We select a certain number of elements randomly. Here, each possible sample has an equal chance of being selected and each item in the entire population also has an equal chance of being selected. 198. (c) (a) This is a true statement. The probability that we associate with an interval estimate is called the confidence interval. This probability, then, indicates how confident we are that the interval estimate will include the population parameter. A higher probability means more confidence. (b) This is a true statement. Using high confidence level is not always desirable, because high confidence levels produce large confidence interval, and such large intervals are not precise; they give very fuzzy estimates. (c) This is a false statement. The number of degrees of freedom used in t-distribution estimation are equal to (n−1), where n is the sample size. We lose one degree of freedom by estimating the standard deviation of the population with the help of the sample standard deviation. (d) This is a true statement. The t-distribution need not be used in estimating if you know the standard deviation of the population. We can use the normal distribution in such case. (e) This is a true statement. The confidence interval is the range of the estimate we are making. An interval estimate describes a range of values within which a population parameter is likely to lie. 199. (d) (a)

This is true for hypothesis testing. The probability that the null hypothesis is true but rejected is α. In other words probability of committing Type I error is α, the level of significance.

(b) This is true for hypothesis testing. The probability that the null hypothesis is false but is accepted is β. In other words, the probability of committing Type II error is β. (c)

This is a true statement. (1–β) is known as the power of test. A high (1–β) implies that the test is doing exactly what it should be doing i.e. Rejecting the null hypothesis when it is false. A low (1–β) indicates poor performance.

(d) This is a false statement. A high (1–β) implies that the test is doing exactly what it should be doing i.e. Rejecting the null hypothesis when it is false. A low (1–β) indicates poor performance. (e)

102

This is a true statement. We can make two types of errors in hypothesis testing namely Type I error and Type II error. There is a trade-off between two types of errors: The probability of making one type of error can be reduced only if we are willing to increase the probability of making the other type of error.

Part I

200. (e) (a)

This is the wrong answer. This will be the null hypothesis for the test.

(b) This is the wrong answer. (c)

This is the wrong answer.

(d) This is the wrong answer. This is a common error made while making the alternative hypothesis. We do not make a hypothesis that none of the proportions are equal to each other. We only say that all the proportions are not all equal, although some of them might be equal to each other it does not mean that null hypothesis is true. (e)

This is the right answer. The alternative hypothesis is that all the three proportions are not all equal.

201. (c) (a)

This is a step followed in the ANOVA. We determine an estimate of the population variance from the variance that exists among the sample means.

(b) This is a step followed in the ANOVA. We determine an estimate of the population variance from the variance that exists within the samples. (c)

This is not a step followed in ANOVA. The contingency table consisting observed and estimated frequency is prepared for testing equality of proportion from more than two samples.

(d) This is a step followed in the ANOVA. We accept the null hypothesis if the two estimates of the population variance are not significantly different. (e)

This is a step followed in the ANOVA. We reject the null hypothesis if the F ratio is too high and falls in the critical region.

202. (d) (a)

This the wrong answer.

(b) This is the wrong answer. (c)


(d) This is the right answer. In two-tailed test of the two variances, we find the left tail of the F distribution using F (n, d,α) is equal to 1/ F (d, n, 1–α). (e)


⎛ x −μ ⎞ 203. (b)Z = (x – μ)/σ then s.d (z) = σ2 = v(z) = v ⎜ ⎟ ⎝ σ ⎠ =

1 σ

2

v (x − μ) =

1 σ

2

v (x) =

1 σ2

σx = 1 =1

When each value is reduced by a constant the mean of the resulting values = previous mean – constant. When the mean itself is subtracted from each value the mean of the resulting values = mean – mean = 0. When each value is reduced by a constant the standard deviation remains unchanged. However, when each value is divided by a constant the standard deviation of the resulting values = previous standard deviation/constant. When each value is divided by the standard deviation the standard deviation of the resulting values = standard deviation/standard deviation = 1.

103


204. (b) i. Simple random sampling may not be appropriate when the population is known to consist of well defined groups such that the elements within each group are heterogeneous and each group is a representative of the population as a whole, because even if the sample is random it may not reflect the nature of the population.

ii.

When the population is known to consist of well-defined groups such that the elements within each group are heterogeneous and each group is a representative of the population as a whole, the cluster sampling is appropriate.

iii.

When the population is known to consist of well-defined groups such that the elements within each group are homogeneous and the groups vary from each other significantly, the stratified sampling is appropriate.

iv.

When the population is known to consist of well-defined groups such that the elements within each group are heterogeneous and each group is a representative of the population as a whole, because even if the sample is random it may not reflect the true nature of the population.

v.

When the population is known to consist of well-defined groups such that the elements within each group are heterogeneous and each group is a representative of the population as a whole, judgmental sampling may not be appropriate because the representativeness of the sample depends upon the knowledge and judgment of the decision-maker.

205. (c) c. d.

A Type I error occurs if the null hypothesis is rejected though it is true. A Type II error occurs if the null hypothesis is accepted though it is false. a, b & e are incorrect interpretations of Type I error.

206. (a) a. If a hypothesis is tested at a 10% significance level then, it means that there is a 10% probability that the null hypothesis will be rejected though it is true. b, c, d & e are incorrect interpretations of the significance level. 207. (b) i. The property of efficiency with regard to an estimator does not refer to the sample size. ii. The property of efficiency with regard to an estimator refers to the size of the standard error. iii. The property of efficiency with regard to an estimator does not refer to the size of the standard deviation of the sample. iv. The property of efficiency with regard to an estimator does not refer to the size of the sample variance. v. The property of efficiency with regard to an estimator does not refer to the size of the sample mean.

Linear Regression 208. (c) b = ∑XY – n(Mean of X) (Mean of Y)/[∑X2 – n(Mean of X)2] = 0.0495 – 5(0.04) (0.138)/ [(0.087) – 5(0.04)2] = 0.0219/0.079 = 0.2772. 209. (b) R2 = [a∑Y + b∑XY – n(Mean of Y)2]/[∑Y2 – n (Mean of Y)2] = 0.127(0.69) + 0.2772 (0.0495) – 5(0.138)2/0.1987 – 5(0.138)2 = 0.0587. 210. (a) Systematic Risk = R2 x Variance of Security Returns = 0.0587 x 0.02587 = 0.0015. 211. (b) Total Risk = Systematic Risk + Unsystematic Risk. Unsystematic Risk = Variance of returns of security – Systematic risk = 0.02587 – 0.0015 = 0.0244. Thus, total risk = 0.0244 + 0.0015 = 0.0259.

104

Part I

212. (b) Unsystematic Risk = Variance of returns of security – Systematic risk = 0.02587 – 0.0015 = 0.0244. 213. (e) R2 = (1 – 1/4) = 3/4. [Hint: R2 is the fraction of variation in Y that is explained by the independent variable X]. 214. (c)

X

Y

XY

X2

Y2

1 3 4 6 8 9 11 14

1 2 4 4 5 7 8 9

1 6 16 24 40 63 88 126

1 9 16 36 64 81 121 196

1 4 16 16 25 49 64 81

56

40

364

524

256

X– X

Y– Y

(6) (4) (3) (1) 1 2 4 7

(4) (3) (1) (1) 0 2 3 4

(X – X )2 (Y – Y )2 36 16 9 1 1 4 16 49

16 9 1 1 0 4 9 16

132

56

X = ΣX/n = 56/8 = 7 Y = ΣY/n = 40/8 = 5

b = [n∑XY – (∑X) (∑Y)]/[n∑X2 – (∑X)2] = [8(364) – (56)(40)]/[8(524) – (56)2] = 672/1056 = 0.6364. 215. (b) r2 = [a∑Y + b∑XY – nY 2 ]/[∑Y2 – nY 2 ]

= [0.55(40) + 0.6364(364) – 8(5)2]/[256 – 8(5)2] = 53.64/56 = 0.957 Where, a = Y – b X = 5 – 0.6364(7) = 5 – 4.45 = 0.55. 216. (a) Y = a + bX = 0.55 + 0.6364X

When X = 16, Y = 0.55 + 0.6364(16) = 10.7278. 217. (d) Se = [(∑ Y 2 − a ∑ Y − b ∑ XY)]/(n − 2) = [(256 − 0.5452( 40) − 0.6364(364))]/(8 − 2) =

=

2.5424/6

0.4237 = + 0.65.

218. (d) If the standard error of the estimate is zero, the estimating equation is said to be perfect estimator of the dependent variable. When the standard error of the estimate is zero, it implies that all the data points are on the regression line. 219. (b) Estimated Return on Security A

= 0.5(7) + 0.5(11) = 9 Estimated Return on Security B = 0.5(13) + 0.5(5) = 9

σ 2A = 0.5(7 – 9)2 + 0.5(11 – 9)2 = 4; then σ A =

4 =2

σ 2B = 0.5(13 – 9)2 + 0.5(5 – 9)2 = 16; then σ B = 16 = 4. 105


220. (b) Portfolio Risk =

2 X 2A σ A + X B2 σ B2 + 2X A X B Cov (A,B)

= (0.75) 2 (4) + (0.25) 2 (16) + 2(0.75)(0.25)(−8) =

0.25 = 0.5.

Where, Cov(A,B) = 0.5(7 – 9) (13 – 9) + 0.5 (11 – 9) (5 – 9) = –8. 221. (c) Portfolio Risk = X 2A σ 2A + X 2B σ 2B + 2X A X B Cov (A,B)

= (0.25) 2 (4) + (0.75) 2 (16) + 2(0.75) (0.25) (−8) = (6.25) = 2.5 Where, Cov(A,B) = 0.5(7 – 9) (13 – 9) + 0.5 (11 – 9) (5 – 9) = –8. 222. (b) Portfolio Risk = X 2A σ 2A + X 2B σ 2B + 2X A X B Cov (A,B)

= (0.5) 2 (4) + (0.5) 2 (16) + 2(0.5)(0.5) (−8) = 1 = 1. Where, Cov(A,B) = 0.5(7 – 9) (13 – 9) + 0.5 (11 – 9)(5 – 9) = –8.

ˆ = 5 + 4X. 223. (e) All of the given alternatives is true for straight line Y 224. (c) P(X2 – X1 > 30) ⎛ 30 − mean (X 2 − X1 ) ⎞ ⎟⎟ = P ⎜⎜ Z ≥ sd (X 2 − X1 ) ⎝ ⎠

⎛ 30 − 25 ⎜ =P⎜Z≥ ⎜ 4 2 + 32 ⎝

⎞ ⎟ ⎟⎟ = P(Z > 1). ⎠

225. (a) Standard deviation of portfolio = W12 σ12 + W22 σ 22 + 2ρ W1 W2 σ1σ 2

But it is given that the returns on the two assets are exhibiting perfect negative correlation, i.e. ρ = –1. Therefore,

0 = W12 σ12 + W22 σ 22 − 2W1 W2 σ1σ 2 , but W2 = 1 – W1. Hence, substituting the value of W2 and the standard deviations of the two assets in the above equation, we get 16W12 − 24W1 + 9 = 0. 226. 227.

228. 229.

Solving the above equation, we get W1 = 3/4, i.e. W1 = 75%. (c) Y = 13 + 0.1X(given) and Z = 0.5X – 50 (given). If Z = 0.5X – 50, then X = (50 + Z)/0.5 = 100 + 2Z and thus, Y = 13 + 0.1(100 + 2Z) = 23 + 0.2Z. (e) Unless data is given, it is not possible to estimate the standard error. An estimating equation is said to be perfect estimator of the dependent variable only when the standard error is zero. If the standard error is zero, all the data points must be on the regression line. (d) P(X > 20) = 0.5. [Hint: The probability that a value fall below or above the mean of a variable is equal to 0.5]. (c) yˆ = a + bx ‘b’ is the regression coefficient of the independent variable. b < 0 implies that the variables x and y are negatively correlated. ∴ Coefficient of correlation = Negative square root of coefficient of determination.

106

Part I

230. (e) If yˆ = 4 – 3x, 4 is the y-intercept and –3 is the slope of regression line. Since the slope of the line is negative, the line represents an inverse relationship. 231. (e) The correlation coefficient describes both the magnitude of the correlation and its direction and, is defined as

r=

Cov (X, Y) or SX SY

∑ (X − X) (Y − Y) ∑ (X − X)2 ∑ (Y − Y)2

i.

–1 ≤ r, r ≤ 1

ii.

r = 0 <=> No relationship between X and Y

iii.

r = 1 <=> Perfect positive linear relationship

iv.

r = −1<=> Perfect negative linear relationship.

232. (e) The standard deviation of a portfolio measures the risk of investing in the portfolio. The

standard deviation (σ p ) is given by σ p = W12 σ12 + W22 σ 22 + 2 W1W2 Cov (1,2) . Markowitz has proposed a Portfolio Theory in which he stated that the risk of a portfolio would be zero, if the investments are made in a portfolio of two securities if the portfolio with investments in two securities with equal weightages has same standard deviations and the covariance between these two securities is perfectly negative (–1). 233. (d) While estimating the standard error of estimate we assume that (i) the observed values for Y are normally distributed around each estimated value of Y and (ii) the variance of the distributions around each possible value of Y is the same. If this second assumption were not true, then the standard error at one point on the regression line could differ from the standard error at another point on the line. 234. (b) If coefficient of determination, r2 = 0.4624, the absolute value of coefficient of correlation, |r| would be equal to r 2 or 0.4625 = 0.68. Since the slope of the regression line is negative (–0.02), the relationship between the two variables is negative. Thus, the value of coefficient of correlation, r would be equal to –0.68. 235. (a) b = (∑ XY − nXY)/(∑ X 2 − nX 2 )

= (193 – 180)/(110 – 100) = 1.3 Where, ∑XY = 15 + 40 + 54 + 84 = 193 X

= (3 + 4 + 6 + 7)/4 = 5

Y

= (5 + 10 + 9 + 12)/4 = 9

∑ X 2 = (9 + 16 + 36 + 49) = 110 nX Y = 4(5)(9) = 180

nX 2 = 4(5)2 = 100. 236. (a) The relationship between money supply and inflation are positive and hence the correlation between money supply and inflation is positive. 237. (a) r = Cov(XY)/σX σY = 12.3/(3.71 x 4.05) = 0.819. 238. (a) Risk in an investment is normally measured by the variance or standard deviation in the returns from the mean or expected return. The higher the value of dispersion (i.e. variance or standard deviation), the higher is the risk.

107


The variance and standard deviation of returns from the security A may be calculated as under: Rates of Probability of Deviations from Return occurrence Expected Rate of Return

Squares of Deviation

Probability x Squares of Deviation

XA − XA

(X A − X A ) 2

Pi (X A − X A ) 2

XA

PA

25

0.36

4.4

19.36

6.9696

20

0.40

(0.6)

0.36

0.1440

15

0.24

(5.6)

31.36

7.5264

Total

14.6400

Expected Value = 25(0.36) + 20(0.4) + 15(0.24) = 20.6 The variance and standard deviation of returns from the security B may be calculated as under: Deviations from Expected Rate of Return

Rates of Return

Probability of occurrence

XB

PB

26

0.50

5.4

29.16

14.58

18

0.30

(2.6)

6.76

2.028

11

0.20

(9.6)

92.16

18.432

Total

35.04

B

B

(X B

− XB )

Squares of Probability x Squares Deviation of Deviation (X B

− XB ) 2

pi ( X B − X B ) 2

Expected Value = 26(0.5) + 18(0.3) + 11(0.2) = 20.6 Since the variance and standard deviation of returns from the security A is less than security B, Mrs. C.F. Anand invests in security A. 239. (c) The coefficient of correlation, r is equal to 0.9. Hence the coefficient of determination, r2 is equal to (0.9)2 = 0.81. 240. (e) R2 is the function of variation in Y explained by the explanatory variable X. Thus, R2 1 – 1/4 = 3/4. 241. (a) The correlation coefficient of a variable with itself equal to 1. 242. (a) In a regression equation Y = a + bX, Y and X represent dependent and independent variables respectively. 243. (d) Covariance between returns of two securities is represented by ρ12 σ1σ2, where ρ12 is the coefficient of correlation between the returns of security 1 and 2, σ1 and σ2 are the standard deviations of the returns of the security 1 and 2 respectively. If two variables were perfectly positively correlated, the value of ρ12 would be equal to 1. Hence, the covariance between the two perfectly positively correlated variables would be equal to 1(σ1 σ2) or σ1σ2. 244. (a) Regression analysis is based on the relationship between two or more variables. The known variable is the independent variable and the variable we are trying to predict is the dependent variable. 245. (c)The correlation between two variables depends on the standard deviations of the two variables and covariance between them. It does not depend on the means of the two variables. 246. (b)The random variables X and Y are said to be independent if the covariance between X and Y Cov(X,Y) = 0 and correlation should be zero. 247. (a) Mathematical model is the general characterization of a process, object, or concept, in terms of mathematics, which enables the relatively simple manipulation of variables to be accomplished in order to determine how the process, object, or concept would behave in different situations. Mathematical models used for solving many questions in finance and economics.

108

Part I

248. (b)

X

Probability

Cumulative Probability

Random numbers allotted

5

0.2

0.20

00 – 19

10

0.5

0.70

20 – 69

18

0.1

0.80

70 – 79

20

0.2

1.00

80 – 99

Since the random number generated was 37 (which falls in between 20 – 69), the value of X in this run is 10. 249. (e) One of the easiest ways of studying the correlation between the two variables is with the help of a scatter diagram.

A scatter diagram can give us two types of information. Visually, we can look for patterns that indicate whether the variables are related. Then, if the variables are related, we can see what kind of line, or estimating equation, describes this relationship. The scatter diagram gives an indication of the nature of the potential relationship between the variables. 250. (a) In a regression equation Y = a + bX, a is the Y-intercept and b is the slope of the regression line. Thus, in this example, the equation can be written as Y = 4 + 3X. 251. (a) If the correlation coefficient between two variables is –1, it means two variables are perfectly negatively correlated and then all the pairs of observations when plotted on a scatter diagram will lie on the regression line. 252. (a) Systematic risk of a share is determined using the beta (or β) of the share. Mathematically, Systematic Risk of a Security = β2 x Variance of the Market Returns. 253. (c) If points (x1, y1) and (x2, y2) lie on a regression line Y = a + bX, then b will be equal to

(y2 – y1)/(x2 – x1) and it represent the slope of the line. 254. (d)Beta of a stock shows the degree of correlation between the market return and the particular stock. Beta equal to 1 means if the market return is change by 10% stock return will also change by 10%. If one is expected that stock market will bullish in the near future he should go for beta of more than one. 255. (a)Unsystematic Risk = Variance of Stock’s Return – Systematic Risk.

Systematic Risk = β2 x Variance of the Market Returns = (1.1)2 (22.8) = 27.588. Thus, unsystematic risk = 33.7 – 27.588 = 6.112. 256. (a) R2 = 1 – (ESS/TSS) = 1 – (9/25) = 0.64 or 16/25. 257. (b) We know that r =

r 2 . Therefore, r is always less than or equal to r2 in magnitude.

258. (b) Regression line should be plotted in such a way that it minimizes the sum of squares of the errors. Hence, it should be drawn in such a way that it minimizes the sum of squares of the vertical distances of each point to the regression line. ∧

∑ (Y − Y ) 2 . 259. (b) The standard error of the estimate of a regression line is given by (n − 2)

260. (c) The coefficient of correlation, r = Cov (X,Y)/SXSY = 0.02/(0.1 x 0.2) = 1. Thus, the two variables in this case are said to be perfectly positively correlated. 261. (e)The information given in the problem is insufficient to determine the direction of correlation between the variables. 262. (b) If the degree of correlation between independent variables increases, the regression coefficients become less reliable.

109


263. (b) Y = a + bX

Y = a + 0(X) [since b = 0 (given)] Y=a The regression line will be parallel to the X-axis. 264. (d) Statisticians sometimes interpret the sample coefficient of determination by looking at the amount of the variation in Y that is explained by the regression line. This implies that the proportion of variation in a dependent variable that is explained by the independent variable is nothing but the sample coefficient of determination, r2. Hence, r2 = 75% or 3/4. 265. (c) Systematic Risk = β2 x Variance of the Market Returns

13.6 = β2 x 12.8 β2

= 13.6/12.8 = 1.0625

Therefore, β = 1.0625 = 1.03. 266. (d)Regression analysis is based on the relationship between two or more variables. The known variable is the independent variable and the variable we are trying to predict is the dependent variable. 267. (c) Sometimes the characteristics, whose possible correlation is being investigated, cannot be measured but individuals can only be ranked on the basis of the characteristics to be measured. We then have two sets of ranks available for working out the correlation coefficient. Sometimes the data on one variable may be in the form of ranks while the data on the other variable are in the form of measurements. In that case, the data on the variable, which are in the form of measurements, can be converted into ranks to estimate the rank correlation coefficient between the two variables. Thus, when both the underlying variables are ordinal or when the data are available in the ordinal form irrespective of the type of variable, we use the rank correlation coefficient to determine the extent of relationship between the two variables. 268. (b) The method of simple linear regression is commonly used to segregate the fixed and variable components of semi-fixed or semi-variable costs. Regression analysis is one of the useful techniques in cost-volume-profit analysis. 269. (c) Y = 3.5(0) + 4.3 = 4.3. 270. (e) R2 = RSS/TSS and TSS = RSS + ESS

R2 = 0.81 = RSS/TSS Therefore, RSS = 0.81(TSS) 207 = TSS – 0.81(TSS) since ESS = TSS – RSS Therefore, TSS = 207/0.19 = 1089.47. 271. (c) The coefficient of correlation, r is equal to 0.00. Hence, the coefficient of determination, r2 is equal to (0.0)2 = 0.00. 272. (c) r = Cov(XY)/SXSY = 1.7 / (3.4)(5.1)

= 1.7/4.164 = 0.408. 273. (b) Statisticians sometimes interpret the sample coefficient of determination by looking at the amount of the variation in Y that is explained by the regression line. This implies that the proportion of variation in a dependent variable that is explained by the independent variable is nothing but the sample coefficient of determination, r2. r2 = RSS = TSS – ESS = 169 – 100 = 69. Therefore, the percentage of changes in the dependent variable is explained by the ESS changes in the independent variable is equal 1 − (simply) to 69/169 x 100 = 40.8%. TSS

110

Part I

274. (b) If the slope of line is downward (see picture) it means there is negative correlation between X and Y

275. (b) Since coefficient of determination, r2 is the square of the coefficient of correlation, r the value of coefficient of determination, r2 would be always positive. 276. (e) The coefficient of determination is given by r2 i.e., the square of the correlation coefficient. It explains to what extent the variation of a dependent variable is expressed by the independent variable. A high value of r2 shows a good linear relationship between the two variables. If r = 1 and r2 = 1, it indicates a perfect relationship between the variables. 277. (c) b = Cov(XY)/V(X) = 550/250 = 2.2. 278. (b) In the equation for standard error of estimate we can observe the sum of the squared deviations is divided by n–2 and not by n. This is because we have lost 2 degrees of freedom in estimating the regression line. 279. (e)

r

= Cov(XY)/(SXSY)

1

= Cov(XY)/(5 x 10)

Cov(XY) = 50. 280. (d) R2 = (0.9)2 = 0.81 or 81%. 281. (d) Beta of a stock shows the degree of correlation between the market return and the particular stock. The return from a share having a beta of –1.5 means stock return increases by 1.5% when the market return decreases by 1%. 282. (b) TSS = RSS + ESS

R2 = RSS/TSS or 1 – (ESS/TSS) = 1 – (30/90) = 2/3. 283. (a) y = a + bx, where a is y-intercept and b is the slope of the regression line. Thus, in this case, y = 3 + 2x (or) 2x + 3. 284. (e) The standard error of the estimate for a regression equation is given by

se =

(

ˆ ∑ Y−Y

)

2

n−2

Where, Y

= values of the dependent variable

Yˆ

= Estimated values from the estimating equation that correspond to each Y value

n

= Number of data points used to fit the regression line.

In the above equation, you can observe the sum of the squared deviations is divided by n–2 and not by n. This is because we have lost 2 degrees of freedom in estimating the regression line. We used the sample to compute a and b. 285. (c) The negative slope of the regression line indicates an inverse correlation between X and Y.

111


286. (b) If coefficient of determination, r2 = 0.4624, the absolute value of coefficient of correlation,

|r| would be equal to r 2 or 0.4624 = 0.68. Since the slope of the regression line is negative (–0.02), the relationship between the two variables is negative. Thus, the value of coefficient of correlation, r would be equal to –0.68. 287. (c) Rx = 5.5 + 1.25(12) = 20.5%. 288. (a) r = Cov(XY)/SXSY = – 0.85 = Cov(XY)/ (16)(9).

Cov(XY) = [ (16)(9) ] (–0.85) = 12(–0.85) = –10.20. 289. (c) R2 = RSS/TSS = 550/625 = 0.88. 290. (e) All of the above statement is true if the regression equation is not a perfect estimator of the dependent variable. 291. (b) a = Y – b (X ) = 12.2 – 1.5(6.8) = 2

Y = a + bX = 2 + 1.5X. 292. (e) Probability of r successes in n trials

= n!/[r!(n – r)!] (p)r (q)n – r = 3!/[1!(3 – 1)!](0.25)1 (0.75)3–1 = 0.4219. 293. (a) β = Cov(XY)/V(X) = 225/180 = 1.25. 294. (c) r2 = 0.476 x 1.036 = 0.493. 295. (e)Mean returns of individual securities are not required for finding out the standard deviation of returns of a portfolio. ∧

296. (b) Se = ∑ (Y − Y ) 2 /(n − 2) = 158.76/9 = 4.2. 297. (d) The relationship between the independent and dependent variables is not necessarily implies cause and effect for example, r = 0 <=> No relationship between X and Y. 298. (d) Since coefficient of determination, r2 is the square of the coefficient of correlation, r the value of coefficient of determination, r2 would be always positive and is not different for positive and negative correlation of the same magnitude. 299. (c)

(a)

This is true for correlation analysis. If the coefficient of correlation between two variables is close to one, then there is a strong correlation between the two variables.

(b) This is true for correlation analysis. If the slope of a regression equation is positive then the coefficient of correlation between the variables involved is positive. (c)

This is false for correlation analysis. Even a highly significant correlation does not necessarily mean that a cause and effect relationship exists between the two variables. Correlation does not necessarily imply causation or functional relationship though the existence of causation always implies correlation or association between two variables.

(d) This is true for correlation analysis. If the slope of a regression equation is negative then the coefficient of correlation between the variables involved is negative. (e)

112

This is a true statement. If the coefficient of correlation between two variables is equal to zero, then there is a no correlation between the two variables.

Part I

300. (b)

(a)

This is a wrong answer. The sum of the squared values of the vertical distances from each plotted point to the line should be minimum (not maximum).

(b) This is the right answer. According to the ‘method of least squares’ criterion, the regression line should be drawn on the scatter diagram in such a way that the sum of the squared values of the vertical distances from each plotted point to the line are minimum. (c)

This is the wrong answer. The sum of the squared values of the vertical (not the horizontal) distances from each plotted point to the line should be minimum (not maximum).

(d) This is the wrong answer. The sum of the squared values of the vertical (not the horizontal) distances from each plotted point to the line should be minimum. (e)

This is the wrong answer. The sum of the squared values of the vertical distances from each plotted point to the line should be minimum (not necessarily zero).

301. (d)

(a)

This is a wrong answer. The standard error of the estimate of a regression line is not the slope of the regression line.

(b) This is a wrong answer. The standard error of the estimate of a regression line is not The square root of the coefficient of determination. (c)

This is a wrong answer. The standard error of the estimate of a regression line is not the Y intercept of the regression line.

(d) This is the right answer. The standard error of the estimate of a regression line is the measure of the variability of the observed values around the regression line. (e)

This is a wrong answer. The standard error of the estimate of a regression line is not the coefficient of correlation between two variables.

302. (b)

(a)

This is the wrong answer. The coefficient of determination r2 measures how good the fit is and gives the variation explained by the model.

(b) This is the right answer. If r be the coefficient of correlation between two variables X and Y, then the variation that is not explained by the regression line is equal to 1 − r2. (c)

This is the wrong answer. ‘r’ is the coefficient of correlation and it signifies the degree of association between two variables.

(d) This is the wrong answer. ‘1− r ’ does not give the variation that is not explained by the model. (e)

This is the wrong answer. ‘1+r2’ does not have any meaning associated with it.

303. (c)

(a)

This is a wrong answer. The given ratio does not follow a normal distribution.

(b) This is a wrong answer. The given ratio does not follow a t-distribution. (c)

RSS /1 follows an F distribution with 1 ESS /(n − 2) degree of freedom in the numerator and (n–2) degrees of freedom in the denominator. A high F ratio signifies that the regression model is working fine and a low value of F means the model should be rejected. This is the right answer. The ratio F =

(d) This is a wrong answer. The given ratio does not follow a chi square distribution. (e)

This is a wrong answer. The given ratio does not follow a binomial distribution.

113


304. (e)

(a)

The covariance is not the average of all the values that may be assumed by both the random variables, considered together.

(b)

The covariance is not the variance of all the values that may be assumed by both the random variables, considered together.

(c)

The covariance is not the standard deviation of all the values that may be assumed by both the random variables, considered together.

(d) The covariance is not the range of all the values that may be assumed by both the random variables, considered together. (e)

The covariance is a single number which measures the extent to which two random variables move together.

305. (a)

(a)

If two random variables are independent of each other then their covariance is equal to 0. b, c, d & e. If two random variables are independent of each other then their covariance cannot be equal to any value different from 0.

306. (c)

i.

The Y-intercept of the regression line does not represent the true value of Y when X= 0.

ii.

The Y-intercept of the regression line does not represent the change in average value of Y per unit change in X.

iii.

The Y-intercept of the regression line represents the mean value of Y when X = 0.

iv.

The Y-intercept of the regression line does not represent the standard deviation of the values of X.

v.

The Y-intercept of the regression line does not represent mean of the values of X.

307. (b)

i.

The slope of the simple regression equation does not represent the mean value of Y when X = 0.

ii.

The slope of the simple regression equation represents the change in average value of Y per unit change in X.

iii.

The slope of the simple regression equation does not represent the true value of Y for a fixed value of X.

iv.

The slope of the simple regression equation does not represent the variance of the values of X.

v.

The slope of the simple regression equation does not represent variance of the values of Y for a fixed value of X.

308. (a)

114

i.

Correlation analysis provides a measure (coefficient of determination) which indicates, how well an estimating equation explains the changes in the estimated (i.e. dependent variable).

ii.

If the coefficient of correlation between two variables is close to 1 then there is a very high correlation between the two variables.

iii.

If the slope of a regression line is positive then, the coefficient of correlation between the variables involved is positive.

iv.

If the slope of a regression line is negative then the dependent variable decreases as the independent variable increases.

v.

Correlation analysis may indicate a possibility of a cause-effect relationship. It does not necessarily indicate that there is a cause-effect relationship between the variables involved.

Part I

309. (c)

i.

If the standard error of estimate for a regression equation is zero, then it does not indicate that the standard deviation of the observed values of the dependent variable is zero.

ii.

If the standard error of estimate for a regression equation is zero, then it does not indicate that the standard deviation of the independent variable is zero.

iii.

If the standard error of estimate for a regression equation is zero, then it indicates that the observed value of the dependent variable will always be equal to its estimated value, for a given value of the independent variable.

iv.

If the standard error of estimate for a regression equation is zero, then it indicates that there is a perfect correlation between the variables.

v.

If the standard error of estimate for a regression equation is zero, then it indicates that there is a perfect correlation between the variables; so the correlation coefficient will be either –1 or 1.

310. (c)

i.

When the slope of a regression line is negative the correlation coefficient need not be 1.

ii.

When the slope of a regression line is negative, there is a negative correlation between the variables; hence the correlation coefficient lies between –1 and 0.

iii.

When the slope of a regression line is negative, there is a negative correlation between the variables.

iv.

When the slope of a regression line is zero, the regression line will be parallel to the horizontal axis.

v.

When the y-intercept of a regression line is zero, the regression line passes through the intersection of horizontal and vertical axes.

311. (a)

i.

The coefficient of determination is the square of coefficient of correlation (r). Hence, it will always be ≥ 0 and ≤ 1.

ii.

From above we can see that the coefficient of determination cannot be less than zero.

iii.

The coefficient of determination is will be equal to 1, only if the coefficient of correlation is equal to –1 or 1; in other cases it will be > 0 and < 1.

iv.

The coefficient of determination is always positive; The coefficient of correlation may be negative.

v.

Since –1 ≤ r ≤ 1, the coefficient of determination which is the square of coefficient of correlation (r), will always have magnitude less than or equal to r.

312. (a)

i.

The graphical plot of the values of the dependent and independent variables, in the context of regression analysis, is called scatter diagram.

ii.

A frequency polygon is a graphical representation of a frequency distribution which uses straight lines to join the top mid-points of the rectangles in a histogram.

iii.

A histogram is a graphical representation of a frequency distribution.

iv.

A p chart is a quality control chart.

v.

An ogive is a graphical plot of a cumulative frequency distribution.

115


Multiple Regression 313. (e) 2 ) R12.3 = [r12 − (r13 )(r23 )]/ (1 − r132 ) (1 − r23

= [0.8196 – (0.7698) (0.7984)] divided by (1 − 0.7698 2 ) (1 − 0.7984 2 ) = 0.5334.

314. (e)

R1.23 = = =

(r122 + r132 − 2r12 r23r13 )/[1 − r232 ]

[(0.8196) 2 + (0.7698) 2 − 2(0.8196) (0.7698) (0.7984)] / [(1 − (0.7984)2 ] (0.7984) / 1 − (0.7984) 2 = 0.8417.

2 315. (a) R 1.23 = (0.8417)2 = 70.85.

316. (b) RSS = TSS – ESS = 22239.4 – 1.3 = 22238.1. 317. (a)

F = [RSS/k]/[ESS/(n – (k + 1))] = [22238.1/4]/[1.3/6 – (4 + 1))] = 5559.525/1.3 = 4276.54. 318. (c) Root M.S.E = Root of Mean Square of Error =

1.3 = 1.14.

319. (e)

R2 = RSS/TSS = 1 – (ESS/TSS) = 22238.1/22239.4 = 0.999 or 100%. 320. (d) 0.046369/0.002742 = 16.91. 321. (d)Multicollinearity reduces the effects of the individual variables in the model; however the variables may together explain the dependent variable. It does not reduce the predictive power of the model. It reduces the effectiveness of sensitivity analysis of the model. Hence (a), (b) and (c) are not true, (d) is true. 322. (a) The standard error of the estimate can be decreased for a regression equation by increasing the number of observations. 323. (e)The coefficient of multiple correlation is dependent on the coefficients of correlation between Y and X1, between Y and X2, and between X1 and X2. Alternatives (a), (b), (c) and (d) are true. 324. (b) R1.23 =

=

2 (r122 + r132 − 2r12 r23 r13 )/(1 − r23 )

[(0.36) 2 + (0.49) 2 − 2(0.36)(0.49)(0.64)]/(1 − (0.64) 2 )

= 0.143918 / 0.5904 =

0.24372 = 0.49372.

325. (d) A normal distribution may be used to approximate the ‘t’ distribution for multiple regression whenever the degrees of freedom are more than 30. 326. (b) The k in the expression n – k – 1 stands for the number of independent variables in the multiple regression. 327. (d) As in simple linear regression, a, b1, b2, ..., bk are unbiased estimators of A, B1, B2, ... Bk respectively. As before, Sbi are the estimators of the standard error of the regression coefficients. In multiple regression, these estimates are difficult to compute, and we will use computer outputs to make the necessary inferences. We will test the hypothesis H0: Bi = 0 and (bi – Bi)/ Sbi ~ tn – (k+1) distribution. To make inference about the regression as a whole, we need to examine whether the sample implies that the Coefficient of Determination R2 is 0 or not.

From this it is clear that we require both degrees of freedom and Sbi to perform our test. 116

Part I

328. (c) From the table, we know that R2 = 0.8908, which implies that 89% of the variation in length of employment is explained by the regression. 329. (d) The degree of freedom for error (= 5) is the one which should be used to test whether the years of school is a significant explanatory variable for longevity. 330. (b) Sb3 is the standard error of regression coefficient b3 (i.e., for the variable “score”) and from the table Sb3 = 0.29. 331. (c) Signs of the possible presence of multicollinearity in a multiple regression are a sharp increase in a t value for the coefficient of an explanatory variable when another variable is removed from the model. 332. (b) Y = a + b1X1 + b2X2 + .... + bnXn, where a, b1, b2 ... bn are constants. From this equation, it is clear that a general multiple regression equation which has n-independent variables, will have n + 1 constants. 333. (a) Multicollinearity refers to existence or presence of correlation among independent variables. 334. (c) Multiple regression analysis is used to relate changes in one dependent variable to more than one independent variable. 335. (b) A general multiple regression equation has only one dependent variable and more than one independent variable. 336. (b) As the degree of correlation between the independent variables increases, the regression coefficients become less reliable. That is, although the independent variables may together explain the dependent variable, but because of Multicollinearity the coefficients of the explanatory variables may be rejected. It can happen that the model may be accepted (through ANOVA and the F test), but the individual coefficients may be rejected (through the t test). This is because the interplay among the independent variables reduces the influence of the individual variables in the model. In the extreme case, if two variables were identical, then the influence of each one in the model would be reduced. 337. (e) (a) This is true for multiple correlation coefficient. It depends on the correlation coefficient between Y and X1. (b) This is true for multiple correlation coefficient. It depends on the correlation coefficient between X1 and X2. (c) This is true for multiple correlation coefficient. It depends on the correlation coefficient between Y and X2. (d) This is true for multiple correlation coefficient. It will take values between 0 and 1. (e) This is false for multiple correlation coefficient. It will take values between 0 and 1. It does not take negative values. 338. (a) (a) This is the right answer. The partial coefficient of correlation between Z and X when Y is kept constant is equal to the simple coefficient of correlation between Z and X, if the coefficient of correlation between Z and Y and coefficient of correlation between X and Y are zero. (b) This is the wrong answer. The partial coefficient of correlation between Z and X when Y is kept constant is not equal to simple coefficient of correlation between Z and Y for the given condition. (c) This is the wrong answer. The partial coefficient of correlation between Z and X when Y is kept constant is not equal to simple coefficient of correlation between X and Y for the given condition. (d) This is the wrong answer. The partial coefficient of correlation between Z and X when Y is kept constant is not equal to partial coefficient of correlation between Z and Y when X is kept constant for the given condition. (e) This is the wrong answer. The partial coefficient of correlation between Z and X when Y is kept constant is not equal to partial coefficient of determination between Z and X when Y is kept constant for the given condition.

117


339. (c)

(a)

This is a wrong answer. The standard error of the estimate is not calculated using the number of degrees of freedom as n – 2.

(b) This is a wrong answer. The standard error of the estimate is not calculated using the number of degrees of freedom as n – k. (c)

This is the right answer. In a multiple regression with n number of observations and k independent variables, the standard error of the estimate is calculated using the number of degrees of freedom as n – k –1.

(d) This is a wrong answer. The standard error of the estimate is not calculated using the number of degrees of freedom as n – k +1. (e)

This is a wrong answer. The standard error of the estimate is not calculated using the number of degrees of freedom as n + k –1.

340. (d)

(a)

This is a wrong answer. This problem is not referred as Random variations.

(b) This is a wrong answer. This problem is not referred as Non-random variations. (c)

This is a wrong answer. This problem is not referred as irregular variations.

(d) This is the right answer. In multiple-regression analysis, the regression coefficients often become less reliable as the degree of correlation between the independent variable increases. This problem is referred by statisticians as Multicollinearity. (e)

This is a wrong answer. This problem is not referred as cyclical variations.

341. (e)

a&b. The negative multiple regression coefficients does not necessarily indicate the presence of multicollinearity. c&d. A strong correlation between the dependent variable and any one of the independent variables does not necessarily indicate multicollinearity. e.

Multicollinearity indicates the presence of a strong correlation between the independent variables in a multiple regression relationship

342. (c)

a.

The coefficient of multiple correlation always takes values in the range of 0 and 1.

b.

It is the positive square root of the coefficient of multiple determination.

c.

It depends upon the coefficient of correlation between the independent variables.

d&e. It depends upon the coefficient of correlation between the dependent variable and each of the independent variables 343. (a) a. b. c. d. e.

A multiple regression equation has only one dependent variable. From above we can see that b is false. A multiple regression equation has more than one independent variable. In a multiple regression equation the regression coefficients need not necessarily be 1. For given values of the independent variables the estimated values of the dependent variable need not necessarily be equal to the observed values

Index Numbers 344. (d) Laspeyres Price Index = ∑[P1Q0/∑P0Q0] x 100 = [15000 + 1400 + 3400 + 800]/[7500 + 800 + 1200 + 500] x 100 = [20600/10000] x 100 = 206. 345. (b) Laspeyres Quantity Index = [∑P0Q1/∑P0Q0] x 100 = [10500 + 560 + 720 + 700]/[7500 + 800 + 1200 + 500] x 100 = [12480/10000] x 100 = 124.80. 346. (a) Paasche’s Quantity Index = [∑P1Q1/∑P1Q0] x 100 = [21000 + 980 + 2040 + 1120]/[15000 + 1400 + 3400 + 800] x 100 = 25140/20600 x 100 = 122.04.

118

Part I

347. (d) Unweighted average of relatives price index = ∑(P1/P0 x 100)/n = [(30/15 x 100) + (14/8 x 100) + (17/6 x 100) + (16/10 x 100)]/4 = 535/4 = 818.33/4 = 204.58. 348. (a) Weighted average of relatives price index (when the base year values are used as weights) = ∑[(P1/P0) x 100)(P0Q0)]/∑P0Q0 = ∑(P1/P0) x P0Q0 x 100/∑P0Q0 = ∑P1Q0/∑P0Q0 x 100, which incidentally is the weighted aggregate price index using Laspeyres method. The weighted average of price relatives using base values is the same as the weighted aggregate of actual prices (Laspeyres method) only when the arithmetic mean is used for averaging the relatives and the base year values are used as weights. If either of these conditions are not satisfied, such a comparison would not be possible. 349. (e) All of the given alternatives are true for Laspeyres, Paasche and Fisher Ideal Index. 350. (c) If Paasche index for a set of prices is higher than the using the Laspeyres method then there is a trend towards more expensive goods as calculation of Paasche index. In general, Paasche index reflects the change in the aggregate value of the current year’s (given period’s) list of goods when valued at base period prices and Laspeyres price index calculates the changes in the aggregate value of the base year’s list of goods when valued at current year prices. 351. (b) Adjusted salary = 11,000 x 100/197 = 5583.75, which is Rs.1,416.25 less than the salary in 1976. Thus, the adjusted salary has decreased by 20 percent (that is, 1,416.25/7,000 x 100). 352. (b) The value index number as described earlier is a combination index which combines price and quantity changes. The value index, an aggregate of all values, measures the changes in values in the base year and the values in the given year. 353. (b) The difference between the Paasche index and Laspeyres index reflects the change in consumption patterns of the commodities. Generally, Laspeyres and Paasche methods tend to produce opposite extremes in index values computed from the same data. The use of Paasche index requires the continuous use of new quantity weights for each period considered. By contrast to Laspeyres index, Paasche index generally tends to underestimate the prices or has a downward bias. Because people tend to spend less on goods when their prices are rising, the use of the Paasche which bases on current weighting, produces an index which does not estimate the raise in prices rightly showing a downward bias. 354. (e) All of the given alternatives are the cause of distortion of index number. 355. (a) Unweighted aggregates price index is equal to ∑P1/∑P0 x 100, where P1 is the price in the current/given year; P2 is the price in the base year; and n is the number of products/items in the composite. 356. (c) The link relatives calculated by using the chain base method, enable comparisons over successive years. There is not much significance in business to the comparisons with the current period and remote past (the base year) and as such the usage of chain index numbers is ideal. 357. (a) An unweighted aggregates index is calculated by totaling the current year/given year’s elements and then dividing the result by the sum of the same elements during the base period. It is the simplest form of index numbers. 358. (e) FEDAI stands for Foreign Exchange Dealers’ Association of India and is not a source of index numbers . 359. (e) Seasonal variations have minimal impact on chain index numbers. 360. (d) Index numbers are one of the most widely used statistical indicators. There are various types of index numbers that are used to measure the economic activity of a country. Since they are very much useful in measuring the economic activity of a country, they are also called barometers of economic activity. Index numbers have a significant role in trade and industry. They are extensively used in indicating the price levels in an economy. 361. (c) Factor reversal test is suggested by Prof. Irving Fisher. According to him, just as each formula should permit the interchange of the two time periods without giving inconsistent results, it ought to permit interchanging of the prices and quantities without giving inconsistent result, i.e. the two results multiplied together should give the true value ratio. Fisher’s ideal index satisfies the factor reversal test.

119


362. (c) (334 – 310)/310 x 100 = 7.74%. 363. (a) Time reversal test has been developed by Prof. Irving Fisher. The test implies that for a price/quantity index, if the time periods are reversed, the resulting index should be the reciprocal of the original price/quantity index. 364. (b)Laspeyres price index is a weighted aggregates price index. Alternatives (a), (c) and (d) are true. 365. (b) 163/141 x 100 = 115.602. 366. (b) 104 x 106 = 105. [Hint: Fisher’s Ideal Index is the geometric mean of the Laspeyre and

Paasche indices]. 367. (b) Index number does not have a unit of measurement. It expressed in the form of numbers only. 368. (d) 102.41 x 103.2 = 102.8 [Hint: Fisher’s Ideal Index is the geometric mean of the

Laspeyre and Paasche indices]. 369. (c) Real Purchasing Power of the rupee = (CPIbase year/CPIcurrent year) x Rs.1 = (100/250) x 1 = 0.4 or 40 paise. 370. (b) 120 x 135 = 127.27 or 127.3 (approximately) [Hint: Fisher’s Ideal Index is the geometric

mean of the Laspeyre and Paasche indices]. 371. (c) Unweighted aggregates price index is equal to ∑P1/∑P0 x 100, where P1 is the price in the current/given year; P2 is the price in the base year; and n is the number of products/items in the composite. 372. (b) 108 x 112.4 = 110.17 or 110.2 (approximately) [Hint: Fisher’s Ideal Index is the

geometric mean of the Laspeyre and Paasche indices]. 373. (c) Fisher’s Ideal Index is the geometric mean of the Laspeyre and Paasche indices. 374. (d) Paasche’s Price Index = ∑P1Q1/∑P0Q1 x 100 = 365/245 x 100 = 148.98. 375. (c) The value index number as described earlier is a combination index which combines price and quantity changes. Because of the difficulties experienced in price and quantity indices, the usage of the value index has been suggested. The value index number can be calculated by the following formula, =

∑ P1 Q1 x 100 ∑ P0 Q 0

Where, P1, Q1, P0 and Q0 follow the usual notations. 376. (d) The consistency of the index numbers has been tested over the years and the most important of these tests are (i) the time reversal test, (ii) the factor reversal test, and (iii) the circular test. 377. (c) The index of industrial production is an example of Quantity index. 378. (b) Real wage = (Nominal wage/Consumer price index) x 100 = (945/108) x 100 = Rs.875. 379. (c) 127 x 130 = 128.49 [Hint: Fisher’s Ideal Index is the geometric mean of the Laspeyre

and Paasche indices]. 380. (c) (a) This is true for index numbers. Index numbers are expressed as a percentage. (b) This is true for index numbers. Index number for the base year is always 100. (c) This is not true for index numbers. Index number is a ratio of the price, quantity or value of the commodities under study and therefore does not have any unit. (d) This is true for index numbers. Index number is calculated as a ratio of the current value to a base value. (e) This is not the correct answer. 120

Part I

381. (e) (a) This is true for index numbers. Index numbers help in establishing trends. Index numbers when analyzed reveal a general trend of the phenomenon under study. (b) This is true for index numbers. Index numbers help in policy making. The dearness allowance paid to the employees is linked to the cost of living index, generally the consumer price index. (c) This is true for index numbers. Usually index numbers are used to determine the purchasing power of the money. Suppose the consumer price index for the urban nonmanual employees increased from 100 in 1984 to 202 in 1992, the real purchasing power of the rupee can be found out as 100/202=0.495. It indicates that if rupee was worth 100 paise in 1984 its purchasing power is 49.5 paise in 1992. (d) This is true for index numbers. Index numbers help in deflating time series data. It plays an important role in adjusting the original data to reflect reality. For example, nominal income can be transformed into real income by using income deflators. (e) This is not true about index numbers. The value index is a combination index. It combines price and quantity changes to present a more spatial comparison. The value index as such measures changes in net monetary worth. Though the value index enables comparison of value of a commodity in a year to the value of that commodity in a base year, it has limited use. Its limited use is owing to the inability of the value index to distinguish the effects of price and quantity separately. 382. (d)

(a)

This is a wrong answer. Marshall-Edgeworth method uses both the current year as well as the base year quantities as weights. Therefore, it is not biased towards the base year or the current year.

(b) This is a wrong answer. Fisher’s ideal index is free from any bias. Both the current year and base year prices and quantities are taken into account by this index. (c)

This is a wrong answer. Laspeyres index tends to overestimate the rise in prices or has an upward bias.

(d) This is the right answer. The Paasche index generally tends to underestimate the prices and has a downward bias. Because people tend to spend less on goods when their prices are rising, the use of the Paasche index which bases on current weighting, produces an index which does not estimate the rise in prices rightly showing a downward bias. Since all prices or all quantities do not move in the same order, the goods which have risen in price more than others at a time when prices in general are rising, will tend to have current quantities relatively small than the corresponding base quantities and they will thus have less weight in the Paasche index. (e)

This is a wrong answer.

383. (c)

(a)

This is the wrong answer. According to the factor reversal test the product of the original index and the index obtained through reversing the factors is not equal to one.

(b) This is the wrong answer. According to the factor reversal test the product of the original index and the index obtained through reversing the factors is not equal to one hundred. (c)

This is the right answer. According to the factor reversal test if we interchange the price and quantity terms in an index number then the product of the original index number and the index number obtained by reversing the factors should be equal to the value index.

(d) This is the wrong answer. According to the factor reversal test the product of the original index and the index obtained through reversing the factors is not equal to the Fisher’s ideal index. (e)

This is the wrong answer. According to the factor reversal test the product of the original index and the index obtained through reversing the factors is not equal to the chain index number. 121


384. (c) a. b. c. d. e. 385. (b) a.

b. c. d. e. 386. (e) a. b. c. d. e.

Fisher’s ideal price index uses both current year and base year quantities. Laspeyres price index uses only base year quantities. Paasche’s price index uses only current year quantities. Marshall-Edgeworth price index uses both current year and base year quantities. Fixed weight aggregates price index uses the weights from a representative period. The fixed weight aggregates price index allows the flexibility to select the base period for comparison of prices, because it uses the weights from a representative period. Paasche’s price index tends to underestimate the rise in prices. Unweighted aggregates price index numbers do not link the price changes to the consumption levels. Laspeyres price index has an upward bias. The weighted average of relatives use values (and not prices) as weights. Fisher’s ideal price index considers base year prices. Fisher’s ideal price index considers base year quantities. Fisher’s ideal price index considers current year prices. Fisher’s ideal price index considers current year quantities. Fisher’s ideal price index is the geometric mean of the Laspeyres and Paasche’s price indices.

Time Series Analysis 387. 388. 389. 390.

(d) (10000/4) x (155/100) = 3875 [Hint: 4 quarters = 1 year]. (a) (3/2.76) x 100 = 108.69. (a) Unadjusted index for the quarter will be simply 100 so the correct answer is (a). (c) Adjusted seasonal index = Unadjusted seasonal index x Adjusting factor = 112.36x = 113.45; thus, x = 113.45/112.36 = 1.009 or 1.01 (approximately).

391. (a) ∑XY = (–14) + (–8) + 0 + 6 + 16 = 0

∑X2 = –22 + –12 + 02 + 12 + 22 = 10 Thus, b = ∑XY/∑X2 = 0/10 = 0.00. 392. (b) Adjusting factor = 0.9234 = 400/x; thus x = 400/0.9234 = 433.18. 393. (c) Irregular variation has been dealt more in a theoretical fashion ascribing to the fact that it is highly inconsistent and non-existence of a mathematical model to deal with it. Analysis for seasonal variation can be done without the isolation of the cyclical trend, we are able to ∧

conduct. The relative cyclical residual can be expressed as

Y−Y ∧

x 100 , so the correct

Y

alternative is (c) only. 394. (a) The value of x for 1996, 1997 and 1998 are –1, 0 and 1 respectively. Thus, the value of x for the year 1999 is 2. Y = 2.65 + 1.1(2) = 4.85. 395. (e) With the help of time series analysis Secular trend, Cyclical fluctuation, Seasonal variation, and Irregular variation can be identified. 396. (a) The moving average of two data points of the observations 4, 5, 7, 8, 5 and 2 are (4 + 5)/2, (5 + 7)/2, (7 + 8)/2, (8 + 5)/2 and (5 + 2)/2, that is 4.5, 6, 7.5, 6.5 and 3.5. ∧

397. (d) When the ratio of actual values (Y) and the corresponding estimated values ( Y ) is multiplied by 100, we are expressing the cyclical variation component as a percent of trend. ∧

Mathematically, we express it as ((Y/ Y ) x 100). When percent of trends are calculated and plotted on a graph, we can observe the variations from the trend line. 122

Part I

398. (a) Under secular trend of variation, we look at the long-term behavior of the variable. This is important in the sense that one should be able to conclude whether the value of the variable has been increasing or decreasing over the years, keeping aside the variation observed in the individual years, which constitute the series. ∧

∧

399. (b) Relative Cyclical Residual = (Y – Y )/ Y . 400. (d) Under secular variation, we look at the long-term behavior of the variable. Secular trend can be utilized for projecting the trend into the future so the correct answer is alternative (d). 401. (e) Irregular variation is, the movement of the variable is random in nature without consistency and therefore, highly unpredictable. Since this type of irregularity exists for very short durations, the period under consideration will be of days, weeks and at the most of months. 402. (a) Upper control limit = x + 3σ x and lower limit = x – 3σ x

Hence, upper control limit – lower control limit = 6σ x i.e., 6σ x = 0.6; i.e., σx = 0.100. 403. (c) Under Seasonal variation, we observe that the variable under consideration shows a similar pattern during certain months of the successive years. An example of seasonal variation would be an increase in water borne diseases during rainy season. That is, this phenomenon seems to be repeating itself every year in a regular fashion. Usually, the time period over which this variation is considered can consist of days, weeks, months and at the most one year. 404. (b) By cyclical variations, we refer to the long-term movement of the variable about the trend line. A more appropriate example of cyclical variation would be the pattern of business cycle. 405. (c) Usually, under Seasonal Variation the time period over which this variation is considered can consist of days, weeks, months and at the most one year. 406. (c) Under secular variation, we look at the long-term behavior of the variable. 407. (e) Relative Cyclical Residual Measure ∧

∧

= (Y – Y )/ Y x 100 = [(86.4 – 84.5)/84.5] x 100 = 2.25. 408. (c) Relative Cyclical Residual Measure ∧

∧

= (Y – Y )/ Y x 100 = [(54 – 50)/50] x 100 = 400/50 = 8.00. 409. (a)

(a)

This is the right answer. A time series consisting of annual data for longer periods is depicted by trend lines. This facilitates us to isolate the component of secular trend variation from the series and examine it for cyclical, seasonal and irregular component. We use “Residual Method” to isolate the cyclical variation component. This method uses one of the two measures namely Percent of trend and relative cyclical residual measure. Both these measures are expressed in terms of percentage.

(b) This is the wrong answer. Cyclical component of the time series is not isolated by Linear regression analysis. (c)

This is the wrong answer. Cyclical component of the time series is not isolated by nonlinear regression analysis.

(d) This is the wrong answer. Seasonal component of the time series is isolated by ratio to moving average method. (e)

This is the wrong answer. Cyclical component of the time series is not isolated by multiple regression analysis. 123


410. (c)

(a)

This is the wrong answer. The variation in the demand of ice creams cannot be explained by secular trend.

(b) This is the wrong answer. The variation in the demand of ice creams cannot be explained by cyclical variation as the variation is observed in different seasons in a year. (c)

This is the right answer. The variation in the demand of ice creams can be well explained by seasonal variation. The variations observed in different seasons of a year can be best explained with the help of seasonal variation component. To find a long term forecasting first the time series has to be deseasonalized to get an accurate estimation.

(d) This is the wrong answer. The variation in the demand of ice creams cannot be explained by Irregular variation. (e)


411. (d)

(a)

This is a wrong answer. The movement of dependent variable above and below the secular trend line over periods longer than one year is not known as Seasonal variation.

(b) This is a wrong answer. The movement of dependent variable above and below the secular trend line over periods longer than one year is not known as Secular variation. (c)

This is a wrong answer. The movement of dependent variable above and below the secular trend line over periods longer than one year is not known as Irregular variation.

(d) This is the right answer. The movement of dependent variable above and below the secular trend line over periods longer than one year is known as Cyclical variation. (e)

This is a wrong answer. The movement of dependent variable above and below the secular trend line over periods longer than one year is not known as Regular variation.

412. (e)

i.

This is true for the ratio to moving average method. This method is used to identify the seasonal trend in a time series.

ii.

This is true for the ratio to moving average method. An index is constructed with a base of 100.

iii.

This is true for the ratio to moving average method. The magnitude of seasonal variations is measured by the individual deviations from the base of 100.

iv.

This is true for the ratio to moving average method. The index is used to nullify the seasonal effects in the time series.

v.

This is not true for the ratio to moving average method. We divide the original data points with the relevant seasonal index to deseasonalize the time series.

413. (d)

i.

The irregular variations cannot be separated from a time series using the percent of trend measure.

ii.

The secular trend is not separated using the percent of trend measure.

iii.

The seasonal variations are not separated using the percent of trend measure.

iv.

The cyclical variations can be separated using the percent of trend measure

414. (c)

124

i.

The seasonal variations show the seasonal patterns in the variable.

ii.

The cyclical variations show the cyclical patterns in the variable.

iii.

Secular trend shows the long term behavior of the variable.

iv.

Irregular variation shows the random variations in the variable

Part I

Quality Control 415. (c) Only alternative (c) is correct. Other alternatives relates to quality control is not correct as they have not significant for quality control. For example, generally high quality items are costly compared to items of lower quality this statement is not correct as high quality items sometimes may be cheaper than lower quality. 416. (e) In x charts, a process is said to be in control, when all the points are within the control limits (i.e. upper and lower limits). A small deviation implies that the point is very much within the control limits. 417. (c) Lower control limit

= R−

⎛ 3d ⎞ 3d 3 R = R ⎜⎜1 − 3 ⎟⎟ d2 d2 ⎠ ⎝

The lower control limit as given by the equation would be zero if the value of ‘n’ that is the sample size is equal to or lower than 6. 418. (d) In statistical process control, the data set which can take on only two values is called an attribute. Thus, we can conclude that both (a) and (b) have the characteristics of an attribute. 419. (b) Lower control limit

= R−

⎛ 3d ⎞ 3d 3 R = R ⎜⎜1 − 3 ⎟⎟ d2 d2 ⎠ ⎝

The lower control limit as given by the equation would be zero if the value of n that is the sample size is equal to or lower than 6. This can also be verified by using the data given; LCL = 5.3 – [(3 x 0.848 x 5.3)/2.534] = 0.

∑x ∑x or x = . k nxk Both gives us the same answer and it can be seen that the second formula is derived from the first one. Here ‘n’ denotes the number of samples that are randomly selected on a day and k denotes the number of days we sample the pistons.

420. (d) The grand mean, that is x , is calculated by employing either x =

421. (e) UCL (p-chart) = p + 3 pq/n = 0.23 + 3 [(0.23 x 0.77)/48] = 0.23 + 3(0.0607)

= 0.23 + 0.18 = 0.41. 422. (e) All of the given alternatives are the example of control charts, so the correct answer is (e). 423 (d) UCL = x + 3σ / n = 1.2 + 3(0.1)/ 5

= 1.2 + 0.134 = 1.334. 424. (c) The value of p is the value of central line. Thus, 0.23 is the value of central line. 425. (e) UCL = x + 3(standard deviation of sampling distribution) = 10 + 3(0.02) = 10.06. [Hint:

σ / n = standard deviation of sampling distribution]. 426. (a) As against the x (mean) and R (range) charts that deal with quantitative aspects, these charts deal with the attributes that are qualitative factors possessed by the data. In statistical process control, the data set that can take only two values is called an attribute. 427. (c) In R chart we looks at control charts which are directed towards monitoring the variability in case of individual characteristics like the variability in the diameter of the pistons. This is based on the logic that less the variability in the products manufactured, more the consistency and hence higher the quality. In these charts, the time variable is represented on the X–axis and the sample range on the Y-axis.

125


428. (b) As against the x and R charts that deal with quantitative aspects, p charts deal with the attributes that are qualitative factors possessed by the data. 429. (b) (a) This is a wrong answer. The x charts are constructed using the mean value as the central limit and three standard deviations from the mean as upper and lower control limits. (b) This is the right answer. R charts are used to monitor the variability in the individual characteristics of the product. The control limit is the mean range and the upper and lower control limits are constructed with the standard deviation in the range values. (c) This is the wrong answer. p charts deal with the attributes that are qualitative in nature. The measure that is applied here is the proportion of the data points satisfying the given criteria and the proportion of the data points that does not satisfy the criteria. (d) This is the wrong answer. (e) This is the wrong answer. 430. (e) (a) This is true about statistical process control. Quality standards in manufacturing process and the service industry can be achieved through the application of statistical methods. (b) This is true about statistical process control. The variability present in the manufacturing process can either be eliminated completely or minimized to the extent possible. (c) This is true about statistical process control. The variation in the specifications of the product due to wearing of the machine is a systematic or non-random variation. (d) This is true about statistical process control. The variation due to an error made by the personnel in the measurement of the product specifications is a random variation. (e) This is not true about statistical process control. If the process is ‘out of control’, it indicates the presence of non-random pattern in the system. The management should first identify the cause of that variation and eliminate it. This elimination or the reduction of the systematic variation results in the process being brought ‘in-control’. Once this is done, the whole process can be redesigned to improve or reduce the incidence of random or inherent variability. 431. (c) (a) This is a wrong answer. If we observe that the sample means are hugging the center line then the deviations are minor and uniform in nature. This indicates that the variation has been reduced to great extent. (b) This is a wrong answer. If the observations are such that it can be described to be ‘hugging the control limits’, then this refers to the deviations being uniform in nature but having large magnitude. This indicates that the means of two different populations are being observed. (c) This is the right answer. If the observations can be described as jumps in the level around which the observations vary, then this pattern indicates that the process mean has been shifting. (d) This is a wrong answer. If some of the points lie outside the upper or lower control limits or if there is an increasing or decreasing trend in the observations, we conclude that the process is out of control. (e) This is a wrong answer. 432. (a) (a) This is the right answer. In a control chart there are some observations that lie outside the upper and lower control limits. These points are referred to as the Outliers. (b) This is the wrong answer. Qualitative variables with only two categories are termed as attributes. This is monitored with the help of p chart. (c) This is the wrong answer. This is another name of random variation. (d) This is the wrong answer. this is the non-random, systematic variability in a process. It usually can be corrected without redesigning the entire process. (e) This is the wrong answer. This is the random variability inherent in the system. It usually cannot be reduced without redesigning the entire process.

126

Part I

433. (b)

i.

‘Hugging the control limits’ indicates that two different populations are being observed.

ii.

‘Hugging the center line’ indicates that the variations have been reduced significantly.

iii.

Cycles indicate the possibility of presence of random variations in the process.

iv.

Increasing trend indicates that the process mean is increasing.

v.

Decreasing trend indicates that the process mean is decreasing.

434. (b)

a.

In a p chart, the center line is drawn on the basis of overall sample proportion.

b.

In a p chart, the lower control limit can only be greater than or equal to zero. c, d & e. The upper and lower control limits are fixed by adding and subtracting the estimated standard error of proportion from the overall sample proportion

Chi-Square Test and Analysis of Variance 435. (e) Only alternative (a) and (b) are correct. F distribution is not a probability distribution so that the area under the curve need not be 1. So alternative(c) is not correct and for small numbers of DOF the curve is skewed extremely to the right and as the number of degrees of freedom increases the distribution tends to become symmetrical so alternative (d) is not correct. 436. (b) F distribution is not a probability distribution so that the area under the curve need not be 1 so the alternative (b) is not correct. 437. (a) The expected frequency for the cell (white collar, neutral) is [(21 + 30 + 25) x (26 + 30 + 28)]/244 = (76 x 84)/244 = 26.163. 438. (e) While calculating between-column variance, it is not necessary that the samples should have the same size. So alternative (a) is not correct. Alternative (c) is also not correct so the correct answer is alternative (e). 439. (d) The value of F = Between-column variance/Within-column variance. Since the value of F statistic is less when between-column variance = 15 and within-column variance = 14, this sample has more likely chances to be accepted. (Because the value of F is very less.) 440. (e) The value of F(n, d, α) = 1/F(d, n, 1 – α), where n is the degree of freedom in the numerator, d is the degree of freedom in the denominator, and α is the significance level. Thus, the value of F(n, d, 0.95) is equal to inverse value of F(d, n, 0.05). 441. (b) DOF = (Number of rows – 1) (Number of columns – 1) = (2 – 1) (2 – 1) = 1. 442. (c)

χ2

= Σ

(f o − f e ) 2 fe

Where, fo is the observed frequency and fe is the expected frequency. 443. (b) ANOVA allows us to test whether the difference(s) among more than two sample means are significant or not. 444. (d) Chi-square statistic = ∑[(f o − f e ) 2 /f e ] = (2 2 /3) + (2 2 /4) = 7/3 . 445. (b) DOF = (Number of rows – 1)(Number of columns – 1) = (3 – 1)(3 – 1) = 4. 446. (d) DOF = (Number of rows – 1)(Number of columns – 1) = (4 – 1)(3 – 1) = 6. 447. (e) DOF = (Number of rows – 1)(Number of columns – 1) = (5 – 1)(4 – 1) = 12. 448. (d) F distribution may have more than one mode, so the alternative (d) is false.

127


449. (d) (a) This is true for chi-square distribution. There will be different chi-square distribution for each degrees of freedom. (b) This is true for chi-square distribution. If the number of degrees of freedom is small, the curve is skewed to the right (c) This is true for chi-square distribution. As the number of degrees of freedom increases the curve also becomes symmetrical. (d) This is not true for chi-square distribution. The area under the chi-square distribution remains equal to one irrespective of the number of degrees of freedom. (e) This is the wrong answer. 450. (d) (a) This is the wrong answer. Analysis of variance is used to test the equality of means for samples drawn from more than two populations. (b) This is the wrong answer. Regression analysis is used to estimate the future value of the dependent variable for a given independent variable with the help of a set of data. (c) This is the wrong answer. Correlation analysis is used to find the degree of association between two variables. (d) This is the correct answer. We use chi-square test as a distributional goodness of fit. The managers can employ the chi-square distribution to verify the appropriateness of the distribution employed by them and conclude whether any significant differences exist between the observed and theoretical distribution they have employed. (e) This is the wrong answer. We use marginal analysis for finding the level at which the expected marginal loss is equal to expected marginal profit. 451. (e) (a) This is a wrong answer. The number of degrees of freedom does not depend on the sample size. (b) This is the wrong answer. The number of degrees of freedom does not depend on the ratio of sample size and the population. (c) This is the wrong answer. The number of degrees of freedom does not depend on the number of rows in the contingency table only. (d) This is the wrong answer. The number of degrees of freedom does not depend on the number of columns in the contingency table only. (e) This is the right answer. When the chi-square distribution is used as a test of independence, the number of degrees of freedom is related to both the number of rows and the number of columns in the contingency table. 452. (b) a. Chi-square test is suitable for testing whether there is a significant difference between more than two sample proportions. b. ANOVA is suitable for testing whether there is a significant difference between more than two sample means. c, d & e. Simple and multiple regression analysis, are techniques for establishing relationship between two or more variables. Correlation analysis is the technique of studying the nature and strength of relationship between two or more variables. 453. (b) a. The observed frequencies may be less than the expected frequencies. Hence, this is not the correct reason. b. The differences between the observed and expected frequencies are squared. Hence, the chi-square statistic cannot be negative. c. The chi-square statistic is not based upon the absolute value of the difference between the observed frequencies and expected frequencies. d & e. The chi-square statistic is not based upon the summation of the squares of the observed or expected frequencies

128

Part I

454. (d) a. The F statistic is not calculated on the basis of the mean of the largest sample. b. The F statistic is not calculated on the basis of the mean of the smallest sample. c. The F statistic is not calculated on the basis of the standard deviation of the smallest sample. d. The F statistic is calculated on the basis of two estimates of the population variance viz., the estimated population variance based on the variance among the sample means and the estimated population variance based on the variance within the samples. e. The F statistic is not calculated on the basis of the variance of the largest sample.

Simulation 455. (c) Daily Demand Probability (’00) 2 0.3 3 0.6 5 0.1

Cumulative Probability 0.3 0.9 1.00

Random numbers allotted 00 – 29 30 – 89 90 – 99

Since the random number generated was 45 (which fall in between 30 – 89), the daily demand is 300. Closing inventory = Opening inventory – sales = 1000 – 300 = 700. 456. (d) Since the random number generated was 90 (which fall in between 89 – 99), the daily demand is 500. Closing inventory = Opening inventory – Sales = 1000 – 500 = 500. 457. (b) When decisions are to be taken under conditions of uncertainty, simulation can be used. Simulation as a quantitative method requires the setting up of a mathematical model which would represent the interrelationships between the variables involved in the actual situation in which a decision is to be taken. 458. (b) System is the segment that is to be to be studied or understood to draw conclusions. In the illustration given above, the market for the product together with the firms’ production process constitutes the relevant system. Only after the system is defined, can we identify the variables which interact with one another in the system and establish their relationships mathematically. 459. (e) Cumulative X Probability Random numbers allotted Probability 5 0.10 0.10 00 – 09 10 0.05 0.15 10 – 14 15 0.05 0.20 15 – 19 20 0.10 0.30 20 – 29 25 0.30 0.60 30 – 59 30 0.20 0.80 60 – 79 35 0.20 1.00 80 – 99 Since the random number generated was 80 (which fall in between 80 – 99), the value of X in this run is 35.

129

Frequently Used Formulae Probability Distribution and Decision Theory 1.

2.

Expected Value, E[x] = ∑ x P(x) Where, x is a random variable and P(x) is probability of x. Covariance a. For a population consisted of paired ungrouped data points {x, y} ∑ (x − μx) (y − μy) Cov xy = N Where, μ x is the arithmetic mean of {x}

μ y is the arithmetic mean of {y}

N is the number of observations in each population. For a paired sample {x, y} Cov xy =

∑ (x − x)(y − y) n −1

Where, x is the arithmetic mean of sample {x}

y is the arithmetic mean of sample {y} n is the number of observations in each sample. b.

For grouped data of paired population Cov xy =

c.

Σf (x − μ x ) (y − μ y )

Σf Where, f is the frequency of the corresponding (x, y) values. Given a probability distribution of paired data {x, y} Covxy = Σ [x – E(x)] [y – E(y)] P(x, y)

Where, P(x, y) is the joint probability of x and y E(x) is the expected value of x E(y) is the expected value of y. 3.

E(a1x1 + a 2 x 2 ) = a1E(x1 ) + a 2 E(x 2 ) .

4.

2 2 V(a1 x1 + a 2 x 2 ) = a1 V(x1 ) + a 2 V(x 2 ) + 2a1a 2 Cov (x1, x 2 )

If Cov (x1, x2) = 0 then x1, x2 are uncorrelated. If x1, x2 are independent,

V(a1 x1 + a 2 x 2 ) = a12 V(x1 ) + a 22 V(x 2 ) . 5.

For a binomial distribution,

⎛ n ⎞ x n−x ⎟ P q ,x =0,1,2,....n ⎝x⎠

P(X = x)= ⎜

Where,

⎛n⎞ n! ⎜ ⎟= x − x)! x!(n ⎝ ⎠ Where, x is the number of success in n trails, p is the probability of success and, q is the probability of failure. 6.

For a hypergeometric distribution,

⎛M⎞ ⎛ N − M ⎞ ⎜ x ⎟ ⎜ n− x ⎟ ⎝ ⎠⎝ ⎠ , x = 0, 1, 2, ..... n P(X = x) = ⎛ N⎞ ⎜n⎟ ⎝ ⎠ Where, x is the number of successes. For a hypergeometric distribution,

μ = E(x) =

nM N

σ 2 = V(x) =

7.

nM(N − M)(N − n) N 2 (N − 1)

Standard Normal Variable, z =

x −μ σ

Where, x is any random variable μ is the mean of the distribution of the random variable σ is the standard deviation of the distribution

Z is the number of standard deviation from x to the mean of the distribution and is known as the z-score or standard score. 8.

In a t-distribution, t =

x− μ S/ n

Where,

x is the sample mean μ is the population mean S is sample standard deviation n is the sample size. 9.

If MP : Marginal Profit ML : Marginal Loss ‘P’ is the probability of generating the additional profit by increasing our activity level by one unit. ‘1 – P’ is the probability of incurring loss by increasing our activity level by one unit. then, Expected (MP) = P x MP Expected (ML) = (1 – P) x ML 131

P* =

ML ML+MP

* P represents the minimum required probability of selling at least one additional unit to justify the stocking of that additional unit.

Statistical Inferences I.

Standard Error for 1.

Sample mean X is σ =

2.

Sample proportion, p

σ

x

σp =

n

pq where q = 1 – p n

σ2 2n

3.

Sample standard deviation, σs =

4.

Difference of two sample means X1 and X 2 i.e. σ x1 − x 2 =

σ12 σ22 + , n1 n 2

Where X1 and X 2 are the means of two random samples of sizes n1 and n 2 drawn from two populations with standard deviation σ1 and σ 2 respectively. 5.

Difference of two proportions:

⎛ ⎞ pˆ qˆ pˆ qˆ ⎜ σ p1 − p ⎟ = n1 + n ⎝ 2 ⎠ 2 Where p1 and p2 are the proportions of two random samples of sizes n1 and n2 drawn from two populations and

pˆ =

n1p1 + n 2 p2 ˆ and qˆ = 1 − p, n1 + n 2

Where pˆ is the estimate of the overall proportion of success in the combined populations using combined proportions for both the samples. 6.

For a finite population of size N, when a sample is drawn without replacement, σx =

σ n

N−n N −1

If the value of the sampling fraction correction factor

II.

Test Statistic, Z =

N−n N −1

n N

is less than 0.05, then finite population

need not be used.

X −μ Standard error of X

Confidence interval for large sample = [X − 2σ, X + 2σ] 132

⎡ ⎛ s Confidence interval for small sample = ⎢ X + t ⎜ ⎝ n ⎣

⎞ ⎛ s ⎞⎤ ⎟,X − t ⎜ ⎟⎥ ⎠ ⎝ n ⎠⎦

Linear Regression 1.

Karl Pearson’s correlation coefficient, r=

cov(X, Y) sXsY Σ (X − X) (Y − Y)

or r =

2

Σ (X − X) Σ (Y − Y)

2.

2

i.

When correlation coefficient = 0 then there is no relationship between X and Y

ii.

When correlation coefficient = – 1 then there is perfect positive linear relationship

iii.

When correlation coefficient = –1 then there is perfect negative linear relationship.

Rank correlation coefficient, n

R = 1−

6∑ Di2 i =1

n3 − n

Where n = Number of individuals ranked, Di is the difference in the ranks of the ith individual, i = 1, 2 ..... n. 3.

ˆ = a + bX, the normal equations are: na +b ∑ X= ∑ Y If the regression line, Y X and aΣX + bΣX 2 = ΣXY and b = n ∑ XY − ( ∑ X) ( ∑ Y) n ∑ X 2 − (∑ X) 2 and a = Y − bX .

4.

Standard error of the estimate for a regression equation is given by

ˆ 2 Σ(Y − Y) n−2

Se =

Where, Y

= Values of dependent variable

ˆ Y

= Estimated values from the estimating equation that correspond to each Y value

n

= Number of data points used to fit the regression line.

Se can also be calculated as 2

Se = 5.

ΣY − aΣY − bΣY n−2

Given the paired observation (X, Y), we can estimate the value Y in two ways i.

Y

ii.

ˆ = a + bx Y

Total variation in Y = Σ (Y − Y)

2

ˆ − Y) Variation explained by the model = Σ (Y

2

133

6.

Total Sum of Squares (TSS) = Σ (Y − Y)

2

ˆ − Y) Regression Sum of Squares (RSS) = Σ (Y

2

ˆ 2 Error Sum of Squares (ESS) = Σ (Y − Y)

TSS = RSS + ESS Coefficient of Determination, 2

R =

aΣY + bΣXY − nY 2

ΣY − nY

2

2

Under ANOVA (Analysis of Variance), F=

7.

RSS /1 ESS /(n − 2)

For a 2-asset portfolio, the standard deviation ( σ ) is given by

σ p = W12 σ12 +W22 σ 22 +2W1W2ρ12 σ1σ 2 Where W1 and W2 are the weights for the assets in the portfolio.

ρ12 is the coefficient of correlation between the returns of the two assets. σ1 and σ 2 are the standard deviations of the returns of the two assets respectively. ρ12 σ1 σ 2 = Covariance between returns of the two assets. 8.

2

Systematic risk = β x variance of the market returns Where β =

C ov (X, Y) V(X)

, where V(X) is variance of market returns.

Variance of returns of the security = V(Y) Unsystematic risk = Variance of returns of security – Systematic risk.

Multiple Regression 1.

In general the multiple regression equation is ˆ = A + B1X1 + B2X2 + B3X3 + ---- + BnXk Y B

2.

Where there are k independent variables X1, X2,--- Xk. ˆ ΣY = na + b ΣX + b ΣX 1

1

2

2

ˆ = aΣX + b ΣX + b ΣX X Σ X1Y 1 1 1 2 1 2 2

ˆ = aΣX + b ΣX X + b ΣX Σ X2Y 2 1 1 2 2 2

2

These three equations are called the “normal equations”. 3.

The standard error of the estimate for a regression equation is given by

ˆ 2 /(n − k −1) Se = Σ(Y − Y) Where, Y is the sample value of the dependent variable

ˆ is the corresponding estimate obtained by using the regression equation Y n = number of observations k = number of independent variables. 134

For the regression equation with two independent variables an alternative formula mainly utilizing data already collected for ascertaining the regression equation is

ΣY 2 − aΣY − b1ΣX1Y − b 2 ΣX 2 Y n − k −1

Se = 4.

Coefficient of multiple correlation between Y and both X1 and X2 is given by 2

2

2

R Y.X X = 1 − (ΣY − aΣY − b1ΣX1Y − b 2 ΣX 2 Y2 ) /(ΣY − (ΣY) / n ) 1

2

R Y.X X is the positive square root of the quantities under the ‘ 1

2

’ sign. It can take values in

the range 0 to 1. We can also find a coefficient of multiple determination R above coefficient of multiple correlation. R 5.

2

2 Y. X1X 2

which is the square of the

also takes values in the range 0 to 1.

Y. X1X 2

In the equation Y = a + b1 X1 + b2 X2, the partial correlation coefficient is given by R12.3 Where R 12.3 =

r12 − r13 . r23

(1 − r132 )(1 − r232 )

is the partial correlation coefficient between Y and X1 when X2 is kept constant.

Index Numbers 1.

Aggregates Method of Constructing an Index a.

Unweighted Aggregates Price Index =

ΣP1 ΣP0

x100

Where, ΣP1 = Sum of all elements in the composite for current year ΣP0 = Sum of all elements in the composite for base year.

b.

Weighted Aggregates Index i.

Laspeyres Price Index =

ΣP1Q 0

x 100

ΣP0 Q 0

Where, P1 = Prices in the current year P0 = Prices in the base year Q0 = Quantities in the base year. Laspeyres Quantity Index =

ΣQ1P0 ΣQ 0 P0

x 100

Where, Q1 = Quantities in the current year. ii.

Paasche’s Price Index =

iii. Fisher’s Ideal Index =

ΣP1Q1 x100 ΣP0 Q1

ΣP1Q0 ΣP0 Q0

Fisher’s Ideal Quantity Index =

x

ΣP1Q1 ΣP0 Q1

ΣQ1P0 ΣQ0 P0

x100

x

ΣQ1P1 ΣQ0 P1

x100

135

Marshall-Edgeworth Index =

iv.

ΣP1Q1

2.

Value Index Number =

3.

Average of Relatives Method

ΣP0 Q 0

Σ(Q 0 + Q1 )P1

x 100

Σ(Q 0 + Q1 )P0

x100

⎛ P1

Σ⎜ a.

Unweighted average of relatives method

=

⎝ P0

⎞

x100 ⎟

⎠

n

⎛ Q1

Σ⎜ Unweighted average of relatives Quantity index =

⎡⎛ P1

Σ ⎢⎜ b.

Weighted average of relatives price index =

⎣⎝ P0

⎝ Q2

⎞

x100 ⎟

⎠

n

⎞

⎤

⎠

⎦

x100 ⎟ (Pn Q n ) ⎥ ΣPn Q n

Where, Pn

= Prices in the fixed period (could also be base or current period)

Qn

= Quantities in the fixed period (could also be base or current period)

PnQn = Value in the fixed period (can be replaced by P0Q0 or P1Q1 as the case may be). Note: If weighted average of relatives price index is to be calculated with the base values, then it will be equal to Laspeyres Index. 4.

Chain Index Numbers

Chain Index for a given year = Averagelink relative of the given year x Chain index of previous year 100

Where, Link relative =

Price in a given period Previous year's price

x 100

Time Series Analysis 1.

Secular Trend ˆ = a + bX Using regression analysis, estimating equation is Y

Where, X is an independent variable and Y is a dependent variable. After coding (or translating time) is done a =Y b =

136

ΣxY Σx 2

Where, x = and x = 2.

(X − X)

if there are odd number of data points

2(X − X)

if there are even number of data points.

Cyclical Variation

i.

Percent of Trend Measure Y x100 ˆ Y ˆ represents estimated values. Where Y represents actual values and Y

Cyclical variation component =

ii.

Relative Cyclical Residual Measure ˆ Y−Y x100. Cyclical Component = ˆ Y

Quality Control 1.

X –Charts

a.

The lower and upper limits for an X –chart are μ x − 3σ x and μ x + 3σ x .

b.

When the mean and standard deviation are not known, then Lower control limit = X − A 2 R Upper control limit = X + A 2 R

Where, X=

1 k

A2 =

2.

ΣX 3

d2 n

R Charts Lower control limit = D3 R Upper control limit = D 4 R Where, ⎛ 3d ⎞ D3 = ⎜ 1 − 3 ⎟ d2 ⎠ ⎝ ⎛ 3d ⎞ D4 = ⎜1 + 3 ⎟ d2 ⎠ ⎝

3.

P Charts Lower control limit = μ p − 3σ p Upper control limit = μ p + 3σ p Where,

σp =

pq and μ p = P n

137

If value of P is not known, then we estimate it by using the overall sample fractions. ΣPj i.e. P = k Where, Pj is the sample fraction in the jth interval of time

and k is the sum of all samples considered.

Chi-Square Test and Analysis of Variance 1.

The chi-square statistic is given by

χ2 = ∑

(f0 − fe )2 fe

Where, f o is the observed frequency f e is the expected frequency.

If this value happens to be smaller we conclude that there is little difference between the actual and the expected frequencies and if the difference is larger we conclude that the actual and the expected values are not equal. 2.

Number of Degrees of Freedom = (Number of rows – 1) x (Number of columns – 1).

3.

The variance among the sample means or Between-Column Variance: s 2 =

4.

The variance within the sample: s 2 = Σ(x − x) n −1

5.

F ratio

=

or

=

F

2

Population variance obtained from the variance among the sample means (between column variance) Population variance obtained from the variance within the individual sample (within column variance) Between column var iance Within column var iance

6.

Degrees of freedom for the numerator = (k – 1).

7.

DOF for Denominator n

∑ (n k =1

Where,

8.

138

j

Σ(x − x) 2 n −1

− 1)

nj

denotes the number of elements in each of the sample and

Σ

symbol indicating that we ought take the sum of number of DOF of each sample.

Chi-square statistic for a sample variance is given by χ 2 =

(n − 1) s 2 σ2

.

Part II: Problems Probability Distribution and Decision Theory Based on the following information answer the questions 1 - 4. A random variable X has the following probability distribution.

1)

7)

3.

1.

X 0 1 2 3 4 5 6 P(X) 0.1 0.15 0.25 0.30 0.10 0.07 0.03 Which of the following is the expected value of X and variance of X?

2)

a.

2.48 and 2.1296.

3)

b.

2.58 and 2.5263.

4)

c.

2.48 and 2.0526.

5)

d.

2.84 and 2.1249.

6)

e.

2.48 and 2.1249.

2.

What is the value of P (1 ≤ X ≤ 5)?

8)

a.

0.75.

9)

b.

0.25.

10) c.

0.15.

11) d.

0.87.

12) e.

0.30.

What is the value of P (2 ≤ X)? 1)

a.

0.7.

2)

b.

0.3.

3)

c.

0.75.

4)

d.

0.25.

5)

e.

0.5.

4. What is the value of P(X < 4)?

6)

1)

a.

0.7.

2)

b.

0.75.

3)

c.

0.80.

4)

d.

0.30.

5)

e.

0.25.

5. Fifteen management graduates cross all hurdles and reach the final phase of a selection process for the recruitment of management trainees by a company. The final phase is an interview and we can gather from past data that, on an average, only 12 percent of the candidates qualifying for the interview get the job. Determine the following probabilities. (Assume that there is no restriction on the number of management trainees to be recruited). What is the Probability that exactly 5 candidates get the job? 7)

a.

0.0205. 139

8)

b.

0.0208.

9)

c.

0.0209.

10) d.

0.0207.

11) e.

0.0208.

12) 6. In the question above (Q. No. 5), what is the Probability that fewer than four candidates get the job? 13) a.

0.9042.

14) b.

0.8570.

15) c.

0.0209.

16) d.

0.0205.

17) e.

0.6570.

Based on the following information answer the questions no. 7 and 8. 1) On a 25-Question, 4-answer multiple choice examination taken by someone who knows absolutely nothing about the subject and must guess, independently each answer, 2)

8)

7.

What is the expected number of correct answers?

3)

a.

6.75.

4)

b.

6.25.

5)

c.

5.26.

6)

d.

5.80.

7)

e.

7.25.

8.

Probability of answering exactly 5 questions correctly is

9)

a.

0.1645

10) b.

0.1550

11) c.

0.20

12) d.

0.25

13) e.

0.33.

14) 9. The owner of a bookshop has observed that the probability that a customer who is browsing through books will make a purchase is 0.3. Suppose that 15 customers browse through books in the section ‘Probability and Statistics’ each hour. Determine the following probabilities. Probability that at least one browsing customer will make a purchase during a specified hour. 15) a.

0.8950

16) b.

0.8725

17) c.

0.90

18) d.

0.9953

19) e.

0.923.

20) 10. If 30 percent of a population owns their own homes, what is the probability that a sample of 7 from this population will contain exactly two homeowners? 21) a.

0.23.

22) b.

0.3125.

140

23) c.

0.3177.

24) d.

0.35.

25) e.

0.3725.

26) 11. Certain refined edible oil is packed in times holding 16 kg each. The filling machine can maintain this but with a S.D. of 0.5 kg. Samples of 25 were taken from the production line. If a sample mean is 16.35 kg and we want to test that at 95% the sample has come from the population of 16 kg tins? What is the test statistics for the above test? 27) a.

3.5.

28) b.

3.8.

29) c.

4.25.

30) d.

4.50.

31) e.

5.125.

32) 12. The marks of 100 students are normally distributed with mean 82 and standard deviation 3. How many students have got marks equal to 82? 33) a.

0.125.

34) b.

0.135.

35) c.

0.145.

36) d.

0.130.

37) e.

0.150.

Based on the following information answer the questions no. 13 and 14. 1) A bank receives about 1460 application per year for a current account with overdraft facility. We can see from past records that the probability of an application being approved is, on the average 0.8. 2)

8)

13. Find the probability that in a particular year not more than 1180 applications are approved. 3)

a.

0.7852

4)

b.

0.6523

5)

c.

0.7125

6)

d.

0.8325

7)

e.

0.9025.

14. Find the probability that in a particular year more than 1180 applications are approved? 9)

a.

0.2045.

10) b.

0.2315.

11) c.

0.2148.

12) d.

0.2545.

13) e. 0.1945. 14) 15. A small publishing house prints calendars during November every year and has the objective of selling them within the first three months of the New Year. After this period the calendars are considered to be worthless. The sales of the past few years had the following distribution: Number of calendars sold (in thousands) Probability 1)

50

60

70

0.10

0.12 0.15

80

90

0.33

0.30

The production cost for each calendar is Rs.7.00 and the publishing house plans to sell each calendar for Rs.15.00. Use the expected value criterion to determine the number of calendars to be produced by the firm for maximizing profits

141

2)

a.

50,000

3)

b.

60,000

4)

c.

70,000

5)

d.

80,000

6)

e.

90,000.

7)

16. Students at a business school are planning to sell t-shirts at a management meet organized by the school. The t-shirts carry the business school and the management meet logo on them. The shirts are custom made by a company for a price of Rs.21.50 each and the students plan to sell them for Rs.43.95 each. Any t-shirts remaining unsold at the end of the management meet are taken back by the company for Rs.19.95 each.

8)

The sales of t-shirts during the business school’s management meets in the past have the following probability distribution. Units Demanded (in hundreds) 20 21 22 23 24 Probability 0.23 0.21 0.22 0.16 0.18

1)

Use the expected value criterion to determine the number of t-shirts to be ordered by the students to maximize their profits and maximum profit.

7)

2)

a.

2400

3)

b.

2300

4)

c.

2200

5)

d.

2100

6)

e.

2000.

17. As part of an air pollution survey, an inspector decides to examine the exhaust of 6 of a company’s 24 trucks. If 4 of the company’s trucks emit excessive amounts of pollutants, what is the probability that none of them will be included in the inspector’s sample? 8)

a.

0.2550.

9)

b.

0.2880.

10) c.

0.2650.

11) d.

0.2440.

12) e.

0.3215.

Based on the following information answer the questions no. 18, 19 and 20. X 06 09 11 12 15 17 1)

7)

Y 11 14 15 19 23 25

P(X,Y) 0.03 0.20 0.07 0.50 0.10 0.10

18. What is the variance of the variables X and Y? 2)

a.

6.3474 and 12.9695.

3)

b.

5.8965 and 12.695.

4)

c.

6.3474 and 13.254.

5)

d.

7.6523 and 14.256.

6)

e.

6.4374 and 12.9596.

19. What is the covariance of X and Y? 8)

a.

8.9650.

142

9)

b.

7.7950.

10) c.

6.5980.

11) d.

9.125.

12) e.

8.8940.

13) 20. What is the mean and variance of the variable Z = 4X - 3Y? 14) a.

7.44 and 4.8279.

15) b.

6.74 and 3.356.

16) c.

- 7.64 and 4.8279.

17) d.

- 7.64 and 4.8279.

18) e.

- 6.85 and 5.6523.

19) 21. ABC company has Rs.12,00,000 to invest. It is considering four alternatives investing in a wildcat well, drilling deeper in a dry but once producing well, or refurbishing a producing well, or investing in municipal bonds at 5 percent annual interest. Investment in the wildcat will cost Rs.12,00,000, drilling deeper costs Rs.10,00,000, and refurbishing the producing well will cost Rs.4,00,000. If an option costing less than Rs.12,00,000 is elected, the remainder of the available money will be invested in municipal bonds at 5 percent. The following table gives relevant probabilities and profits on an annual basis. Wildcat Drilling Deeper Refurbishing Probability Profit/( Probability Profit/(Loss) Probability Profit/(Loss) Loss) P(oil) = 0.10 Rs.77, P(oil) = 0.20 Rs.2,40,00,000 P(improve) = 0.60 Rs.4,00,000 00,000 P(gas) = 0.20 Rs.10, P(gas) = 0.05 Rs.15,00,000 P(no change) = 0.30 Rs.4,00,000 00,000 P(dry) = 0.70 (Rs.12, P(dry) = 0.75 (Rs.10,00,000) P(worse)= 0.10 (Rs.8,00,000) 00,000 ) 1)

7)

Which alternative gives the maximum expected annual profit? 2)

a.

Wildcat.

3)

b.

Drilling Deeper.

4)

c.

Refurbishing.

5)

d.

Wildcat and Drilling Deeper same and maximum expected profit.

6)

e.

Wildcat and Refurbishing have same and maximum expected profit.

22. Multiple choice test consists of eight questions and three answers to each question (of which only one is correct). If a student answers each question by rolling a balanced die and marking the first answer if he gets 1 or 2, the second answer if he gets 3 or 4 and the third answer if he gets 5 or 6, what is the probability that he will get exactly four correct answers? 8)

a.

0.1502.

9)

b.

0.1707.

10) c.

0.1657.

11) d.

0.1723.

12) e.

0.1805.

13) 23. The output of a production process is 10 percent defective. What is the probability of selecting exactly two defectives in a sample of five? 14) a.

0.0813.

15) b.

0.0729.

143

16) c.

0.6123.

17) d.

0.9125.

18) e.

0.7932.

19) Based on the following information answer the questions no. 24 and 25. 20) City motor is planning the staff of the garage facilities. From information provided by the manufacturer, the estimated number of annual mechanic hours that the garage will likely need is Hours Probability

10,000 0.2

12,000 0.3

14,000 0.4

16,000 0.1

1) The manager plans to pay each mechanic Rs.9.00 per hour and to charge his customers Rs.16.00. Mechanics will work a 40-hour week and get an annual 2-week vacation. 2)

24. Determine how many mechanics should be hired? 3)

a.

5.

4)

b.

6.

5)

c.

7.

6)

d.

8.

7) e. 4. 25. How much should CM pay to get perfect information about the number of mechanics they need? 9) a. Rs.12,272. 10) b. Rs.10,712. 11) c. Rs.13,712. 12) d. Rs.11,712. 13) e. Rs.11,217. 14) 26. Bike Wholesale parts was established in the early 1970s in response to demands of several small and newly established bicycle shops that needed access to wide variety of inventory but were not able to finance it themselves. The company carries a wide variety of replacement parts and accessories but does not maintain any stock of completed bicycles. Management is preparing to order 27" x 1 1/4" rims from the Flexspin Company in anticipation of a business upturn expected in about 2 months. Flexspin makes a superior product, but the lead time required necessitates that wholesalers make only one order, which must last through the critical summer months. In the past, Bike Wholesale parts has been quite successful and plans to move its operations to a larger plant during the winter. Management feels that the combined cost of moving some items such as rims and the existing cost of financing them is at least equal to the firm’s purchase cost of Rs.7.30. Accepting management’s hypothesis that any unsold rims at the end of the summer season are permanently unsold, determine the number of rims the company should order if the selling price is Rs.8.10, pattern of sales follows normal distribution with mean of 120 and standard deviation of 28. 15) a. 84 rims 16) b. 72 rims 17) c. 78 rims 18) d. 89 rims 19) e. 74 rims. 8)

20) Based on the following information answer the questions no. 27 and 28 21) During a meeting a discuss scheduling plant capacity, production manager for Keystone Foundry and Metal works were concentrating on planning for the manufacture of castings used by small diesel-engine manufacturers. Because of a recent recession, demand by the engine manufacturers has been sluggish, but there were signs that it might increase in the coming year. The managers had made these estimates on the expected sales of castings: lbs.

Probability

144

Weak sales Average sales Strong sales

45,000 50,000 55,000

0.40 0.35 0.25

1) To produce the castings, Keystone had a fixed start-up cost of Rs.30,000 and a variable manufacturing cost of Rs.0.50 per pound. On the average, sales prices were Rs.1.50 per pound. 2) 27. Using expected profit as the decision criterion, what is the average cost of production and at what level should Keystone produce? 3) a. Average cost of production is 55,000 Keystone should produce 50,000 lbs. 4) b. Average cost of production is 56,000 Keystone should produce 50,000 lbs. 5) c. Average cost of production is 55,000 Keystone should produce 55,000 lbs. 6) d. Average cost of production is 45,000 Keystone should produce 45,000 lbs. 7) e. Average cost of production is 55,000 Keystone should produce 45,000 lbs. 8) 28. How much should Keystone pay to find out exactly what sales will be? 9) a. Rs.2,050. 10) b. Rs.2,150. 11) c. Rs.2,350. 12) d. Rs.2,250. 13) e. Rs.2,000. 14) 29. Anjali, head of a small business-consulting firm, must decide how many MBAs to hire as fulltime consultants for the next year. Anjali knows from experience that the probability distribution on the number of consulting jobs her firm will get each year is as follows: Consulting jobs Probability 1)

27 0.2

30 0.4

33 0.1

Anjali also knows that each MBA hired will be able to handle exactly three consulting jobs per year. The salary of each MBA is Rs.60,000. Each consulting job is worth Rs.30,000 to Anjali’s firm. Each consulting job that the firm is awarded but cannot complete costs the firm Rs.10,000 in future business lost.

2)

8)

24 0.3

How many MBAs should be hired so as to maximize the expected profit? 3)

a.

8.

4)

b.

9.

5)

c.

10.

6)

d.

11.

7)

e.

Either 9 or 10.

30. In the above question what is the value of perfect information? 9)

a.

Rs. 54,000.

10) b.

Rs. 44,000.

11) c.

Rs. 48,000.

12) d.

Rs. 50,000.

13) e.

Rs. 52,000.

14) 31. The heights of persons in a region are assumed to be normally distributed with mean 60" and standard deviation 5". How high a doorway should be erected so that at least 95% of the people can pass through it without having to bend? 15) a.

65.3.

16) b.

68.2.

17) c.

66.6.

18) d.

60.50.

145

19) e.

59.50.

20) 32. Net sales of ABC Ltd. are normally distributed with mean of Rs.3.7 crore and standard deviation of Rs.0.4 crore. Net sales of XYZ Ltd. are also normally distributed with mean of Rs.4.0 crore and standard deviation of Rs.0.8 crore. If sales of ABC Ltd. and XYZ Ltd. are independent. 21)

What is the probability that in a given year net sales of ABC Ltd. will be greater than net sales of XYZ Ltd.? 22) a.

0.3669.

23) b.

0.3520.

24) c.

0.4523.

25) d.

0.1334.

26) e.

0.1331.

27) 33. In the above question (Q.No.32), If net sales of XYZ Ltd. in a given year are 4.5 crore what is the probability that ABC Ltd. sales will be less than XYZ Ltd. sales in that year? 28) a.

0.4772.

29) b.

0.9772.

30) c.

0.8530.

31) d.

0.9952.

32) e.

0.4522.

33) 34. The average cost of a flat in Hyderabad is Rs.4,00,000 with a standard deviation of Rs.30,000. What is the probability that a flat in Hyderabad costs at least 4,15,000 rupees? (Assume that cost of flats follow a normal distribution) 34) a.

0.3085.

35) b.

0.2563.

36) c.

0.3550.

37) d.

0.3010.

38) e.

0.3265.

39) 35. Four investors A, B, C and D invested in four portfolios P1, P2, P3 and P4 consisting of two securities X and Y. The relevant data about these portfolios is as below. 40)

1)

A.

Weights/Portfolio

B.

P1 Weight of Security X 100% Weight of Security Y 0% Rate of Returns and Standard Deviation Portfolio P1 P2 P3

1)

P3 0% 100%

P4 25% 75%

Standard Deviation 4% 5% 1/2%

Calculate the rate of return and standard deviation of returns for portfolio P4 assuming the same time horizon for all the portfolios. 2)

a.

V(P4) =

17.84375(%)2 and E(P4) 2

=

3)

b.

V(P4) =

5.74355(%) and E(P4) = 45%

4)

c.

V(P4) =

13.7775(%)2 and E(P4) = 40%

5)

d.

V(P4) =

12.8875(%) 2 and E(P4)

6) 7)

Expected Rate of Return 20% 30% 40%

P2 50% 50%

e.

V(P4) =

2

16.84375(%) and E(P4)

35%

=

35%

=

35%.

36. The central university is preparing for the annual day celebrations. As a fund raising device, the

146

committee plans to sell souvenir T-shirts. It plans to buy heat-transfer patches from a supplier for Rs.7.50 and plain white T-shirts for Rs.15.00. A local merchant supplies the appropriate heating device and also purchases all unsold white cotton T-shirts. The committee plans to sell the shirts for Rs.32.50. Transfer of the color to the shirt will be completed when the sale is made. In the past year, similar shirt sales have averaged 200 with a standard deviation of 34. There will be no market for the patches after the celebration. How many patches should the committee buy? 8)

a.

200.

9)

b.

206.

10) c.

205.

11) d.

204.

12) e.

202.

13) 37. A newly started merchant banking company has carried out a market survey to ascertain the volume of business it is likely to get in the current year. The survey reveals the following information: Volume of business (Rs. in crore) 5 8 10 15 Probability 0.50 0.35 0.10 0.05 1)

7)

Based on the information obtained above, the company has to plan the strength of professional staff. One professional staff is able to look after the business of Rs.25 lakh annually and he is to be paid Rs.1.5 lakh per annum as salary. Every one crore of business results in Rs.10 lakh profits to the company and every one crore of business refused is likely to cost the company Rs.5 lakh in terms of future losses. What is the optimum strength of the professional staff the company should hire? 2)

a.

20.

3)

b.

32.

4)

c.

40.

5)

d.

60.

6)

e.

Either 20 or 32.

38. In the above question no.37 what is the value of perfect information? 8)

a.

Rs.17.25 lakh.

9)

b.

Rs.14.25 lakh.

10) c.

Rs.18.00 lakh.

11) d.

Rs.17.75 lakh.

12) e.

Rs.19.25 lakh.

13) 39. An auto finance company has a number of schemes of car finance to suit its customers. In almost all cases the returns of the company’s investment in car finance vary from 20% to 24% with a mean return of 22%. Suppose a customer walks in, what is the probability that a deal will be struck with him which will give the company a return between 21% to 23% on his account? Assume that the probability of striking the deal with the customer is 20%. 14) a.

0.1750

15) b.

0.1733

16) c.

0.1825

17) d.

0.1650

18) e.

0.1925.

19) 40. A company manufactures a product A which sells in the market for Rs.50. The cost of production of the same is Rs.40. What is the probability of selling an additional unit that will justify its production? 20) a.

0.80.

21) b.

0.70.

147

22) c.

0.75.

23) d.

0.85.

24) e.

0.78.

25) 41. The placement division of ICFAI is interested in finding out the annual earnings of CFAs numbering around 1500. How large a sample it should take in order to determine the mean annual earnings within plus and minus Rs.2,000 and at 95 percent confidence level? You may assume that the earnings of CFAs follow normal distribution with a standard deviation of Rs.5,000. 26) a.

21

27) b.

22

28) c.

23

29) d.

25

30) e.

30.

31) 42. New Age Bank is interested in planning, how much cash to be kept in the vault. The branch manager of the bank is trying to study the profile of average deposits of its customers for the purpose. The information collected by him is as follows: Deposits (Rs.) Less than 5,000 5,000 - 10,000 More than 10,000 Observed frequency 22 61 17 1) What are the expected frequencies if the data are normally distributed with mean of 6,000 and standard deviation of Rs.3,000? 2)

a.

37,54 and 9.

3)

b.

35,51 and 11.

4)

c.

32, 54 and 9.

5)

d.

37, 54 and 12.

6)

e.

39, 54 and 9.

7) Based on the following information answer the questions no. 43 and 44. 8) A venture capital finance company has analyzed its past data and found out that 2 out of every 27 projects financed by it have succeeded and have given abnormal returns. It has 6 projects under consideration at the moment. Assume that the past trend is likely to be repeated in future (Hint: You may use Binomial distribution). 9)

43. What is the probability that only 2 projects will succeed? 10) a.

0.08.

11) b.

0.05.

12) c.

0.04.

13) d.

0.07.

14) e.

0.06.

15) 44. What is the probability that none of the projects will succeed? 16) a.

0.52.

17) b.

0.69.

18) c.

0.92.

19) d.

0.63.

20) e.

0.70.

21) 45. The stock markets in India have been giving an annual return of 21.5% on an average with the standard deviation of 9.72%. The last 4 year returns from the stock markets are as follows: Year Stock Market Return

148

1996 1997 1998 1999

(%) -1.30 12.22 -24.63 50.32

1)

Based on the above information and at a significance level of 5%, you are required to find out whether or not the long-term average returns from stock markets have declined in recent years.

2)

a. Long-term average return from stock markets have decline in recent year and the value of Z is -2.54.

3)

b. Long-term average return from stock markets have decline in recent year and the standard error of mean is 4.86.

4)

c. Long-term average return from stock markets have decline in recent year and the standard error of mean is 5.62.

5)

d. Long-term average return from stock markets have decline in recent year and the value of Z is -2.25.

6) e. Both (a) and (b) above. 7) Based on the following information answer the questions no. 46 and 47. 8) E-store.com has set up a new electronic department store on the internet. It has observed that customer who visits its site and browses for more than a minute is likely to buy something one out of four times. There are approximately 10 customers in an hour how visit that site for more than a minute. 9)

46. What is the probability that at least six customers will buy something from the store in a specified hour? 10) a.

1.97%.

11) b.

1.95%.

12) c.

1.82%.

13) d.

1.80%.

14) e.

1.99%.

15) 47. What is the probability that no customer will buy anything from the store in a specified hour? 16) a.

4.52%.

17) b.

4.40%.

18) c.

4.25%.

19) d.

5.70%.

20) e.

5.63%.

21) 48. A stock broker has identified that 80% of the telephone calls he receives during business hours are for buying or selling securities and rest of the calls are for other purposes. 22)

What is the probability that out of the first 10 telephone calls received in a day exactly five calls are for trading in securities? 23) a.

2.25%.

24) b.

2.40%.

25) c.

2.64%.

26) d.

2.80%.

27) e

3.00%.

28) 49. In the above question no.48 What is the probability that out of the first 10 telephone calls received in a day more than seven calls are for trading in securities?

149

29) a.

57%.

30) b.

58%.

31) c.

67.7%.

32) d.

62.5%.

33) e.

66.86%.

34) 50. An investor is planning to invest a certain sum of money in any one of three different portfolios. He projects three views of the economic conditions that may prevail in future - optimistic, normal and pessimistic. Once the investment is made, any one of these conditions will prevail. However, he has insufficient knowledge to assign any probabilities of occurrence to these economic conditions. According to him, the likely gains from the portfolios under these three conditions in future will be as follows: Portfolio Optimistic Normal Pessimistic A 50,000 25,000 5,000 B 40,000 35,000 -2,000 C 65,000 30,000 -5,000 1)

7)

If the investor uses the regret criterion then which portfolio will he select for making the investment? 2)

a.

A.

3)

b.

B.

4)

c.

C.

5)

d.

Either A or B.

6)

e.

Either A or C.

51. If the investor uses the maximin criterion then which portfolio will he select for making the investment? 8)

a.

Portfolio A.

9)

b.

Portfolio B.

10) c.

Portfolio C.

11) d.

Either A or C.

12) e.

Either B or C.

13) 52. Johnson Foods is a firm involved in the business of supplying processed food products as per the tender specifications of various purchasing parties. On the basis of its past experience it has estimated that there is a 45% probability that it can secure the supply contract with the party inviting the tender. The firm has presently filed quotations in response to tenders invited by 4 parties. 14)

What is the probability that it will secure supply contracts with at least 3 of the parties inviting the tender? 15) a.

0.2415.

16) b.

0.2450.

17) c.

0.2510.

18) d.

0.2560.

150

19) e.

0.2390.

20) 53. In the above question, what is the probability that it will not secure any of the supply contracts? 21) a.

0.0965.

22) b.

0.0975.

23) c.

0.0875.

24) d.

0.0825.

25) e.

0.0915.

26) 54. Shyam Fabricators purchased a used milling machine from Kakatiya Engineering Works. It was communicated to Shyam Fabricators that there would be a probability of 5% that the output from the machine will be defective. It is expected that this rate of defective output will continue in future. What will be the probability that maximum one out of every ten jobs done in that machine will be defective? 27) a.

0.9131.

28) b.

0.9250.

29) c.

0.9190.

30) d.

0.8960.

31) e.

0.9350.

32) 55. The owner of Vijaya Stores is deliberating upon the number of 500 ml packets of homogenized milk it must stock. Each 500 ml packet of homogenized milk costs Rs.6.50 and can be sold at Rs.8.00. A packet which remains unsold in a day is considered stale and cannot be sold. The probability distribution of the number of packets demanded in a day is given below: No. of Packets Probability 200 0.06 201 0.10 202 0.15 203 0.30 204 0.20 205 0.10 206 0.09 1 1)

7)

What is the minimum required probability of selling at least one additional packet in order to justify stocking that packet? 2) a. 0.8225. 3) b. 0.8125. 4) c. 0.7950. 5) d. 0.8325. 6) e. 0.8550. 56. In the above question, find out the number of packets to be stocked so that the expected profit in a day is maximum? 8) a. 200. 9) b. 201. 10) c. 202. 11) d. 203.

151

12) e. 204. 13) 57. A fair coin is tossed five times. Find the probabilities that a head appears exactly three times? 14) a.

0.3125.

15) b.

0.3090.

16) c.

0.3225.

17) d.

0.3190.

18) e.

0.3150.

19) 58. 2% of eggs are brown. If 6 eggs are packed, at random, into a carton, what is the probability that none are brown? 20) a.

0.8256.

21) b.

0.8754.

22) c.

0.8858.

23) d.

0.8990.

24) e.

0.9125.

25) 59. A jury consists of 12 citizens, selected at random, from a population which is 55% female. What is the probability that the jury will have at least two female members? 26) a.

0.998920.

27) b.

0.997854.

28) c.

0.965821.

29) d.

0.954212.

30) e.

0.912563.

31) 60. Suppose a batter (in baseball) gets a hit with a probability of 0.3, and gets out the rest of the time. What is the probability of that batter getting 0 hits in 10 at bats? 32) a. 0.0282 33) b. 0.0325 34) c. 0.0521 35) d. 0.0625 36) e. 0.0125. 37) 61. What is the probability of flipping a fair coin eight times and getting only two heads? 38) a. 0.2032. 39) b. 0.1094. 40) c. 0.1231. 41) d. 0.1331. 42) e. 0.1525. 43) 62. A company manufactures shoes, which sell in the market for Rs.120. The cost of production of the same is Rs.90. What is the probability of selling an additional unit that will justify its production? 44) a. 0.80. 45) b. 0.70. 46) c. 0.75.

152

47) d. 0.85. 48) e. 0.78. 49) 63. The height X of 500 soldiers are found to have normal distribution. Of them 258 are found to be within 2 cm. pf the mean height of 170 cm find the s.d of X? 50) a. 2.525. 51) b. 2.857. 52) c. 3.250. 53) d. 3.565. 54) e. 2.252. 55) 64. Restaurant manager is considering a new location for her restaurant. The projected annual cash flow for the new location is: Annual Cash Flow Rs.10, 000 Rs. 30,000 Rs.70, 000 Rs.90, 000 Rs.100,000 Probability 0.10 0.15 0.50 0.15 ? 1)

7)

1) 2) 3) 4) 5) 6) 7)

The expected cash flow for the new location is: 2) a. Rs.12,800 3) b. Rs.64,000 4) c. Rs.70,000 5) d. Rs.60,000 6) e. Rs.50,000. 65. Consider the following data relating to the share price of ABC Ltd.: Price (Rs./share) 10 20 33 45 65 Probability 0.1 0.3 0.4 0.15 0.0 The expected price of the share of ABC Ltd. is a. Rs.30.20 b. Rs.31.80 c. Rs.32.00 d. Rs.33.00 e. Rs.32.20. 66. The current price of the stock is Rs.110. The stock does not pay any dividends. End of year price Probability Rs.100 0.1 Rs.105 0.2 Rs.110 0.4 Rs.120 0.2 Rs.125 0.1

1)

The expected return is

2)

a.

1.25%

3)

b.

1.36%

4)

c.

1.75%

5)

d.

2.31%

e.

2.40%.

6) 7)

67. For the probability distribution given below what is the standard deviation? 0 0.5 1 1.5 2 Total Value of X Probability 0.16 0.4 0.33 0.1 0.01 1.0 1)

a.

0.205

153

6)

2)

b.

0.325

3)

c.

0.125

4)

d.

0.221

5)

e.

0.392.

68. A variable X, which has a normal distribution, has a mean of 24 and a standard deviation of 5. The probability that the value of X lies in the interval 20 to 28 is approximately. 7)

a.

24.7%

8)

b.

31.5%

9)

c.

38.5%

10) d.

49.5%

11) e.

57.6%.

12) 69. If the probability of success in a Bernoulli experiment is 0.70 and the test is repeated 60 times, then the standard deviation of the results of this experiment is 13) a.

3.54

14) b.

3.61

15) c.

3.76

16) d.

3.92

17) e.

4.01.

18) 70. A portfolio consists of two stocks A and B in which funds have been invested in the ratio of 2:3. If the expected returns on stocks A and B are 15% and 25% respectively then, the expected return on the portfolio is 19) a.

14%

20) b.

17%

21) c.

18%

22) d.

21%

23) e.

23%.

24) 71. ABC Fabricators purchased a used milling machine from Kakatiya Engineering Works. It was communicated to Shyam Fabricators that there would be a probability of 5% that the output from the machine will be defective. It is expected that this rate of defective output will continue in future. What is the probability that maximum three out of every ten jobs done in that machine will be defective? 25) a. 99.89%. 26) b. 97.5%. 27) c. 96.85%. 28) d. 95%. 29) e. 92.6%. 30) Based on the following information answer the questions no. 72 and 75. 31) A random variable X has the following probability distribution.

154

X P (X) 1)

0 0.05

1 0.1

2 0.15

3 0.20

4 0.25

5 0.10

6 0.15

Which of the following is the expected value of X and variance of X? a. 3.40 and 3.84. b. 2.90 and 2.5263. c. 3.40 and 3.75. d. 3.25 and 3.84. e. 3.48 and 3.9249. 7) What is the value of P (1 ≤ X ≤ 5)? a. 0.80. b. 0.25. c. 0.15. d. 0.10. e. 0.30. 13) 74. What is the value of P (2 ≤ X)? 72. 2) 3) 4) 5) 6) 73. 8) 9) 10) 11) 12)

14) a.

0.7.

15) b.

0.3.

16) c.

0.85.

17) d.

0.25.

18) e. 0.5. 19) 75. What is the value of P(X < 4)? 20) a.

0.05.

21) b.

0.10.

22) c.

0.15.

23) d.

0.50.

24) e. 0.20. 25) 76. A die is tossed 1200 times. Find the Probability that the number of ‘sixes lies between 190 and 210. (Given that the area of Z between 0 and 0.78 is 0.2823 and that between 0 and 0.81 is 0.2910. 26) a.

0.625

27) b.

0.582

28) c.

0.486

29) d.

0.528

30) e. 0.669. 31) 77. An analyst’s estimate that a stock of a company XYZ has the following probability distribution of annual return depending on the news about its profits. News Good Average Bad 1)

Return (x) 15% 13% 7%

Probability f(x) 0.1 0.6 0.3

What is the Expected return: E(x) of the stock? 2)

a.

11.4.

3)

b.

12.25.

4)

c.

11.6.

155

7)

5)

d.

13.

6)

e.

12.

78. What is the standard deviation of return of the stock for the above question no.76? 8)

a.

2.25.

9)

b.

2.62.

10) c.

2.94.

11) d.

2.99.

12) e.

2.10.

13) 79. Out of 800 families with 4 children each, how many would be expected to have 2 boys and 2 girls assume that probability of boys and girls are equal? 14) a.

200.

15) b.

250.

16) c.

300.

17) d.

350.

18) e.

400.

19) 80. The average selling price of houses in a city is Rs.50,000 with a standard deviation of Rs.10,000, Assuming the distribution of selling process to be normal, what is the percentage of houses that sell for more than Rs.55,000? (Area between z = 0 and z = 1 is 0.3413 and the area between z = 0 and z = 0.5 is 0.1915, where z is the standard normal values. 20) a.

30.85%.

21) b.

28%.

22) c.

29.5%.

23) d.

32.6%.

24) e.

33.6%.

25) 81. In the above question no.79, what is the percentage of houses selling between Rs.45,000 and Rs.50,000? 26) a.

52.5%.

27) b.

53.28%.

28) c.

50.65%.

29) d.

49.65%.

30) e.

48.25%.

31) 82. The average number of road accidents in Hyderabad is found to average to 16 per month. Assuming that accidents follow Poisson distribution, find the probability that in any given month, the number of road accidents exceed 23.

156

32) a.

0.0301

33) b.

0.0302

34) c.

0.0304

35) d.

0.0309

36) e.

0.0310.

37) Based on the following information answer the questions no. 83 and 84 38) Futuristic Electronics Ltd. Manufactures electronics components for TVs and VCRs. The company planners expect a growth in demand for the company’s products over the next 8 year period. There are three alternatives for meeting the growth in demand 39) i. Expand the existing plant 40) ii. Build a new plant 41) iii. Subcontract out the excess production required. 42) The growth in demand can be - low, high or moderate. The following table provides the estimated payoffs from the alternative courses of action under the different levels or demand. Alternative Expand Build Subcontract 1)

Low 1/10/00 1/5/00 1/8/00

Demand Moderate 15 18 10

High 25 30 20

83. Which of the following will be selected by the decision-makers based on the maximax criterion? 2)

a.

Expand.

3)

b.

Build.

4)

c.

Sub-contract.

5)

d.

Either Expand or Build.

6)

e.

Either Sub-contract or Expand.

7)

84. Which alternative will be selected by the decision-makers based on the maximin criterion? 8) a. Expand. 9) b. Build. 10) c. Sub-contract. 11) d. Either Expand or Build. 12) e. Either Sub-contract or Expand. 13) 85. The owner of XYZ Stores is deliberating upon the number of 1 kg packets of panner it must stock. Each 1 kg packets of panner costs Rs.80 and can be sold at Rs.100.0. A packet which remains unsold in a day is considered stale and cannot be sold. The probability distribution of the number of packets demanded in a day is given below: No. of Packets Probability 100 0.06 101 0.10 102 0.15 103 0.30 104 0.20 105 0.10 106 0.09 Total 1 1)

What will be the minimum required probability of selling at least one additional packet in order to justify stocking that packet? 2)

a.

0.80.

157

7)

3)

b.

0.7154.

4)

c.

0.7852.

5)

d.

0.8365.

6)

e.

0.8525.

86. In the above question no.85 what will be the number of packets to be stocked so that the expected profit in a day is maximum? 8)

a.

100.

9)

b.

102.

10) c.

101.

11) d.

103.

12) e.

104.

13) 87. The Manager of Popular Fast Foods is deliberating on the number of pieces of a popular snack, “Magic Puff,” to prepare for the evening. Each piece of “Magic Puff” costs Rs.10 to prepare and is sold at Rs.16. 14)

Any unsold pieces cannot be stored because the item is highly perishable. The manager of the shop has arrived at the following probability distribution of sales on the basis of past experience. Number of pieces that Probability can be sold 150 0.05 151 0.10 152 0.15 153 0.30 154 0.20 155 0.12 156 0.08

1)

The minimum required probability of selling one or more additional pieces of the item in order to justify preparing them for the evening is

7)

2)

a.

0.60

3)

b.

0.6154

4)

c.

0.6852

5)

d.

0.6365

6)

e.

0.625.

88. In the above question no. 87, what will be the number of pieces of the item to be prepared for the evening so that the profit in the evening is maximum? Number of pieces Probability Cumulative probability that sales that can be sold will be at this level or more 150 0.05 1.00 151 0.10 0.95 152 0.15 0.85 153 0.30 0.70 154 0.20 0.40 155 0.12 0.20 156 0.08 0.08 1)

a.

153

2)

b.

154

3)

c.

155

4)

d.

156

158

5) 6)

e.

151.

89. If there is a 50% probability that the price of a share will be Rs.40, a 20% probability that the price will be Rs.25 and a 30% probability that the price will be Rs.30, then the expected price of the share is 7)

a.

Rs.30

8)

b.

Rs.32

9)

c.

Rs.34

10) d.

Rs.36

11) e.

Rs.38.

12) 90. The probability of getting exactly 2 heads in 3 tosses of a fair coin is 13) a.

30.5%

14) b.

32.5%

15) c.

35.5%

16) d.

37.5%

17) e.

40.0%.

18) 91. In a Bernoulli experiment the probability of success is 0.2 and the number of trials is 7. The approximate probability of getting one success is 19) a.

0.294

20) b.

0.367

21) c.

0.425

22) d.

0.550

23) e.

0.625.

92. The owner of a movie theatre wants to determine the average number of moviegoers in the coming month. From the past experience he assigns probabilities to the number of moviegoers per day for the coming month. No. of moviegoers Probability of the number of per day moviegoers per day 500 0.15 550 0.20 600 0.30 650 0.35 Which of the following is the expected number of moviegoers per day for the coming month? 1)

a.

553.

2)

b.

563.

3)

c.

573.

4)

d.

583.

5) e. 593. 93. A biased coin has the probability of giving a head when tossed equal to 0.60. What is the probability of getting exactly three heads in 4 tosses? 1)

a.

0.2546.

2)

b.

0.3456.

3)

c.

0.4654.

159

6)

4)

d.

0.5346.

5)

e.

0.6354.

94. A normal variable has the mean of 12 and a standard deviation of 5. What is the probability that the normal variable will take a value in the interval of 5.6 and 21.8? 7)

a.

80.54 %.

8)

b.

85.56 %.

9)

c.

86.64 %.

10) d.

87.47 %.

11) e.

95.52 %.

12) 95. The latest nationwide political poll indicates that for persons who are randomly selected, the probability that they are pro NDA is 0.55, the probability that they are pro Congress is 0.30, and the probability that they are pro Left Front is 0.15. Assuming that these probabilities are accurate, what is the probability that out of randomly chosen group of 10 persons atleast eight are pro NDA? 13) a.

0.0995.

14) b.

0.1980.

15) c.

0.4400.

16) d.

0.5500.

17) e.

0.9950.

18) 96. An investment analyst must make a decision about investing in one of the five investment portfolio packages. Each investment portfolio contains different proportions of common stock, industrial bonds and real estate. The table below gives the gain of investing in these portfolios under different states of nature S1, S2 and S3 based on the projection of changes in stock prices, yields on bonds and appreciation in real estate values. Gains from investment (in Rs.) Portfolio State S1 State S2 State S3 A 30,000 4,000 2,000 B 40,000 25,000 -15,000 C 50,000 20,000 -30,000 D 40,000 10,000 5,000 E 20,000 5,000 -5,000 1)

7)

1)

If the analyst specifies his degree of optimism as 0.7, then according to Hurwicz criterion which portfolio will he invest in? 2)

a.

Portfolio A.

3)

b.

Portfolio B.

4)

c.

Portfolio C.

5)

d.

Portfolio D.

6) e. Portfolio E. 97. Mithra Motors has recently started its vehicle showroom in the city. Its chief asset is a franchise to sell automobiles of a major Indian manufacturer. Mithra’s general manager is planning the staffing of the dealership’s garage facilities. From information provided by the manufacturer and from other nearby dealerships, he has estimated the number of annual mechanic hours that the garage will be likely to need. Hours 10,000 12,000 14,000 16,000 Probability 0.2 0.3 0.4 0.1 The manager plans to pay each mechanic Rs 9.00 per hour and to charge his customer Rs.16.00.

160

Mechanics will work a 40-hour week and get an annual 2-week vacation. Assuming that the mechanics do not get paid holidays, how many mechanics Mithra Motors should hire and what is the maximum amount it should pay to get perfect information about the number of mechanics it needs? (Assume 1 year = 52 weeks)

7)

2)

a. Mithra Motors should hire 4 mechanics and should pay maximum Rs.20,800 to get perfect information

3)

b. Mithra Motors should hire 5 mechanics and should pay maximum Rs.19,600 to get perfect information

4)

c. Mithra Motors should hire 6 mechanics and should pay maximum Rs.12,000 to get perfect information

5)

d. Mithra Motors should hire 7 mechanics and should pay maximum Rs.14,000 to get perfect information

6)

e. Mithra Motors should hire 8 mechanics and should pay maximum Rs.28,800 to get perfect information.

98. The probability distributions of independent random variables, X and Y, are given below: X 10 15 20 25 Probability 0.20 0.40 0.30 0.10 Y Probability

8)

8 0.25

12 0.55

1)

If a random variable, Z, is defined as: Z = 2X + 5Y

2)

What is the expected value of the random variable Z?

3)

a.

6.6.

4)

b.

11.8.

5)

c.

33.

6)

d.

59.

7)

e.

92.

16 0.20

99. A random sample of 15 people is taken from a group in which 40% favor a particular political stand. What is the probability that at least 4 individuals in the sample favor this political stand? 9)

a.

0.0047.

10) b.

0.0219.

11) c.

0.0634.

12) d.

0.0905.

13) e.

0.9095.

14) 100. The standard deviation of the number of successes in a binomial distribution is probability of success in a single trial is 1/4.

3 . The

15) What is the probability of obtaining exactly eight successes? 16) a.

0.0197.

17) b.

0.1001.

18) c.

0.25.

19) d.

0.75. 161

20) e.

0.80.

21) 101. The test scores of 600 students are normally distributed with a mean of 76 and standard deviation of 8. Approximately what is the number of students scoring between 70 and 82? 22) a.

164.

23) b.

150.

24) c.

328.

25) d.

272.

26) e.

450.

27) 102. The following details are available with regard to a product: 28) Cost per unit of the product

= Rs.120

29) Selling price per unit of the product

= Rs.210

30) Salvage value of each unsold unit of the product 31)

= Rs.30

What is the minimum required probability of selling an additional unit of the product which justifies stocking that unit? 32) a.

0.25.

33) b.

0.33.

34) c.

0.50.

35) d.

0.67.

36) e.

1.00.

37) Statistical Inferences 38) Based on the following information answer the questions no. 103 and 104. The weights of persons using a lift are normally distributed with a mean of 70 kg and a standard deviation of 9 kg. The lift has a maximum permissible load of 300 kg. 1)

7)

103. Four persons are in the lift. What is the probability that the maximum load is exceeded? 2)

a.

0.1526.

3)

b.

0.1335.

4)

c.

0.1225.

5)

d.

0.8665.

6)

e.

0.8474.

104. If one person is in the lift but accompanied by luggage weighing three times his own weight, what is the probability that the maximum load is exceeded? 8)

a.

0.2877.

9)

b.

0.2254.

10) c.

0.2456.

11) d.

0.3251.

12) e.

0.3521.

13) 105. From past experience it is known that the standard deviation of the number of days of absence per worker per year in an industry is 15 days. What is the probability that a random sample of 100 workers will yield a mean value that differs from the true population value by more than 3 days?

162

14) a.

0.0325.

15) b.

0.0456.

16) c.

0.0525.

17) d.

0.0625.

18) e.

0.0575.

19) 106. A soft drink vending machine is set so that the amount of drink dispensed is a random variable with a mean of 200 ml and a standard deviation of 15 ml. What is the probability that the average (mean) amount dispensed in a random sample of size 36 is at least 204 ml? 20) a.

0.0548.

21) b.

0.0456.

22) c.

0.0512.

23) d.

0.0635.

24) e. 0.0425. 25) 107. Goodwill Chemicals (GC) wants to investigate the mean yield of a new process for making chemical B. A small pilot plant that used the process during experimentation has supplied data that the standard deviation of the hourly yields is 1 ton. 26) The process is now in operation, running at full scale. Goodwill wants to take a random sample of hourly yields of B in order to estimate μ, the mean hourly yield in tons. How large should the sample be, so that, the probability that X will be within 0.2 ton of μ is at least 0.95? 27) a. 91. 28) b. 92. 29) c. 93. 30) d. 94. 31) e. 97. 32) 108. A company producing contact lenses introduces a new type which it claims has a longer life than the old version. Six people were asked to test the new lens and this gave a mean life of 4.6 years with a standard deviation of 0.49 years. Construct a 90 percent confidence interval for the mean life of the new type of lens. 33) a. 4.502 to 5.254. 34) b. 4.196 to 5.004. 35) c. 4.860 to 5.094. 36) d. 5.196 to 6.125. 37) e. 5.196 to 6.525. 38) Based on the following information answer the questions no. 109 and 110. 39) The management of a firm wishes to test whether the average number of days of absence per worker through sickness has changed this year compared with the average value in the past which was 8 days, with a standard deviation ( σ ) of 14 days. A random sample of 49 workers is taken giving a mean of 10.6 days of absence per worker this year. Consider the 5 percent level of significance 40) 109. The acceptance region falls between 41) a. 4.20 and 11.80 42) b. 4.0 and 12 43) c. 3.5 and 12.5 44) d. 4.08 and 11.92 45) e. 5 and 11. 46) 110. For test whether the absenteeism this year is significantly higher than it has been in the past the acceptance region is

163

47) a. less than 11.30 48) b. less than 11.90 49) c. less than 12.50 50) d. less than 12.00 51) e. less than 11.60. 52) 111. A manufacturer of TV tubes has for many years used a process giving a mean tube life of 4700 hours and a standard deviation of 1460 hours. A new process is tried to see if it will increase the life significantly. A sample of 100 new tubes gave a mean life of 5000 hours. Is the new process better than the old at the 1 percent level of significance? 53) a.

Yes and the value of

μ + ZσX

5040.18.

54) b.

Yes and the value of

μ + ZσX

is 4984.7.

55) c.

No and the value of

μ + ZσX

56) d.

No and the value of

μ + ZσX

is 5040.18. is 4300.

μ + ZσX is 4700. 57) e. Yes and the value of 58) Based on the following information answer the questions no. 112 and 113.

59) A brand of matches is sold in boxes on which it is claimed that the average contents are 40 matches. A check on a pack of 5 boxes gives the following results: 1)

41, 39, 37, 40, 38 112. Test the manufacturer’s claim at 5% level of significance keeping the interests of both the manufacturer and the customer in mind, which of the following is the correct statement to justify your answer? 2) i. The null and alternative hypothesis are i. ii. iii. iv. 1) 2) 3) 4) 5)

6)

H0: μ = 40 H1: μ ≠ 40 Standard error of the sampling distribution of the mean is 0.7971 The acceptance region lies in between 38.0371 to 41.9629 Sample mean lies in the acceptance region we accept the null hypothesis so the manufacturer’s claim is justified. a. Only (i) and (iv) are correct. b. Only (ii) and (iv) are correct. c. Only (iii) and (iv) are correct. d. Both (i), (iii) and (iv) are correct. e. All (i), (ii), (iii) and (iv) are correct.

113. If as a customer we test the manufacturer’s claim at 5% significance level then which of the following statement is correct?

7)

i.

8) 9)

The null and alternative hypothesis are H0: μ = 40 and H1: μ < 40

μH 0 − t σ X is 38.49 Value of iii. The appropriate t value for this one tailed test at 4 degrees and 5% level of significance is 2.232. a. Only (i). b. Only (ii). c. Only (iii). d. Both (i) and (ii).

ii.

10) 11) 12) 13)

164

14) e. All (i), (ii) and (ii). 15) 114. Two brands of truck tyres are being compared by a transport firm. A random sample of 50 tyres of brand X (group 1) had an average life of 45000 kilometers, while the average life of a random sample of 40 brand Y tyres (group 2) was 46500 kilometers. Assuming that σ1 was 2000 km and σ2 was 1500 km, What is the value of σX1 − X 2 ?

16) a.

369.1206.

17) b.

371.2365.

18) c.

379.2106.

19) d.

385.5612.

20) e. 370.2345. 21) 115. In the above question no.114 is there any significant difference in quality of two tyres at 1% significance level? σˆ X − X 1 2 which is ± 952.3011. 22) a. Yes as the difference in mean is greater than (μ1 - μ2) ± Z σˆ X − X 1 2 which is ± 1082.3311. 23) b. Yes as the difference in mean is greater than (μ1 - μ2) ± Z σˆ X − X 1 2 which is ± 1228.3651. 24) c. Yes as the difference in mean is greater than (μ1 - μ2) ± Z σˆ X − X 1 2 which is ± 1252.3051. 25) d. No as the difference in mean is less than (μ1 - μ2) ± Z σˆ X − X 1 2 which is 48.3845. 26) e. No as the difference in mean is greater than (μ1 - μ2) ± Z 27) Based on the following information answer the questions no. 116 - 118. 28) A sample of 100 electric light bulbs produced by manufacturer A showed a mean lifetime of 1190 hours and a standard deviation of 90 hours. A sample of 75 bulbs produced by manufacturer B showed a mean lifetime of 1230 hours and a standard deviation of 120 hours. 29) 116. What is the standard deviation of Mean Value? 30) a.

15.2225.

31) b.

14.5225.

32) c.

16.5227.

33) d.

14.5227.

34) e.

14.8227.

35) 117. Is there any difference between the mean lifetimes of the two brands of bulbs at significance levels of 5% σˆ X − X 1 2 which is ± 32.3845. 36) a. Yes as the difference in mean is greater than (μ1 - μ2) ± Z σˆ X − X 1 2 which is ± 35.2153. 37) b. Yes as the difference in mean is greater than (μ1 - μ2) ± Z σˆ X − X 1 2 which is ± 36.3845. 38) c. Yes as the difference in mean is greater than (μ1 - μ2) ± Z σˆ X − X 1 2 which is ± 42.3845. 39) d. No as the difference in mean is less than (μ1 - μ2) ± Z σˆ X − X 1 2 which is ± 32.3845. 40) e. No as the difference in mean is greater than (μ1 - μ2) ± Z 41) 118. Is there any difference between the mean lifetimes of the two brands of bulbs at significance levels of 1%? σˆ X − X 2 1 which is ± 32.3845. 42) a. Yes as the difference in mean is greater than (μ1 - μ2) ± Z σˆ X − X 2 1 which is ± 35.2153. 43) b. Yes as the difference in mean is greater than (μ1 - μ2) ± Z 165

σˆ X − X 2 1 which is ± 36.3845. Yes as the difference in mean is greater than (μ1 - μ2) ± Z σˆ X − X 2 1 which is ± 42.3845. 45) d. No as the difference in mean is less than (μ1 - μ2) ± Z ˆσ X − X 2 1 which is ± 42.6286. 46) e. No as the difference in mean is less than (μ1 - μ2) ± Z 47) 119. A manufacturer claimed that at least 95% of the equipment that it supplied to a factory conformed to specifications. An examination of a sample of 200 pieces of equipment revealed that 18 were faulty. Test this claim at significance levels of 1% and state which of the following statement is correct?

44) c.

48) i. 49) ii.

The null and alternative hypothesis for this problem are H0: p = 0.95 and H1: p < 0.95 σp

= 0.0154

50) iii.

p H0 − Zσ p

51) iv.

Manufacturer’s claim not satisfied.

52) a.

Only (i) and (ii).

53) b.

Only (i) and (iii).

54) c.

Only (i), ii and (iv).

55) d.

Only (i), (iii) and (iv).

56) e.

Only (i), (ii) and (iii).

= 0.9540

57) 120. Random samples of 200 bolts manufactured by machine A and of 100 bolts manufactured by machine B showed 19 and 5 defective bolts, respectively. Test the hypothesis that the two machines are showing different qualities of performance and state which of the following statement is correct? 58) i. The null and alternative hypothesis are H0: p1 - p2 = 0 59) H1: p1 - p2 ≠ 0 pˆ = 0.08, Where pˆ is the combined proportion 60) ii. pˆ p −p 1 2 = 0.0432 61) iii. 62) iv. The upper and lower limits of the acceptance region are ± 0.0651 63) v. Two machines are not showing different qualities of performance. 64) a. Only (i), (ii), (iii), and (iv). 65) b. Only (i), (ii), (iv) and (v). 66) c. Only (i) and (iv). 67) d. Only (i), (iii), (iv) and (v). 68) e. Only (i), (ii) and (iii). 69) 121. Your Managing Director demands that you determine, with 99 percent confidence and to an accuracy of 1 percent either way the percentage of bills sent out by your firms which are in error. At present no one has any idea what this figure may be. How many bills would you need to examine? 70) a. 16641. 71) b. 15462. 72) c. 17524. 73) d. 16782. 74) e. 15986.

75) 122. Hinton Press hypothesizes that the life of its largest web press is 14,500 hours, with a known standard deviation of 2,100 hours. From a sample of 25 presses, the company finds a sample mean of 13,000 hours. At a 0.01 significance level, test whether the company conclude that the average life of the presses is less than the hypothesized 14,500 hours and state which of the following statement is

166

correct? 76) a. Standard error of the test is 420 and the acceptance region’s lower limit is 13,521. 77) b. Standard error of the test is 420 and the acceptance region’s lower limit is 11,521. 78) c. Standard error of the test is 455 and the acceptance region’s lower limit is 14,152. 79) d. Standard error of the test is 455 and the acceptance region’s lower limit is 13,521. 80) e. Standard error of the test is 420 and the acceptance region’s lower limit is 12,521. 81) 123. A ketchup manufacturer is in the process of deciding whether to produce a new extra-spicy brand. The company’s marketing research department used a national telephone survey of 6,000 households and found that the extra-spicy ketchup would be purchased by 335 of them. A much more extensive study made 2 years ago showed that 5 percent of the households would purchase the brand then. At a 2 percent significance level, test whether the company concludes that there is an increased interest in the extra-spicy flavor and state which of the following statement is correct? 82) i. Null hypothesis H0: p = 0.05 83) H1: p < 0.05 84) ii. S.E of the proportion is 0.00281 85) iii. The acceptance region’s upper limit is 0.05578 86) iv. The company can conclude that there is an increased interest in the extra-spicy flavor. 87) a. Only (i), (ii) and (iv). 88) b. Only (i), (ii), (iii) and (iv). 89) c. Only (ii), (iii) and (iv). 90) d. Only (i) and (iv). 91) e. Only (i) and (iii). 92) 124. Given a sample mean of 83, a sample standard deviation of 12.5, and a sample size of 22, tests the hypothesis that the value of the population mean is 70, against the alternative that it is more than 70. Use the 2.5% significance level and state which of the following statement is correct?

93) i.

H0: μ=

70 is the null hypothesis

94) H1: μ > 70 is the alternative hypothesis 95) ii. Standard error is 2.665 96) iii. The upper limit of the acceptance region is 78.54 97) iv. H0 is rejected. 98) a. Only (i), and (ii). 99) b. Only (i), (iii) and (iv). 100) c. Only (i), (ii), and (iii). 101) d. Only (i), (ii) and (iv). 102) e. Only (i), (iii) and (iv). 103) 125. A television documentary on overeating claimed that Americans are about 10 pounds overweight on average. To test this claim, eighteen randomly selected individuals were examined, and their average excess weight was found to be 12.4 pounds, with a sample standard deviation of 2.7 pounds. At a significance level of 1%, is there any reason to doubt the validity of the claimed 10 pound value? 104) a.

Yes and the standard error of the mean is 0.6364.

105) b. 106) c.

No and the standard error of the mean is 0.6864. Yes as the lower acceptance limit are 8.156.

107) d.

No and the upper acceptance limit is 11.84.

108) e.

Yes and the upper acceptance limit is 13.54.

109) Based on the following information answer the questions no. 126 and 127.

110)Two independent samples of observations were collected. For the first sample of 60 elements, the mean was 86 and the standard deviation 6. The second sample of 75 elements had a mean of 82 and a 167

standard deviation of 9. 111) 126. What is the standard error of the difference between the two means? 112) a. 1.296. 113) b. 1.345. 114) c. 1.425. 115) d. 1.125. 116) e. 1.1010. 117) 127. Using α = 1%, test whether the two samples can reasonably be considered to have come from populations with the same mean and state which of the following statement is true? 118) a. Upper limit of the acceptance region are 3.331. 119) b. Upper limit of the acceptance region are 5.331. 120) c. Lower limit of the acceptance region are - 3.331. 121) d. Both a and c. 122) e. Both b and c. 123) Based on the following information answer the questions no. 128 - 130. 124)Two different areas of a large eastern city are being considered as sites for day-care centers. Of 200 households surveyed in one section, the proportion in which the mother worked full-time was 0.52. In the other section, 40 percent of the 150 households surveyed had mothers working at full-time jobs. Use 4% level of significance. ˆ 125) 128. What is the best estimates of combined proportion (i.e. p ) for the above test? 126) a. 0.4686. 127) b. 0.4925. 128) c. 0.4486. 129) d. 0.4725. 130) e. 0.4605. 131) 129. What is the estimated standard error of the difference between two proportion using combined estimates from both samples? 132) a. 0.0562. 133) b. 0.0525. 134) c. 0.0550. 135) d. 0.0535. 136) e. 0.0539. 137) 130. Is there any significant difference in the proportions of working mothers in the two areas of the city? 138) a. Yes and the upper limit of the acceptance region is 0.1105 the lower limit of acceptance is0.1105. 139) b. No and the upper limit of acceptance is 0.1205 and the lower limit of acceptance is 0.1205. 140) c. Yes and the upper limit of the acceptance region is 0.1005 the lower limit of acceptance is - 0.1005. 141) d. Yes and the upper limit of the acceptance region is 0.1095 the lower limit of acceptance is 0.1095. 142) e. No and the upper limit of acceptance is 0.1315 and the lower limit of acceptance is 0.1315. 143) Based on the following information answer the questions no. 131 - 133.

A coal-fired power plant is considering two different systems for pollution abatement. The first system has reduced the emission of pollutants to acceptable levels 68 percent of the time, as determined from 200 air

168

samples. The second, more expensive system has reduced the emission of pollutants to acceptable levels 76 percent of the time, as determined from 250 air samples. If the expensive system is significantly more effective than the inexpensive system in reducing pollutants to acceptable levels, then the management of the power plant will install the former system. ˆ 1) 131. What is the best estimates of combined proportion (i.e., p ) for the above test? 2) a. 0.9244. 3) b. 0.6245. 4) c. 0.7244. 5) d. 0.5225. 6) e. 0.6225. 7) 132. What is the estimated standard error of the difference between two proportion using combined estimates from both samples? 8)

a.

0.04625.

9)

b.

0.04530.

10) c.

0.04362.

11) d.

0.05215.

12) e.

0.04219.

13) 133. Which system will be installed if management uses a significance level of 0.02 in making its decision? 14) a. Management can install the less expensive system and the lower limit of the acceptance region is - 0.0983. 15) b. Management can install the more expensive system and the lower limit of the acceptance region is - 0.0925. 16) c. Management can install the less expensive system and the lower limit of the acceptance region is - 0.0953. 17) d. Management can install the more expensive system and the lower limit of the acceptance region is - 0.0993. 18) e. Management can install the more expensive system and the lower limit of the acceptance region is - 0.01083. 19) 134. Aquarius Health Club has been advertising a rigorous program for body conditioning. The club claims that after 1 month in the program, the average participant should be able to do eight more pushups in 2 minutes than he or she could do at the start. What is the standard deviation of the sample? Participant Before After

6)

1)

a.

3.859.

2)

b.

2.985.

3)

c.

3.652.

4)

d.

3.525.

5)

e.

3.925.

01 38 45

02 11 24

03 34 41

04 25 39

05 17 30

06 38 44

07 12 30

08 27 39

09 32 40

10 29 41

135. In the question no. 122. Test whether the random sample of ten program participants given below support the club’s claim? Use the 0.025 level of significance and state which of the following statement is true?

169

7)

i. The appropriate hypothesis is H0: μ1 - μ2 = 8(Null Hypothesis: The difference in increase is 8.) H1: μ1 - μ2 < 8

8)

ii.

ˆ S.E σ x = 1.22

9)

iii.

The upper limit of the acceptance region is 10.76

10) iv.

H0 is rejected hence the club’s claim is supported.

11) a.

Only (i), (ii), and (iv).

12) b.

Only (i) and (iii).

13) c.

Only (i) and (iv).

14) d.

Only (iii) and (iv).

15) e.

Only (ii), (iii), and (iv).

16) 136. A car retailer thinks that a 40,000 mile claim for tyre life by the manufacturer is too high. She carefully records the mileage obtained from a sample of 64 such tyres. The mean turns out to be 38,500 miles. The standard deviation of the life of all tires of this type has previously been calculated by the manufacturer to be 7,600 miles. Assuming that the mileage is normally distributed, determine the largest significance level at which we should accept the manufacturer’s mileage claim, that is, at which we would not conclude the mileage is significantly less than 40,000 miles. 17) a.

0.0571

18) b.

0.0457

19) c.

0.517

20) d.

0.0525

21) e.

0.0565.

22) 137. A large portfolio manager purchases 250 scrips through stockbroker X and 300 scrips through stockbroker Y. When the manager sent the scrips for name transfer to respective companies, he found that 12 of the 250 scrips bought through broker X were returned as bad deliveries while 20 of the 300 scrips bought through broker Y were returned as bad deliveries. What is the best estimates of ˆ combined proportion (i.e., p ) for the above test? 23) a.

0.052.

24) b.

0.062.

25) c.

0.058.

26) d.

0.049.

27) e.

0.054.

28) 138. In the above question no. 137. What is the standard error of the proportion?

29) a.

0.02.

30) b.

0.0195.

31) c.

0.0245.

32) d.

0.031.

33) e.

0.0365.

34) 139. A sample of eight workers in a cloth manufacturing company gave the following figures for the amount of time (in minutes) needed to mount a collar on a shirt: 10, 12, 13, 9, 8, 14, 10, and 11. At the 5% level of significance, would it be justified in saying that the mean time to mount a collar is more

170

than 10 minutes? (Assumption: X is normally distributed) 35) i.

ˆ Yes and σ x = 0.718066

36) ii.

Yes and t observed = 1.22

37) iii.

No and the appropriate hypothesis is μ 38) H0: = 10, H1: μ > 10

39) iv.

No and the t observed = 1.22

40) v.

ˆ No and σ x = 0.718066

41) a.

Only (i) and (ii).

42) b.

Only (i).

43) c.

Only (ii).

44) d.

Only (iii), (iv) and (v).

45) e. Only (iii) and (iv). 46) 140. In a survey of banking habits of the population, 1000 persons were interviewed. Of these 480 had an account with nationalized banks only, 320 had an account with private sector banks only and remaining did not have any account in any of the banks. 47)

Assuming that this is a random sample, construct a 95% confidence interval for the true proportion of the population having banking relations with either the nationalized banks or the private sector banks.

48)

(Also assume that the events of having accounts with nationalized bank and private sector banks are mutually exclusive.) 49) a.

77.52%, 82.48%

50) b.

75.25% and 80.65%

51) c.

75.9% and 83.65%

52) d.

77.9% and 85.25 %

53) e.

78.65% and 84%.

54) 141. A Bank Manager believes that the Monday Morning cash balance of his bank should not fall significantly below Rs.5 crore because this would increase the risk of cash-out. At the same time the cash balance should not be significantly more than Rs.5 crore because this would increase the loss of interest on idle cash balances. He checks a sample of 100 Monday Morning balances and finds the average to be Rs.5.04 crore. If the Manager uses a significance level of 5%, is this average significantly different from the required cash balance of Rs.5 crore? Long experience reveals that the standard deviation of Monday balances is Rs.30 lakh. 55) a.

No and the σ x is Rs.3.5 lakh

56) b.

No and the acceptance region is 494.12 and 505.88

57) c.

No and the acceptance region is 493.85 and 506.15

58) d.

Yes and the acceptance region is 496.25 and 503.75

59) e.

No and the acceptance region is 493.25 and 506.75.

60) Based on the following information answer the questions no. 142 - 143.

61) In order to test whether declaration of dividends has any effect on the market price of a share of a company a random sample of 8 companies was taken from companies which have declared at least 15% dividends. The data regarding share prices of the sample companies is Market price 10 days before dividends were declared

1 70

2 65

3 112

4 58

5 25

6 147

7 95

8 68

171

Market price 10 days after the declaration of dividends

7)

80

85

110

1)

142. What is the standard deviation of the distribution?

2)

a.

2.3569.

3)

b.

2.2542.

4)

c.

2.1023.

5)

d.

2.6523.

6)

e.

2.8245.

64

32

159

100

70

143. What is the value of t for the appropriate test (with 7 degrees of freedom)?

8)

a.

3.182.

9)

b.

3.125.

10) c.

3.525.

11) d.

3.805.

12) e. 4.025. 13) 144. A money market mutual fund manages a Rs.100 crore fund invested in money market securities. It has assessed its average daily cash needs as 10% of the corpus to meet the redemptions pressure even in difficult times. Last 100 days average cash balance was found to be 10.08% of the corpus. The experience of mutual fund manager has revealed the standard deviation of average daily cash balances as Rs.30 lakh. What is the value of standard error ( σ x ) for the test? 14) a.

Rs.0.0295 crore.

15) b.

Rs.0.0354 crore.

16) c.

Rs.0.03 crore.

17) d.

Rs.0.0284crore.

18) e.

Rs.0.0325 crore.

19) 145. An index fund manager compiled comprehensive data on weekly returns from the stock market index for last 10 years. On studying the data, he found that the index provided a weekly return of 0.4% on an average with a standard deviation of 0.35% during the 10-year period and a return of 0.32% with a standard deviation of 0.45% in the last nine months. He wants to find out whether the average weekly return from the index has declined from its long-term average. Help him in finding out the same using the test of hypothesis and state which of the following is the rejection level. Assume a level of significance of 5%. You may also assume that long-term average weekly returns from the index follow a normal distribution.

20) a.

Below 0.308

21) b.

Below 0.325

22) c.

Below 0.350

23) d.

Above 0.325

24) e.

Above 0.308.

25) 146. A soft drink manufacturer has allegedly been under filling its bottles. Upon inspection by the authorities it is found from a sample of 81 bottles that the average volume of contents is 298.5 ml. The manufacturer claims that its bottles contain 300 ml. The population standard deviation is known to be 2.7 ml. Assume 5% level of significance. What is the standard error of mean ( σ x )? 26) a.

0.30.

27) b.

0.25.

28) c.

0.28.

29) d.

0.29. 172

30) e.

0.27.

31) 147. In the above question no.134 does the finding justify the allegation?

32) a.

Yes and the acceptance value of Z is -1.64 and Z value for the observed data is -5.00.

33) b.


34) c.


35) d.


36) e.


Based on the following information answer the questions no. 148 and 149.

1)

Orient Spares has been supplying bolts of a specific size to Western Machine Tools Ltd. The standard deviation of the diameter of the bolts is known to be 4.0 mms. Orient Spares has recently supplied a lot of 10,000 bolts to the company and it claims that the bolts have a mean diameter of 4.0 cms. The quality control department of the company has taken a sample of 25 bolts and identified the mean diameter of the bolts to be 4.4 cms.

2)

You are required to find out whether or not the lot should be accepted at a level of significance of 1%. 3)

148. What is the standard error of the mean for the appropriate test?

4)

a.

0.0825.

5)

b.

0.07.

6)

c.

0.09.

7)

d.

0.075.

8)

e.

0.08.

9)

149. State whether or not the lot should be accepted at a level of significance of 1%?

10) a.

No and for the observed sample, Z = 5.

11) b.

No and for the observed sample Z = 4.

12) c.

No and the acceptance region are Z = ± 2.57.

13) d.

Both a and c only.

14) e.

Both b and c only.

15) 150. Kitchen-care Products Ltd. has a sales force of 561 sales persons. The sales manager of the company wants to know the estimated interval within which the average commission paid to its salespersons over the last year falls. A sample of 49 salespersons are taken and the average is found to be Rs.48,000. The standard deviation of the sample is Rs.4,200. What is the standard error of the mean for the finite population? 16) a. 576.51. 17) b. 573.71. 18) c. 523.75. 19) d. 585.23. 20) e. 562.35. 21) 151. In the above question 150 you are required to find out a 95 percent confidence interval for the average commission paid in the last year? 22) a. The average commission paid to the sales persons last year falls between Rs.48,250 and Rs.46,875. 23) b. The average commission paid to the sales persons last year falls between Rs.49,124.47 and Rs.46,875.52. 24) c. The average commission paid to the sales persons last year falls between Rs.50,125.75 and Rs.45,215.25. 25) d.

The average commission paid to the sales persons last year falls between Rs.51,124.47 and 173

Rs.47,880.25. 26) e. The average commission paid to the sales persons last year falls between Rs.54,124.47 and Rs.49,975.63. 27) 152. The senior consultant of Robinson & Smith Advisory Services is interested to know whether the mean daily wages paid to the semi-skilled workers are same in two industries. The following information has been collected: Industry Mean daily Standard deviation of Sample size wages (Rs./day) sample (Rs./day) Industry 1 64 4.40 121 Industry 2 70 5.20 100 What is the Estimated standard error of the difference between the two means? 1) a. 0.656. 2) b. 0.625. 3) c. 0.526. 4) d. 0.685. 5) e. 0.612.

6)

153. In the above question no.140 at a 5% level of significance. What is the standardized value of the difference between sample means to find out whether the mean daily wages paid to the semi-skilled workers are same in both the industries?

7)

a.

-9.65.

8)

b.

-9.75.

9)

c.

-8.25.

10) d.

-8.75.

11) e.

-9.15.

12) 154. By throwing 5 dice 30 times, a person obtained success 23 times securing a 6 which was considered success. One wants to whether there is any difference between observed result and expected result as being significantly different, what is the test statistics for this appropriate test? 13) a.

-0.498.

14) b.

-0.438.

15) c.

0.398.

16) d.

-0.456.

17) e.

-0.412.

18) 155. The mean lifetime of 100 electric bulbs produced by a manufacturing company is estimated to be 1570 hours with a standard deviation of 120 hours. If μ is the mean life time of all the bulbs produced by the company, and one wants to test the hypothesis μ = 1600 hrs, using a level of significance of 0.05 then, which of the following is the test statistics value for the above test? 19) a.

-2.35.

20) b.

-2.62.

21) c.

-2.75.

22) d.

-2.50.

23) e.

-2.85.

24) 156. Certain refined edible oil is packed in times holding 16 kg each. The filling machine can maintain this but a with a S.D of 0.5 kg. Samples of 25 were taken from the production line. If a sample mean is 16.35 kg and one wants to test that sample has come from 16 kg tins what is the test statistics for the appropriate test? 25) a.

3.5. 174

26) b.

3.12.

27) c.

3.25.

28) d.

3.85.

29) e.

2.95.

30) 157. The mean produce of rice of a sample of 100 fields is 200 ib per acre with a standard deviation of 10 Ib. Another sample of 150 fields gives the mean at 220 Ib with a standard deviation of 12 Ib. Assuming the standard deviation of the mean field at 11 Ib, for the universe find at 15 level if the two results are consistent, what is the appropriate test statistics of the above test? 31) a.

-12.

32) b.

-13.

33) c.

-14.

34) d.

-15.

35) e.

-16.

36) 158. If the population standard deviation is 20 and the sample size is 16 then the standard error of mean will be 37) a.

1.12

38) b.

1.25

39) c.

5.00

40) d.

6.75

41) e.

9.23.

42) 159. In a certain State A, 450 people were considered regular consumer of Pepsi out of a sample of 1000 persons. In another State B 400 out of 800 consumed Pepsi regularly. One wants to test whether there is any difference between the two states as far as Pepsi drinking habit, what is the appropriate test stastics for the above test? 43) a.

1.978.

44) b.

-1.215.

45) c.

-2.11.

46) d.

-2.45.

47) e.

-3.11.

48) 160. Measurements of the diameters of a random sample of 200 ball bearings made by a certain machine during one week showed a mean of 0.824 inches and standard deviation of 0.042 inches. At 95% confidence, what is the lower limits for the mean diameter of all the ball bearings? 49) a.

0.81818.

50) b.

0.80812.

51) c.

0.82635.

52) d.

0.84521.

53) e.

0.86521.

54) 161. In a sample of 600 parts manufactured by a factory the number of defective parts was found to be 45. The company however claimed that only 5% of their product is defective. What is the test statistics of the test?

55) a.

2.31.

56) b.

2.45.

57) c.

2.05.

58) d.

2.81.

175

59) e.

2.67.

60) 162. The mean I.Q. of a sample of 1600 children was 99. Is it likely that this was a random sample from a population with mean I.Q. 100 and standard deviation 15, what is the test statistics value for the test? 61) a.

-2.67.

62) b.

-2.50.

63) c.

-2.25.

64) d.

2.15.

65) e.

2.10.

66) 163. Two sets of 10 students selected at random from a college were taken. One was given memory tests as they were and the other set was given a memory test after two weeks training and the scores were given below: Set A: 10 8 7 9 8 10 9 6 7 8 Set B: 12 8 8 10 8 11 9 8 9 9 1) 2) 3) 4) 5) 6)

Do you think there is any significant effect due to training? (Given, t = 2.10 at d.f = 18, =0.05) a. - 7. b. - 6. c. - 5. d. - 8. e. - 7.5.

7)

164. The heights of men are normally distributed with mean, m = 69 inches, and standard deviation, s = 4 inches. What proportion of men are taller than 72 inches?

8)

a.

0.2154.

9)

b.

0.2266.

10) c.

0.2365.

11) d.

0.2564.

12) e.

0.2015.

13) 165. The heights of men are normally distributed with mean, m = 69 inches, and standard deviation, s = 4 inches. What proportion of men are between 65 and 67 inches in height? 14) a.

0.1225.

15) b.

0.1498.

16) c.

0.1525.

17) d.

0.1625.

18) e.

0.1563.

19) 166. The heights of 1000 Iron rods maked with a certain mix have a normal distribution with a mean of 5.75 cm. And a standard deviation of 0.75 cm. Find the number of iron rods having heights between 5 cm and 6.25 cm. For a standard normal variate z, the area between z = -1 and z = 0 is 0.6286, the area between z = 0 and z = 0.67 is 0.2486 and the area between z = 0 and z = 0.84 is 0.3000. 20) a.

590

21) b.

580

22) c.

600

23) d.

585

24) e.

595.

176

25) 167. In the above question, find the maximum height of the flattest 200 cakes. 26) a.

5.50 cm

27) b.

5.12 cm

28) c.

5.30 cm

29) d.

5.21 cm

30) e.

5.85 cm.

31) 168. In a CFA examination mean of marks scored by 400 students is 45 with a standard deviation of 15. Assuming the distribution to be normal, what is the number of students securing marks between 30 and 60? 32) a.

250.

33) b.

256.

34) c.

259.

35) d.

273.

36) e. 279. 37) 169. A simple random sample of size 49 is drawn from a finite population consisting of 205 items. If the population standard deviation is 12.5, then what is the standard error of sample mean when the sample is drawn without replacement? 38) a.

1.56.

39) b.

1.65.

40) c.

1.78.

41) d.

1.87.

42) e. 2.01. 43) Based on the following information answer the questions no. 170 and 171. 44) A company is interested in knowing if here is a difference in the average salary received by foremen in two divisions. Accordingly sample of 12 foremen in the first division and 10 foremen in the second division are selection at random. Based upon experience foremen’s salaries are known to be approximately normally distributed, and the standard deviations are about the same.

Sample size Monthly salary of foreman Standard deviation of salaries

First Division 12 1,050 60

Second Division 10 980 74

1) 2) 3) 4) 5) 6) 7)

170. What is the pooled estimate of standard deviation for the above test? a. 70.76. b. 71.56. c. 72.65. d. 69.50. e. 70.15. 171. What is the test statistics value of the above test? 8) a. 2.51. 9) b. 2.31. 10) c. 2.10. 11) d. 1.97. 12) e. 3.22. 13) 172. A simple sample of heights of 8,000 boys has a mean of 168 cms. And a standard deviation 6,8 cms. While a simple sample of height of 1,600 girls has a mean of 172 cm and a standard deviation of

177

6.2 cm. What is the value of test statistic of the above test? 14) a. 11.58. 15) b. 13.50. 16) c. 10.25. 17) d. 10.9. 18) e. 14.25. 19) 173. You are given the following information relating to purchase of bulbs from two manufacturers A and B. Manufacturer No. of bulbs bought Mean life S.D A 50 1980 hours. 80 hrs. B 70 2010 hours. 60 hrs. 1) 2) 3) 4) 5) 6) 7)

Is the difference between standard deviations significant at 10% level? a. Yes and the value of test statistics is -2.29. b. Yes and the value of test statistics is -2.24. c. Yes and the value of test statistics is -2.92. d. No and the value of test statistics is 2.29. e. Yes and the value of test statistics is 2.24. 174. The manufacturer of a new brand of bulb guarantees that, with the aid of company’s unique marketing strategies only 22% of the bulb marketed will not sell quickly. Does an experiment in which 30 of 100 bulbs marketed did not sell quickly tend to refuse the company’s claim at 1% level? 8)

a.

Claim of the manufacturer is justified and value of test statistics is 1.818.

9)

b.

Claim of the manufacturer is justified and value of test statistics is 1.

10) c.

Claim of the manufacturer is justified and value of test statistics is 1.215.

11) d.

Claim of the manufacturer is not justified and value of test statistics is 2.56.

12) e.

Claim of the manufacturer is not justified and value of test statistics is 2.95.

13) 175. A random sample of size 16. The sum of the squares of the deviations taken from mean is 160. Can this sample be regarded as taken from the population having 56 as mean? What is the test statistics for the above data? 14) (For, v = 15, t0.05 = 2.131, tp.01 = 2.947) 15) a. -3.63. 16) b. -3.25. 17) c. -2.98. 18) d. -2.75. 19) e. - 4.00. 20) 176. Calculate 95% confidence limits of the mean of the population given that (For variance = 15, t0.05 = 2.131, tp.01 = 2.947) 21) a. 51.30 and 54.70 22) b. 54.65 and 51.35 23) c. 54.10 and 50.90 24) d. 55 and 51 25) e. 58 and 49. 26) 177. In a CAT examination mean of marks scored by 400 students is 45 with a standard deviation of 15. Assuming the distribution to be normal, what are the limits between which marks of the middle 50% of the students lie? 27) a. 35 and 55. 28) b. 40 and 60.

178

29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44)

45) 46) 47) 48) 49) 50)

55)

c. 42 and 62. d. 49 and 69. e. 39 and 59. 178. For a normal distribution with mean 3 and variance 16, find the value y of the variate such that the probability of the variate lying in the internal (3, y) is 0.4772. [You are given P(Z ≤ 2) = 0.9772] a. 11 b. 12 c. 13 d. 14 e. 10. 179. In a sample of 120 workers in a factory the mean and standard deviation of wages were Rs.11.35 and Rs.3.03 respectively. Find the percentages of workers getting wages between Rs.9 and Rs.17 in the whole factory assuming that the wages are normally distributed. a. 74% b. 73% c. 72% d. 75.09% e. 76.25%. 180. The mean diameter of a sample of 200 bolts produced by a Machine is 5.02 mm and the standard deviation is 0.05 mm. Maximum tolerance in the diameter allowed ranges from 4.9 mm to 5.08 mm (otherwise, the bolts are defective). What is the percentage of defective bolts produced by the machine? (Assume that the diameters are normally distributed and area under standard normal curve between z = 0 and z = 1.2 is 0.3849.) a. 22%. b. 23%. c. 22.5%. d. 23.5%. e. 24%. 181. A simple random sample of size 49 is drawn from a finite population consisting of 205 items. If the population standard deviation is 12.5, then what is the standard error of sample mean when the sample is drawn without replacement? 51) a. 1.56. 52) b. 1.65. 53) c. 1.78. 54) d. 1.87. e. 2.01.

56) 182. In a test of ANOVA, the estimate of the population variance based on the variance among the sample means is 3.362 and the estimate of the population variance based on the variance within samples is 1.124. What is the F ratio for this test? 57) a. 0.334. 58) b. 1.944. 59) c. 2.243. 60) d. 2.991. 61) e. 3.779. 62) 183. The academic council of Ohio University has decided to bring some changes to its existing grading system. The university wants to determine what proportion of the students is in favour of new grading system. The total number of students enrolled in the university is 60,000 and the university feels that atleast 36,000 of them support the changes. What should be the sample size that will enable

179

the university to be 89.9 percent certain of estimating the true proportion in the favour of new system within plus and minus 0.02? 63) a. 1,075. 64) b. 1,411. 65) c. 1,613. 66) d. 1,680. 67) e. 1,711. 68) 184. John Wright, an overzealous graduate student, has just completed a first draft of his 700-page dissertation. John has typed this paper himself and is interested in knowing the average number of typographical errors per page, but does not want to read the whole paper. Knowing a little bit about business statistics, John selected 50 pages at random to read and found that the average number of typographical errors per page was 4.5 and the sample standard deviation was 1.2 typographical errors per page. What will be the interval estimate for the average number of typographical errors per page in his paper at the 90 % confidence level? 69) a.

4.5 ± 0.2683.

70) b.

4.5 ± 0.2783.

71) c.

4.5 ± 0.3206.

72) d.

4.5 ± 0.3326.

73) e.

4.5 ± 0.3665.

74) 185. Atlas Sporting Goods has implemented a special trade promotion for its tennis balls and feels that the promotion should result in a price benefit for the consumer. Atlas knows that before the promotion began, the average retail price of the ball was Rs. 44.95, with a standard deviation of Rs.5.75. Atlas samples 25 of its retailers after the promotion was launched and finds the mean price for the ball is now Rs.42.95. Atlas wants to examine whether the price of balls has significantly reduced after the trade promotion. Which of the following conclusions can be drawn on the basis of the above information at a significance level of 2%? 75) a.

The average retail price to the consumers has increased significantly.

76) b.

The average retail price to the consumers has not decreased significantly.

77) c.

The average retail price to the consumer has decreased significantly.

78) d.

We cannot draw any conclusion, as the sample size is too small.

79) e.

We cannot draw any conclusion, as information is insufficient.

80) 186. Anand Electronics runs a chain of stores selling stereo systems and components. It has been very successful in many university towns, but it has had some failures. Analysis of its failures has led it to adapt a policy of not opening a store unless it can be reasonably certain that more than 15 percent of the students in town own stereo systems costing Rs.5,000 or more. A survey of 300 students at BHU, Varanasi has revealed that 57 of them own stereo systems costing atleast Rs. 5,000. Anand Electronics is willing to run a 5 percent risk of failure, and wants to make a decision on the basis of the sample result. Which of the following are correct for this test of hypothesis? i.

A two-tailed test of hypothesis should be done.

ii.

A left tail test of hypothesis should be done.

iii.

A right tail test of hypothesis should be done.

iv.

The alternative hypothesis is p ≠ 0.15.

v.

The alternative hypothesis is p < 0.15.

vi.

The alternative hypothesis is p > 0.15.

vii.

The standard error of proportion is 0.19.

viii. The standard error of proportion is 0.02375. ix.


180

x.

The critical values for the test are -1.96 and +1.96.

xi.

The critical value for the test is -1.64.

xii.

The critical value for the test is +1.64.

xiii. The standardized value of sample proportion is − 0.21. xiv. The standardized value of sample proportion is 1.684. xv.

The standardized value of sample proportion is 1.766.

xvi. At a 5 percent risk of failure Anand Electronics should open a store in Varanasi. xvii. At a 5 percent risk of failure Anand Electronics should not open a store in Varanasi.

6)

1)

a.

Only (i), (iv), (vii), (x), (xiii) and (xvii).

2)

b.

Only (ii), (v), (viii), (xi), (xiv) and (xvi).

3)

c.

Only (ii), (v), (ix), (xi), (xv) and (xvi).

4)

d.

Only (iii), (vi), (ix), (xii), (xv) and (xvi).

5)

e.

Only (iii), (vi), (viii), (xii), (xv) and (xvii).

187. The Securities Exchange Board of India (SEBI) was considering a proposal to require companies to report the potential effect of employees’ stock options on earnings per share (EPS). A random sample of 41 high-technology firms revealed that the new proposal would reduce EPS by an average of 13.8 percent, with a standard deviation of 18.9 percent. A random sample of 35 producers of consumer goods showed that the proposal would reduce EPS by 9.1 percent on average, with a standard deviation of 8.7 percent. On the basis of these samples, is it reasonable to conclude at significance level of 10 percent that the SEBI proposal will cause a greater reduction in EPS for high technology firms than the producers of consumer goods? Which of the following are correct for this test of hypothesis? i. A two-tailed test of hypothesis on difference between means is appropriate. ii. A one tailed test of hypothesis on difference between means is appropriate. iii. The estimated standard error of difference between the means is 2.559. iv. The estimated standard error of difference between the means is 3.768. v. The estimated standard error of difference between the means is 3.298. vi. The standardized difference between the sample means is 1.836. vii. The standardized difference between the sample means is 1.247. viii. The standardized difference between the sample means is 1.425. ix. At a significance level of 10 percent, we can conclude that the SEBI proposal will not cause a greater reduction in EPS for high technology firms than the producers of consumer goods. x. At a significance level of 10 percent, we can conclude that the SEBI proposal will cause a greater reduction in EPS for high technology firms than the producers of consumer goods.

1)

a.

Only (i), (iii), (vi) and (ix).

2)

b.

Only (ii), (v), (viii) and (x).

3)

c.

Only (ii), (iv), (vii) and (ix).

4)

d.

Only (i), (iv), (vii) and (x).

5)

e.

Only (ii), (v), (vi) and (ix).

6)

188. Goofy Toys is trying to establish a relationship between the number of toys that can be sold in a given month and the price that can be realized for one of its new toys, Scooby Doo. The marketing head of the company has provided the following information: Price (in Rs.) 28 26 25 24 21 19 Quantity that can be sold 135 160 170 180 215 235

1)

The dealer establishes a linear regression equation based on the above information to predict the quantity of toys he can sell for a given price. The dealer also wishes to test the accuracy of the slope

181

of the regression line. Which of the following statements are true about the regression equation? The estimated number of toys that can be sold by the dealer at a price of Rs 30 is 2) i. approximately 114. 3) ii. The estimated number of toys that can be sold by the dealer at a price of Rs 30 is approximately 125. 4) iii. The estimated number of toys that can be sold by the dealer at a price of Rs 30 is approximately 130. 5) iv. The standard error of estimate for the regression line is approximately 4.256. 6) v. The standard error of estimate for the regression line is approximately 10.568. 7) vi. The standard error of estimate for the regression line is approximately 24.346. 8) vii. The standard error of b coefficient is 0.5698. 9) viii. The standard error of b coefficient is 1.6785. 10) ix. The standard error of b coefficient is 4.3568. 11) a.

Only (i), (iv), and (vii)

12) b.

Only (ii), (v) and (viii)

13) c.

Only (iii), (vi), and (ix)

14) d.

Only (ii), (iv) and (vii)

15) e. Only (i), (v) and (ix). 16) 189. A research company has designed three different systems to clean up oil spills. The following table contains the results, measured by how much surface area (in square meters) is cleared in one hour. The data were found by testing each method in several trials. System A 55 60 63 56 59 55 System B 57 53 64 49 62 System C 66 52 61 57 1)

The research team wants to test the results whether the three systems are equally effective at a significance level of 5%. Which of the following are true for the above test? 2)

i. The estimate of the population variance based on the variance among the sample means is 4.467.

3)

ii. The estimate of the population variance based on the variance among the sample means is 4.501.

4) 5)

iii. iv.

The estimate of the population variance based on the variances within the samples is 26.00. The estimate of the population variance based on the variances within the samples is 22.28.

6) 7)

v. vi.

The critical value for the given test is 3.89. The critical value for the given test is 3.29.

8)

vii. The critical value for the given test is 3.49.

9)

viii. At a significance level of 5 percent, the research team should conclude that there is no significant difference in the surface area cleaned in one hour by the three systems.

10) xi. At a significance level of 5 percent, the research team should conclude that there is a significant difference in the surface area cleaned in one hour by the three systems. 11) a. Only (ii), (iv), (vi) and (ix) 12) b.

Only (i), (iii), (v) and (viii)

13) c. 14) d.

Only (i), (iii), (vii) and (ix) Only (ii), (iv), (v) and (viii)

15) e.

Only (i), (iii), (vi) and (viii).

16) 190. For a sample randomly collected from a population, the following details are available: 17) Sum of the squares of the observations = 1840

182

18) Sum of the observations

= 160

= 16 19) Number of observations 20) What is the estimated standard error of mean? 21) a.

1.

22) b. 23) c.

2. 4.

24) d.

10.

25) e.

16.

26) 191. From an association consisting of 540 individuals, a sample of 60 individuals is taken. From this sample, the average age of the individuals is found to be 31 years and the standard deviation is found to be 6.84 years. 27) A 95 percent confidence interval for the mean age of the individuals in the association has to be constructed. The lower and upper confidence limits of the confidence interval are 28) a.

24.16 years and 37.84 years respectively

29) b.

28 years and 33 years respectively

30) c.


31) d.


32) e. 17.6 years and 44.41 years respectively. 33) 192. A computer store purchases untested microchips for computers, from a wholesaler. It has to estimate the proportion of faulty microchips in the consignments supplied by the wholesaler. From a large consignment sent by the wholesaler a sample of 200 microchips were tested and 20 microchips were found to be faulty. 34) A 98 percent confidence interval has to be constructed for the true population proportion of faulty microchips. The lower and upper confidence limits of the confidence interval are 35) a. 1 percent and 19 percent respectively 36) b. 5.08 percent and 14.92 percent respectively 37) c. 7.88 percent and 12.12 percent respectively 38) d. 87.88 percent and 92.12 percent respectively 39) e. 85.08 percent and 94.92 percent respectively. 40) 193. According to a recently completed survey in a city, the mean number of hours of television viewing per household is 7.25 hours per day with a standard deviation of 2.4 hours per day. The survey involved a sample of 200 households spread in different parts of the city. One of the private television channels assumes that the mean number of hours of television viewing per household in the city, is 6.70 hours per day. It is to be tested whether the mean number of hours of television viewing per household in the city is more than 6.70 hours per day. 41) At a significance level of 5 percent, what is the conclusion? 42) a. The sample mean is incorrect. 43) b. The sample standard deviation is incorrect. 44) c. The mean number of hours of television viewing per household in the city is equal to 6.70 hours per day. 45) d. The mean number of hours of television viewing per household in the city is less than 6.70 hours per day. 46) e. The mean number of hours of television viewing per household in the city is more than 6.70 hours per day. 47) 194. A random sample of 300 loans was collected from the loans made during the recent five year period, by a financial institution which finances small scale enterprises. This sample showed that 120 of the loans were made to women entrepreneurs. A complete census of all the loans made by the financial institution five years ago showed that 42 percent of the loans were made to women

183

entrepreneurs. It is to be tested whether the proportion of loans made to the women entrepreneurs by the financial institution has reduced in the past five years. 48) At a significance level of 2 percent, what is the conclusion? 49) a. The sample information is incorrect. 50) b. The population proportion is incorrect. 51) c. The proportion of loans made to the women entrepreneurs has reduced in the past five years. 52) d. The proportion of loans made to the women entrepreneurs has increased in the past five years. 53) e. The proportion of loans made to the women entrepreneurs has not reduced in the past five years. 54) 195. A commodity merchant knows that the mean retail price of a specific variety of rice three months ago was Rs.14.50 per kg. In the current month the merchant has collected the information on the price charged for the same variety of rice by 16 randomly selected merchants in the same city. It was found from the sample that the mean retail price was Rs.15.00 per kg and the standard deviation was Rs.1.25 per kg. It is to be tested whether the mean retail price of the rice in the current month is more than Rs.14.50 per kg. 55) At a significance level of 5 percent, what is the conclusion? 56) a. The sample standard deviation is incorrect. 57) b. The sample mean is incorrect. 58) c. The mean retail price of the rice in the current month is more than Rs.14.50 per kg. 59) d. The mean retail price of the rice in the current month is not more than Rs.14.50 per kg. 60) e. The mean retail price of the rice in the current month is less than Rs.14.50 per kg.

61) Linear Regression 62) 196. If the variances of X and Y are 2.3 and 4.1, and the covariance between them is 2.1 then the coefficient of correlation between X and Y is 63) a. 0.098 64) b. 0.223 65) c. 0.408 66) d. 0.684 67) e. 0.748. 68) 197. The slope and the intercept on y-axis for a simple regression line are 3 and 5 respectively. If the independent variable has a value of 5 then the estimated value of the dependent variable is 69) a. 3 70) b. 5 71) c. 20 72) d. 27 73) e. 28. 74) 198. For a simple linear regression equation Σ(Y - Yˆ)2 = 178.56 and there are 11 pairs of observations. The standard error of estimate is 75) a. 4.03 76) b. 4.45 77) c. 12.60 78) d. 13.36 79) e. 19.84. 80) 199. For a regression equation, Yˆ = 3 - 5X, the coefficient of determination (R2) is 0.36. The correlation coefficient between X and Y is equal to 81) a. - 0.60 184

82) 83) 84) 85) 86) 87) 88) 89) 90) 91) 92) 93) 94) 95)

b. c. d. e. 200.

- 0.36 - 0.06 0.06 0.36. Given that ∑X = 118, ∑Y=141, n = 10 ∑XY=1677, ∑X2 = 1404, What is the regression co-efficient of Y on X? a. 1.37. b. 1.25. c. 1.29. d. 1.42. e. 1.65. 201. Given the regression equation of y on x and x on y are respectively y = 2x and 6x - y = 4. What is the correlation coefficient between x and y?

96) a. 97) b. 98) c. 99) d. 100) e.

0.578. 0.625. 0.595. 0.653. 0.675.

101) 202. Given that the variance of x = 9 and the regression lines are: 102) 8x - 10y + 66 = 0 103) 40x - 18y = 214 104) What will be the Coefficient of correlation? 105) a.

0.6.

106) b.

0.59.

107) c.

0.85.

108) d.

0.72.

109) e.

0.65.

110) 203. In the above question, what is the standard deviation of y? 111) a.

4.

112) b.

4.25.

113) c.

4.5.

114) d.

3.99.

115) e.

4.63.

116) 204. The lines of regression of y on x and x on y are respectively y = x + 5 and 16x - 9y = 94. 117) What will be the standard deviation of x if the variance of y is 16. 118) a.

3.

119) b.

3.5.

120) c.

3.8.

121) d.

4.

122) e.

4.2.

185

123) 205. In the above question, what will be the covariance of x and y? 124) a.

3.

125) b.

8.

126) c.

4.

127) d.

5.

128) e.

9.

129) 206. For a given data set of 10 observations of two variables X and Y, the covariance between X and Y is -121.5 and the standard deviation of the variable X is 9.0. The sums of the observations of X and Y in the given data set is 55 and 33 respectively. If a regression line is plotted for estimating the value of Y for a given value of X, then what is the estimated value of Y for X equal to 5? 130) a.

2.15.

131) b.

3.25.

132) c.

4.05.

133) d.

5.15.

134) e. 6.25. 135) 207. The conditional returns on the securities of XYZ Ltd. and PQR Ltd for year 2003 with their respective probabilities are as follows: Market Condition

Probability

Return (%)

Bullish Stable Bearish

40 % 50 % 10 %

XYZ 40 25 -15

PQR 35 25 20

1)

What is the covariance of returns?

2)

a.

67.0% squared.

3)

b.

73.0% squared.

4)

c.

77.5% squared.

5)

d.

87.5% squared.

6) 7)

e. 93.0% squared. 208. The probability distribution of returns of stock of M/s. ACRO Ltd. and the returns on market are given below: Probability (P) 0.30 0.35 0.15 0.20

1) 2) 3) 4) 5) 6) 7) 8) 9)

Returns of stock of M/s. ACRO Ltd. (in %) 7 8 14 16

Market returns (in %) 9 5 10 14

The variance associated with the market returns is 10.6875(%)2. The risk-free rate of return is 6%. According to CAPM, the risk premium for the stock of M/s. ACRO Ltd., is a. 1.77% b. 2.43% c. 2.56% d. 2.72% e. 3.39%. 209. The correlation coefficient between return of M/s. X Ltd. and the market return is 0.45. The variance of M/s. X Ltd. is 3.33 and that for the market is 10. What is the value of beta? a. 0.40. b. 0.56.

186

10) c. 0.26. 11) d. 1.00. 12) e. 1.25. Based on the following information answer the questions no. 210 - 212. x y Rs. Rs. Arithmetic average 6 8 Standard deviation 5 40/3 8 . 15 Coefficient of correlation between x and y =

1)

210. What is the regression coefficient of y on x?

2)

a.

64/45.

3)

b.

32/45.

4)

c.

16/45.

5)

d.

8/45.

6)

e.

64/90.

7)

211. What is the regression equation of x on y?

8)

a.

y = 5x - 32.

9)

b.

y = 5x - 22.

10) c.

y = 5x +32.

11) d.

y = 3x - 32.

12) e.

y = 8x -15.

13) 14) 15) 16) 17) 18)

What is the most likely value of y when x = 100 Rupees? 139.25. 141.70. 142.62. 143.52. 149.63.

212. a. b. c. d. e.

19) Based on the following information answer the questions no. 213 and 214. X Y

1 1

3 2

4 4

6 4

1) 2) 3) 4) 5) 6)

213. a. b. c. d. e.

What is the covariance between X and Y? 11. 12. 12.25. 13. 14.25.

7) 8) 9) 10) 11) 12)

214. a. b. c. d. e.

What is the coefficient of correlation? 0.999. 0.977. 0.925. 0.915. 0.895.

8 5

9 7

11 8

14 9

13) Based on the following information answer the questions no. 215 - 217. 14) Consider the following data on the number of hours which ten persons studied for a statistics test and their

187

scores on the test. Hours Studied (X) Test Score (Y) 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)

4 9 10 14 4 7 12 22 1 17 31 58 65 73 37 44 60 91 21 84

What is the regression equation of the form Y = a + bX for the above table? Y = 18.521 + 2.525X. Y = 20.954 + 3.521X. Y = 22.635 + 3.856X. Y = 21.693 + 3.4707X. Y = 23.632 + 3.925 X. What is the Standard error of the estimate of the above data? 5.2813. 5.1325. 5.7582. 5.9862. 5.2515. Which of the following is the coefficient of determination and what does it convey? a. 0.9230 and we can say that 92.3% of the variation in the dependent variable, test scores is explained by the independent variable. 15) b. 0.9530 and we can say that 95.3% of the variation in the dependent variable, test scores is explained by the independent variable. 16) c. 0.8530 and we can say that 14.7% of the variation in the dependent variable, test scores is explained by the independent variable. 17) d. 0.9230 and we can say that 7.7% of the variation in the independent variable, test scores is explained by the dependent variable. 18) e. Both a and d above. 215. a. b. c. d. e. 216. a. b. c. d. e. 217. 14)

19) 218. Using the following information find the linear regression of y on x? 20)

2 Σx = 424; Σxy = 12815; Σy = 15123

21)

Σy

22)

X means expenditure on milk (‘000 Rs)

23)

Y means expenditure on egg (’00)

24) a.

Y = - 0.65x + 63.86.

25) b.

Y = - 0.35x + 68.70.

26) c.

Y = - 0.65x + 83.86.

2 = 363; Σx = 21926; n = 10

27) d.

Y = - 0.55x + 73.86.

28) e.

Y = - 0.45x + 85.86.

29) 219. In the above question, what is the expenditure on egg when the expenditure on milk is 28,000? 30) a.

4,500.

31) b.

4,566.

32) c.

4,488.

33) d.

4,785.

34) e. 4,850. 35) 220. Consider the following data: Output (in lakh of units)

Total cost (in lakh of rupees)

188

05 07 09 11 14 17 20 22 24 28

Identify the fixed and variable cost components using the least squares method.

1)

7)

140 155 170 180 200 230 240 260 275 310

2)

a.

Fixed costs = Rs.105.836 lakh, and Variable cost per unit = Rs.9.1089

3)

b.


4)

c.


5)

d.


6) e. Fixed costs = Rs.104.106 lakh, and Variable cost per unit = Rs.9.2089. Consider the following historical rate of return information is provided for Efficient Company Limited and the stock market and answer the Q. No. 221 - 224.

1)

Year Efficient Co. 1992 12 1993 09 1994 -11 1995 08 1996 11 1997 4 221. What is equation for efficient characteristic line?

2)

a.

ˆ = 5.7735 - 0.1515 X. Y

3)

b.

ˆ = 4.7735 - 0.1615 X. Y

4)

c.

ˆ = 7.7735 - 0.1715 X. Y

5)

d.

ˆ = 8.7735 - 0.1815 X. Y

6)

e.

ˆ = 6.7735 - 0.1415 X. Y

7)

222. What is efficient’s beta?

8)

a.

0.67735.

9)

b.

- 0.1415.

10) c.

0.1415.

11) d.

- 0.67735.

12) e.

0.

Market 15 13 14 -9 12 09

13) 223. What is the systematic risk for the above data?

14) a.

1.64183.

15) b.

1.7425.

16) c.

1.6253.

17) d.

1.63521.

18) e.

1.7125.

19) 224. What is the coefficient of determination for the above data?

189

20) a.

0.02125.

21) b.

0.02256.

22) c.

0.02244.

23) d.

0.0296.

24) e.

0.0216.

25) 225. What is the value of bxy based on information given below? 26) n = 20; Σ x = 80; Σ y = 40; Σ x = 1680; Σ y = 320; Σ xy = 480 2

27) a.

1/3.

28) b.

2/3.

29) c.

4/3.

30) d.

5/3.

31) e.

2.

2

32) 226. In the above question what is the value of byx? 33) a.

16/65.

34) b.

4/3.

35) c.

5/3.

36) d.

4/5.

37) e. 6/25. 38) Based on the following information answer the questions no. 227 - 230. 39) Cost accountants of a company have collected information on overhead expenses and units produced at different plants of the company. They want to estimate a regression equation to predict future overheads.

3)

Overheads 191 170 272 155 280 173 234 116 153 178 Units 40 42 53 35 56 39 48 30 37 40 227. Which of the following represent the regression equation for the cost accountants? ˆ = - 90.4430 + 6.4915 X. Y a. ˆ = - 87.4430 + 7.4915 X. Y b.

4)

c.

5)

d.

ˆ = - 82.4430 + 6.4915 X. Y ˆ = - 80.4430 + 6.4915 X. Y

6)

e.

ˆ = - 80.4430 + 5.4915 X. Y

7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)

228. a. b. c. d. e. 229. a. b. c. d. e.

Predict overheads when 50 units are produced for the above regression equation? 244.1320. -244.1320. 324.575. -324.575. 425.036. What is the standard error of the estimate? 10.232. 9.525. 9.7512. 10.525. 10.254.

1) 2)

190

19) 230. Find 95% confidence intervals for the overhead expense estimate, when units produced = 165. 20) a.

967.50 to 1013.67

21) b.

967.0595 to 1014.2495

22) c.

967.0595 to 1014.2495

23) d.

967.0595 to 1014.2495

24) e.

967.0595 to 1014.2495.

25) 231. The variance of variables, x and y are 36 and 25 respectively, and the coefficient of correlation between them is 0.5. What is the covariance between x and y. 26) a.

12.

27) b.

15.

28) c.

18.

29) d.

24.

30) e.

Cannot be determined.

ˆ (Y − Y) 31) 232. For a simple linear regression equation ∑ = 125 and the number of observation is 11. What is the standard error estimate? 2

32) a.

13.01.

33) b.

12.85.

34) c.

16.62.

35) d.

13.87.

36) e. 21. 37) Based on the following information answer the questions no. 233 - 237. 38) The following data shows monthly advertising expenditure (in lakh of rupees) and sales (in lakh of rupees) over a period of six months. Estimate the relationship between sales (Y) and advertising expenditure (X). Sales 3 Advertising Expenditure 1

15 2

6 3

20 4

9 5

25 6

1)

233. What is the regression equation of the form Y = a + bx to the above.

2)

a.

ˆ = 3.3522 + 3.5146 X Y

3)

b.

ˆ = 2.3999 + 3.0286 X Y

4)

c.

ˆ = 2.4949 + 3.2286 X Y

5)

d.

ˆ = 2.6956 + 3.0286 X Y

6)

e.

ˆ = 2.3999 + 3.5286 X. Y

7)

234. What is the standard error of the estimate given by the regression line?

8)

a.

6.0973.

9)

b.

7.0825.

10) c.

7.0973.

11) d.

8.0923.

12) e.

6.9974.

13) 235. Give 95% confidence intervals for the sales estimate given by the regression line when the advertising expenditure is Rs.7 lakh. 14) a.

3.07 to 43.302

191

15) b.

3.897 to 43.302

16) c.

3.897 to 45.302

17) d.

5.897 to 43.302

18) e.

4.236 to 41.236.

19) 236. Test if the advertising expenditure has a positive impact on the sales level at 5% level of significance and state which of the following is the correct statement? 20) i.

Standard error for the regression coefficient b is 1.6966.

21) ii.

Standardized value of regression coefficient b is t = 1.8851.

22) iii.

The critical t value at 0.05 level of significance and with degree of frodom = 4 is 2.132.

23) iv. Appropriate hypothesis for the test is Null hypothesis, H0: B = 0 against the alternative hypothesis H1: B > 0. 24) a.

Only i, ii, and iii

25) b.

Only i, ii and iv

26) c.

Only ii, iii and iv

27) d.

Only i, iii and iv

28) e.

All i, ii, iii, iv.

29) 237. What is the coefficient of determination? And what does it convey? 30) a. 0.4434 and it convey that 44.34% of the variation in sales is explained by variation in the advertising expenditure. 31) b. 0.4225 and it convey that 42.25% of the variation in sales is explained by variation in the advertising expenditure. 32) c. 0.4434 and it convey that 55.66% of the variation in sales is explained by variation in the advertising expenditure. 33) d. 0.4225 and it convey that 57.75% of the variation in sales is explained by variation in the advertising expenditure. 34) e. 0.4434 and it convey that 44.34% of the variation in sales is unexplained by variation in the advertising expenditure. 35) 238. For a certain bivariate data the following results are given: 36) n = 25; Σx = 125; Σy = 100 2 2 37) Σx = 650; Σy = 436; ΣXY = 520

38) Find the value of bxy? 39) a.

5/9.

40) b.

4/5.

41) c.

4/11.

42) d.

5/13.

43) e. 5/16. 44) Based on the following information answer the questions no. 239 - 243. 45) One of the phases in the selection process for the recruitment of sales persons by a company is an aptitude test. The aptitude test results and the first month’s sales data of eight sales persons are given below.

Sales person A B

Number of units sold Aptitude Test in the first month Results 30 6 49 9 192

C D E F G H

18 42 39 25 41 52

3 8 7 5 8 10

1)

239. What the linear regression line relating the tests scores with the number of units sold in the first month?

2)

a.

1.2277 + 5.1389 X.

3)

b.

1.0277 + 5.1389 X.

4)

c.

1.5255 + 5.2222 X.

5)

d.

1.9277 + 5.6539 X.

6)

c.

1.0277 + 5.1580 X.

7)

240. What is the standard error of the sales estimate given by the regression line?

8)

a.

1.5623.

9)

b.

1.6982.

10) c.

1.8922.

11) d.

1.9685.

12) e.

1.5263.

13) 241. Find 95% confidence intervals for the estimate of the first month’s sales by the regression line if a person’s aptitude test score is 4.

14) a.

17.427 to 24.537

15) b.

19.427 to 25.738

16) c.

15.427 to 25.738

17) d.

17.427 to 25.738

18) e.

18.427 to 25.738.

19) 242. Test the linear relationship between the aptitude test score and the first month’s sale at 5% level of significance and state which of the following statement is correct? 20) i.

Standard error for the regression coefficient b is 0.283.

21) ii.

Standardized value of regression coefficient b is t = 18.1587.

22) iii.

The critical t-value at 0.05 level of significance and with d.f = 6 is 2.447.

23) iv. Appropriate hypothesis for the test is Null hypothesis, H0: B = 0 against the alternative hypothesis H1: B ≠ 0. 24) a.

Only i, ii, and iii.

25) b.

Only i, ii and iv.

26) c.

Only ii, iii and iv.

27) d.

Only i, iii and iv.

28) e.

All i, ii, iii, iv.

29) 243. What the coefficient of determination. What does it convey?

30) a. 0.9821 and its convey that 98.21% of the variation in the first month’s sales is explained by the variation in the aptitude test results.

193

31) b. 0.9821 and its convey that 1.79% of the variation in the first month’s sales is explained by the variation in the aptitude test results. 32) c. 0.9541 and its convey that 95.41% of the variation in the first month’s sales is explained by the variation in the aptitude test results. 33) d. 0.9541 and its convey that 4.59% of the variation in the first month’s sales is explained by the variation in the aptitude test results. 34) e. 0.9821 and its convey that 98.21% of the variation in the first month’s sales is unexplained by the variation in the aptitude test results. 35) Consider the following data regarding the commission earnings last month achieved by 9 salesmen in relation to their years of experience in selling and Answer the question no. 244 to 248.

Sales Commission (in Number of years Rs. ’00) experience 50 06 100 13 140 27 110 15 70 09 90 11 140 21 100 14 80 12 1)

244. What is the best fit regression equation between these two sets of data, assuming that sales commission is dependent on years of selling experience?

2)

a.

36.334 + 4.46099 X.

3)

b.

34.334 + 5.27 X.

4)

c.

34.334 + 4.46099 X.

5)

d.

33.525 + 4.5525 X.

6)

e.

38.134 + 4.4759 X.

7)

245. What is the value of Standard error of the regression coefficient?

8)

a.

0.5125.

9)

b.

0.5236.

10) c.

0.5426.

11) d.

0.5896.

12) e.

0.5797.

13) 246. What is the degree of correlation between the two data sets? 14) a.

0.9457.

15) b.

0.9253.

16) c.

0.9856.

17) d.

0.9452.

18) e.

0.9632.

19) 247. What proportion of the variation in commission earnings is explained by factors other than years of selling experience? 20) a.

0.1056.

21) b.

0.1125.

194

22) c.

0.1225.

23) d.

0.1023.

24) e.

0.1325.

25) 248. What commission would be expected for a salesman with 10 years of experience? 26) a.

75.

27) b.

78.943.

28) c.

80.25.

29) d.

77.563.

30) e. 71.521. 31) 249. Consider the following data. Return on stock 10 12 08 15 09 11 08 10 13 11

Return on market 11 15 03 18 10 12 06 07 18 13

1)

A beta greater than 1 indicates that the stock is relatively sensitive to changes in the market; a beta less than 1 indicates that the stock is relatively insensitive. For the above data one wants to test to see if it is significantly less than 1. Use α = 5%.

2)

The standardized statistic is t for the above test is?

3)

a.

- 10.20.

4)

b.

- 9.50.

5)

c.

-10.80.

6)

d.

- 8.90.

7) 8)

e. -10.50. 250. What is the systematic risk of the stock-using coefficient of determination for above question no .249?

9)

a.

4.563205.

10) b.

5.125632.

11) c.

4.236523.

12) d.

4.63253.

13) e.

4.188803.

14) 251. The following table shows the assessed values and the selling prices of eight houses, constituting a random sample of all the houses sold recently in a metropolitan area: Assessed value (in lakh) 4.03 7.20 3.25 4.48 2.79 5.16

Selling price (in lakh) 6.34 11.83 5.52 7.40 4.88 8.11

195

8.04 5.80

12.32 9.25

1)

Fit a least squares line which will enable us to predict the selling price of a house in terms of its assessed value.

2)

a.

ˆ Y

= 0.6817 + 1.4929 X

b.

ˆ Y

= 0.6017 + 1.5925 X

c.

ˆ Y

= 0.6017 + 1.4929 X

5)

d.

ˆ Y

= 0.5827 + 1.5427 X

6) 7)

ˆ = 0.5027 + 1.4021 X. Y e. 252. For the above question no.251 test the null hypothesis B = 1.30 against the alternative hypothesis B > 1.3 at 0.05 level of significance State which of the following statement is correct? Test hypothesis is H0: B = 1.3, H1: B > 1.3. 8) i. 9) ii. Standard error of the estimate given by the regression line (Se) is 0.2756. 10) iii. Standard error of the estimate given by the regression coefficient (Sb) 0.2756. 11) iv. Acceptance region of the hypothesis is 1.4098. a. Only i, ii. b. Only i, ii and iv. c. Only i, ii and iii. d. Only ii, iii and iv.

3) 4)

12) 13) 14) 15)

16) e.

Only i, iii and iv.

Based on the following information answer the questions no. 253 - 257.

1)

Consider the table below which gives the returns on a scrip Y (denoted by Y) and the returns on a particular market index (denoted by M). Period Y M

1 0.30 0.20

2 0.30 0.00

3 0.40 0.40

4 5 -0.10 0.30 -0.20 0.20

6 0.60 0.60

1)

253. Which of the following is the regression line, with M as the independent variable?

2)

a.

ˆ = 0.25 + 0.75M. Y

3)

b.

ˆ = 0.15 + 0.85M. Y

4)

c.

ˆ = 0.165 + 0.645M. Y

5)

d.

ˆ = 0.175 + 0.725M. Y

6)

e.

ˆ = 0.15 + 0.75M. Y

7)

254. What is the Standard error for the regression line? Y 0.3 0.3 0.4 - 0.10 0.30 ˆ 0.3 0.15 0.45 0 0.30 Y 2 0 0.0225 0.0025 0.01 0 ˆ ( Y − Y)

1)

a.

0.0935

2)

b.

0.0912

3)

c.

0.1025

4)

d.

0.1011

0.60 0.60 0

196

5)

e.

0.0875.

6)

255. Which of the following is standard error for regression coefficient?

7)

a.

0.1479.

8)

b.

0.1279.

9)

c.

0.1389.

10) d.

0.1425.

11) e.

0.1629.

12) 256. If we test at 5% significance level, for B = 0 (where B is the coefficient of M in the population regression line) as opposed to B not equal to 0. What will be the acceptance region (i.e critical points)? 13) a.

± 2.876.

14) b.

± 2.776.

15) c.

± 2.972.

16) d.

± 2.675.

17) e.

± 2.575.

18) 257. At M = 0.00 give a 95% prediction interval for the returns on scrip Y?

19) a.

- 0.1096, 0.4096.

20) b.

- 0.06037, 0.3604.

21) c.

- 0.0360, 0.3360.

22) e.

- 0.053, 0.3033.

23) e.

- 0.987, 0.3987.

24) 258. The data regarding expected rates of return and standard deviation of rates of return for securities X and Y is as given below. (X and Y are perfectly negatively correlated.) Security Expected Rate of Return Standard Deviation of Rate of Return X 30% 5% Y 28% 4% 1)

If a portfolio of these securities is formed with weights of X and Y being 75% and 25% respectively, what would be the expected rate of return and standard deviation of rate of return for this portfolio?

2)

a.

29.5% and 2.75%.

3)

b.

29.5% and 2.81%.

4)

c.

29% and 2.75%.

5)

d.

28.66 and 2.79%.

6)

e.

29.5% and 2.78%.

7)

259. In the above question no. 258, What is the equation for a riskless portfolio?

8)

a.

P = 0.49X + 0.56Y.

9)

b.

P = 0.44X + 0.65Y.

10) c.

P = 0.34X + 0.52Y.

11) d.

P = 0.46X + 0.59Y.

12) e.

P = 0.44X + 0.56Y.

Based on the following information answer the questions no. 260 to 261.

1)

At SRC Hyderabad, it was felt that in the bear market of 1995, “non-specified scrips with steadily

197

increasing demand” would give good returns. Moreover, it was thought that the returns on such scrips were linearly related to the trading volumes. 2)

To test this hypothesis, ten representative scrips were selected (in December, 1995) where there were steadily increasing trading volumes over the previous few weeks. It was felt that this trend would persist for a couple more weeks.

3)

In the table below, the first column displays the scrips, the second column presents the (weekly) change in volumes for the week ending November 3, 1995, and the third column presents the corresponding (weekly) returns. (Numbers have been altered slightly to make some computations easier.) Name Change in Volume (%) Returns (%) Clio Fin. 27 -15 Asian Net 49 004 Smelters 171 028 Heera 191 032 Shri Rang 193 036 Napa 199 034 Vantage 228 043 Sagar 422 052 Kenwood 500 056

1)

Five weekly trading volumes of Gujarat Cotton were considered starting from the week ending on 6.10.95 (with the fifth week ending on 3.11.95). The trading volume for the week ending 6.10.95 was 50. The trend of this weekly trading volume was given by 50 + 60 x t.

2)

What do you think the weekly return should be for the week ending on 10.11.95? 3)

260. Which of the following statement is correct for the above data?

4)

i.

Average of returns is 30%.

5)

ii.

Average of trading volumes is Rs.220 lakh.

6)

iii.

The standard deviation of the returns is 22.6108.

7)

iv.

The standard deviation of the trading volumes is 154.1792.

8)

a.


9)

b.


10) c.

All (i) ,(ii), (iii) and (iv) above.

11) d.

Only (i), and (iii).

12) e. Only (ii), (iii) and (iv). 13) 261. State which of the following statement is correct? The covariance of the returns and the trading volumes is 3,094.125. 14) i. 15) ii. The correlation coefficient is Cov (V,R)/(STD (V) x STD (R)) = 0.8876 16) iii. The linear relationship is given R = 1.3643 + 0.1302 x V. 17) a. Only (i) above 18) b. Only (ii) above 19) c. Only (iii) above 20) d. Only (i) and (ii) above 21) e. All (i), (ii) and (iii) above. 22) 262. Since the above ten scrips were considered representative of all non-specified scrips for which trading volumes were increasing steadily, it was felt that the model developed in Part (g) above was applicable to all such scrips. Five weekly trading volumes of Gujarat Cotton were considered starting from the week ending on 6.10.95 (with the fifth week ending on 3.11.95). The trading volume for the week ending 6.10.95 was 50. The trend of this weekly trading volume was given by 50 + 60 x t. 23) What do you think the weekly return should be for the week ending on 10.11.95? 24) a. 42%. 198

25) 26) 27) 28)

b. c. d. e.

43%. 44%. 45%. 47%.


1)

A certain company would like to predict how the sales trainees in its salesmanship course will perform. At the beginning of their 2-month course, the trainees are given an aptitude test. This is denoted by X-score in the table below. Records are kept of the sales of each salesman. The Y-values given below are the values of sales (for the year following the training period) of each of the individuals who took the aptitude test. Salesman 1 2 3 4 5 6 7 8 9 10 11 12 X 65 63 67 64 68 62 70 66 68 67 69 71 Y(in ’000 Rs.) 68 66 68 65 69 66 68 65 71 67 68 70 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)

263. Which of the following represent linear regression equation that best fits this data? a. 35.72 + 0.5079X. b. 33.72 + 0.5125X. c. 33.72 + 0.5079X. d. 39.12 + 0.6089X. e. 29.75 + 0.4850X. 264. Which of the following is the standard error of the estimate for this relationship? a. 1.315. b. 1.454. c. 1.525. d. 1.356. e. 1.852. 265. What is the correlation coefficient between X and Y? a. 0.7233 b. -0.7596 c. -0.7133 d. 0.7825 e. 0.7725. 266. Following are the monthly closing BSE-Sensex figures and prices of Ind Jyoti mutual fund for one year. Based on the quarterly returns State which of the following statement is correct?

Month and Year BSE Sensex Sep. 96 3205 Oct. 96 2930 Nov. 96 2847 Dec. 96 2713 Jan. 97 3096 Feb. 97 3316 Mar. 97 3663 Apr. 97 3825 May 97 3755 Jun. 97 4133 Jul. 97 4189 Aug. 97 3876 Sep. 97 3925 1)

i.

Ind Jyoti 7.60 7.10 7.50 8.20 7.70 7.80 7.75 8.10 8.15 8.25 8.50 8.30 9.00

Standard deviation of the quarterly return is 19.13.

199

2)

ii.

iii.

Beta for Ind Jyoti is -0.273.

4)

iv. The investment is not advisable in the fund as the average returns are lower than the market returns.

5)

a.

Only (i), (ii) and (iv).

6)

b.


7)

c.

All of (i), (ii), (iii) and (iv) above.

8)

d.

Only (ii), (iii) and (iv).

9)

e.

Only (ii) and (iv).

3)

Covariance between Ind Jyoti and BSE is -99.89.


1)

The data regarding securities A and B is given as follows: Security A B

1)

Expected Returns Proportions in the Portfolio M N 18% 0.40 0.60 23% 0.60 0.40

Variance Covariance Matrix A B -3.88 -2.59 -2.59 -8.87

Suppose we construct two portfolios M and N with securities A and B in the proportions as indicated above. 2)

267. What will be the expected returns and standard deviation of portfolio M?

3)

a.

21% and 1.6034.

4)

b.

20.5% and 1.6034.

5)

c.

21% and 1.6125.

6)

d.

20.5% and 1.6325.

7)

e.

21% and 1.6525.

8)

268. What will be the expected returns and standard deviation of portfolio N?

9)

a.

20% and 1.254.

10) b.

21% and 1.254.

11) c.

21% and 1.2225.

12) d.

23.8% and 1.254.

13) e. 23.8% and 1.2125. 14) 269. Which portfolio would you prefer? Why? 15) a.

As the risk per unit of return for portfolio N is more, one should prefer portfolio N.

16) b.

As the risk for portfolio N is less, one should prefer portfolio M.

17) c.

There is no in difference is between portfolio M and N.

18) d.

As the risk for portfolio N is more, one should prefer portfolio M.

19) e.

As the risk per unit of return for portfolio N is less, one should prefer portfolio N.

20) 270. Following are the risk and return characteristics of different kinds of securities: Security Type A B C D E F Return 13 15 17 20 23 26 Risk 1.19 2.37 3.49 5.10 7.27 8.95 1)

What is the risk free return from the above data?

2)

a.

11%.

3)

b.

11.12%.

4)

c.

10.8%.

5)

d.

11.5%.

200

6)

e.

7)

271. Mr. Ganesh Pendse, an amature equity research analyst is trying to observe the relationship between the returns from a security with the sales of the company. He collects the data about the average yearly returns of the security as well as the yearly sales of the company for the last ten years, which are shown below: Year Sales (in Rs. Share Price hundred crore) (in Rs.) 1989 2.10 11 1990 2.70 13 1991 2.90 18 1992 2.90 24 1993 3.10 28 1994 3.00 32 1995 3.30 38 1996 3.70 42 1997 4.00 47 1998 4.40 53

1)

What is the regression equation between the sales and the share price of the company?

2)

a.

Y = -38.176 + 20.491 X.

3)

b.

Y = -35.176 + 21.491 X.

4)

c.

Y = -35.176 + 20.491 X.

5)

d.

Y = -39.125 + 22.635 X. Y = -425.925 + 20.231 X.

11.75%.

6)

e.

7)

272. In the above question no. 271 The correlation coefficient between sales and share price of the company?

8)

a.

92.25.

9)

b.

97.64.

10) c.

95.25.

11) d.

91.36.

12) e. 99.25. 13) 273. In a bivariate Sample, the sum of squares of differences in ranks of observed values of two variables is 231 and the Correlation Coefficient between them is -0.4. Find the number of pairs. 14) a.

10

15) b.

12

16) c.

13

17) d.

14

18) e.

15.

Based on the following information answer the questions no. 274 and 275.

X Y

: 1 : 150

2 180

1)

274. What is the covariance between X and Y?

2)

a.

15.

3)

b.

16.

4)

c.

17.

5)

d.

18.

6)

e.

19.

3 140

4 5 180 200

201

7)

275. What is the Regression Coefficient of Y on X, for the above data?

8)

a.

8.

9)

b.

7.

10) c.

8.5.

11) d.

9.

12) e.

9.23.

13) 276. The lines of regression of y on x on y are respectively y = x + 5 and 16x - 9y = 94. What will be the variance of x if the variance of y 16? 14)

a.

3/4.

15)

b.

1/2.

16)

c.

1/3.

17)

d.

2/5.

18)

e.

1/5.

19) 277. The probability distribution of returns from two securities A and B in different economic scenarios is given below: Economic scenario Probability Returns from A Returns from B (%) (%) Recession 0.20 -10.00 -3.00 Slow recovery 0.25 --5.00 - 2.00 Moderate recovery 0.30 -25.00 -15.00 Boom 0.25 -35.00 -20.00 1) From the above data, Which of the following is the Mean returns and standard deviation of returns from securities A? 2)

a.

15.5% and 16.725%.

3)

b.

15% and 16.725 %.

4)

c.

15.5% and 15.625%.

5)

d.

6.6% and 12.25%.

6)

e.

112.5% and 15%.

7)

278. In the above question no. 277, which of the following is the Mean returns and standard deviation of returns from securities B.? a. 9.4% and 9.992%. b. 9.4% and 9.052 %. c. 9 % and 9.052%. d. 10% and 2.25%. e. 10.02% and 10.095%. 279. In the above question no.277 what is Pearson’s correlation coefficient for securities A and B? a. 0.9927. b. 0.9656. c. 0.9763. d. 0.9265. e. 0.9965.

8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)

Based on the following information answer the questions no. 280-282.

1)

Rajkiran Projects Ltd. is considering three independent projects: A, B and C. The probability distribution of the NPV of the projects (in Rs. lakh) is given below: Probability

0.1

0.2

0.45

0.25

Project 202

1)

A -11 0.5 B -40 2.0 C -100- 5.0 280. What is expected NPV of the Project A, B and C?

10 10 20

21 30 35

2)

a.

8.75%, 8.40% and 8.75%.

3)

b.

8.75%, 8.55% and 8.75%.

4)

c.

8.75%, 8.75% and 8.75%.

5)

d.

8.25%, 8.40% and 8.40%.

6)

e.

8.65%, 8.45% and 8.80%.

7)

281. What is the standard deviation in NPV of project A, B and C?

8)

a.

9.53, 18.25 and 32.45.

9)

b.

10.25,18.98 and 35.26.

10) c.

9.53, 18.75 and 37.61.

11) d.

8.25,18.40 and 38.40.

12) e.

9.53, 18.98 and 37.61.

13) 282. Which project should be selected by the company?

14) a.

Project A.

15) b.

Project B.

16) c.

Project C.

17) d.

Project A or B.

18) e.

Project A or C.

19) 283. A simple regression relationship was developed between two variables X and Y, with X as the independent variable. 2 ˆ 2 = 250; n = 7 ∑ Y = 816; ∑ Y = 85254; ∑ (Y − Y) 20) 21)

The coefficient of correlation between X and Y is

22) a.

0.2717

23) b.

0.8744

24) c.

0.3731

25) d.

0.2563

26) e.

0.7863.


1)

Following are the average prices of Sterlite Industries and the values of BSE Sensitive Index for the last 6 years. Year Stock Price BSE Index Sterlite (Rs.) 1999 245 3070 1998 255 3220 1997 240 3370 1996 390 3097 1995 655 3518

203

1994

395

3636

1)

284. What is the equation of the regression line between Sterlite price and the BSE Index?

2)

a.

ˆ = -831.33 + 0.42X. Y

3)

b.

ˆ = -825.22 + 0.36X. Y

4)

c.

ˆ = -831.33 + 0.36X. Y

5)

d.

ˆ = -830.13 + 0.29X. Y

6)

e.

ˆ = 841.33 - 0.39X. Y

7)

285. What is the standard error of the estimate?

8)

a.

152.28.

9)

b.

149.36.

10) c.

155.36.

11) d.

159.78.

12) e. 156.85. 13) 286. For following 20 as the working mean for prices and 70 as the working mean for demand what is the covariance between price and demand for the above data? Price (x) Demand (y) 14 84 16 78 17 70 18 75 19 66 20 67 21 62 22 58 23 60 1) a. - 19.4. 2) b. - 18.5. 3) c. - 17.8. 4) d. - 20.5. 5) e. - 19.62. 6)

287. What is the coefficient of correlation for the above data?

7)

a.

- 0.82.

8)

b.

- 0.95.

9)

c.

- 0.97.

10) d.

- 0.92.

11) e. 0.89. 12) 288. Following information is available regarding the capital receipts and capital expenditure of Government of India for the last five years. Year 1998-99 1997-98 1996-97 1995-96

Capital Receipts Capital (Rs. in crore) Expenditure (Rs. in crore) 105933 57865 196731 53046 174728 42074 58338 38415

204

1)

1994-95 68695 38627 Which of the following is correlation coefficient between capital receipt and capital expenditure of Government of India for the last five years?

2)

a.

0.9525.

3)

b.

0.9814.

4)

c.

0.9963.

5)

d.

09754.

6)

e.

0.9253.

7)

289. A term lending institution has collected the following data relating to six units in the textile industry.

Sl. No.

Fixed assets as % of Long-term debt total investment as % of total investment 65 64 67 69 72 77 69 73 71 75 68 72

1. 2. 3. 4. 5. 6. 1)

Assuming that long-term debt of a company depends on the extent to which the investment is required in the fixed assets of the company. What is the regression equation describing the relationship between the above two variables?

2)

a.

ˆ = 1.792 X - 51.40 Y

3)

b.

ˆ = 1.792 X + 51.40 Y

4)

c.

ˆ = 1.792 X - 52.95 Y

5)

d.

ˆ = 1.825 X - 51.25 Y

6)

e.

ˆ = 1.652 X - 51.40. Y

7)

290. In the above question, what is the correlation coefficient between the two variables?

8)

a.

0.975.

9)

b.

0.925.

10) c.

0.963.

11) d.

0.950.

12) e. 0.945. 13) 291. Fine Products (P) Ltd. has collected the following information regarding its advertising expenditure and the sales for the last 7 years.

1)

Year

Advertising Expenditure (’000 Rs.)

1993 1994 1995 1996 1997 1998 1999

25 28 33 38 45 49 56

Sales (Rs. lakh) 7 9 13 16 18 23 29

The company is targeting sales of Rs.35 lakh in the year 2000. With the help of regression analysis, you are required to find out the amount of advertising expenditure that the company should budget for

205

the next year. 2)

a.

Rs.65,250

3)

b.

Rs.66,850

4)

c.

Rs.76,985

5)

d.

Rs.56,362

6)

e.

Rs.66,600.

7)

292. Popular Toys Company is trying to formulate a relationship between the quantity that can be sold and the price for one of its new products, the “Cinderella”-doll. Its marketing consultants have provided the following data:

8)

Price (Rs.) 28.0 26.0 25.0 24.0 21.0 19.0

Quantity that can be sold (’000 units) 130 145 170 190 215 225

1)

You are required to Formulate the relationship between the price and the quantity that can be sold with the help of a suitable statistical method and state which of the following is the correct formulation that represent the relationship between the price and the quantity?

2)

a.

Y = 438.68 - 10.89X.

3)

b.

Y = 448.25 - 10.89X.

4)

c.

Y = 438.68 - 11.25X.

5)

d.

Y = 439.28 - 12.89X.

6)

e.

Y = 429.18 - 13.89X.

7)

293. Crown Bakery Products Company is studying the effect of its sales promotion drives for one of its products conducted at different times. The following data has been collected by the company about its sales promotion drives conducted in the past. Percentage of discount offered Sales before the Sales after the (%) promotion drive promotion drive (’000 units) (’000 units) 05 24 30 10 10 10 15 08 12 20 16 22 25 22 30 30 10 16 40 12 20 50 22 34

1)

Which of the following is the coefficient of correlation between percentage of discount offered and the change in sales?

2)

a.

0.695.

3)

b.

0.792.

4)

c.

0.925.

5)

d.

0.750.

6)

e.

0.825.

7)

294. In the above question. Find out the percentage of change in sales that is explained by the variations in percentage of discount offered?

206

8)

a.

60.5%.

9)

b.

62.7%.

10) c.

63.5%.

11) d.

64.32%.

12) e. 65.60%. 13) 295. The following data are available for the demand at various prices of a product: Price (Rs. per unit) 10.00 19.00 18.00 17.00 16.00 15.00 14.00 13.00

Demand (’000 units) 30 44 63 72 90 108 125 140

1)

Which of the following is the relationship between the price and demand for the product using linear regression technique?

2)

a.

ˆ = 185.77 - 16.71X. Y

3)

b.

ˆ = 198.77 - 15.81X. Y

4)

c.

ˆ = 186.77 - 19.81X. Y

5)

d.

ˆ = 184.25 - 14.18X. Y

6)

e.

ˆ = 186.77 - 15.81X. Y

7)

296. In the above question, the demand for the product at a price of Rs.7.50 per unit is

8)

a.

65250 units

9)

b.

61950 units

10) c.

64236 units

11) d.

66185 units

12) e. 68195 units. 13) 297. Following are the car registrations and the Urban Population Index in the country for the last six years. Year 1995 1996 1997 1998 1999 2000

Urban Population Index 158 163 175 177 178 195

Car Registrations (’000) 490 539 628 708 810 898

1)

What is the coefficient of the correlation between the Urban Population Index and the car registrations with the help of the above data?

2)

a.

0.951.

3)

b.

0.936.

4)

c.

0.981.

5)

d.

0.924.

6)

e.

0.963.

7)

298. The finance manager of Broadway Industries Pvt. Ltd. is trying to establish a relationship

207

between sales and operating costs for the purpose of forecasting the same in the forthcoming year. He has collected the following data from past records: Year ended Sales (Rs. in lakh) Operating costs (Rs. in lakh) 31/03/1995 54 21 31/03/1996 60 22 31/03/1997 55 22 31/03/1998 64 24 31/03/1999 70 26 31/03/2000 82 30 1)

What is the relationship between sales and operating costs using linear regression technique?

2)

a.

ˆ = 3.89 + 0.316 X. Y

3)

b.

ˆ = 3.50 + 0.316 X. Y

4)

c.

ˆ = 3.99 + 0.333 X. Y

5)

d.

ˆ = 3.75 + 0.342 X. Y

6)

e.

ˆ = 3.96 + 0.345 X. Y

7)

299. In the above question no.298 Project the level of operating costs for the year ended 31/03/2001, given projected sales of Rs.85 lakh?

8)

a.

32.75 lakh.

9)

b.

33.25 lakh.

10) c.

30.75 lakh.

11) d.

39.63 lakh.

12) e.

31.63 lakh.

13) 300. The following data show the marks of 10 students in Maths and Science in an examination: Marks in Maths : 45 70 65 40 80 40 50 70 85 60 Marks in Science : 35 80 70 40 90 45 60 80 80 50 1)

What is the covariance of marks between Maths and science?

2)

a.

250.

3)

b.

262.

4)

c.

253.

5)

d.

261.

6)

e.

259.

7)

301. What is the correlation between marks in Maths and science?

8)

a.

0.91.

9)

b.

0.99.

10) c.

0.89.

11) d.

0.95.

12) e. 0.86. Based on the following information answer the questions no. 302 and 304. The finance manager of Crystal Technologies Ltd. has collected the following data from past records: Average receivables balance Bad debts (Rs. in (Rs. in lakh) ’000) 49 194 54 125 56 118

208

60 64 71

117 150 140

1)

302. The coefficient of correlation between average receivables balance and bad debts?

2)

a.

0.7284.

3)

b.

0.8224.

4)

c.

0.8963.

5)

d.

0.8363.

6)

e.

0.8425.

7)

303. The percentage of variations in bad debts that is explained by the variations in average receivables balance?

8)

a.

67.64%.

9)

b.

68.25%.

10) c.

61.25%.

11) d.

74.62%.

12) e.

58.62%.

13) 304. What is the percentage of variations in bad debts that remains unexplained? 14) a.

30.25%.

15) b.

32.36%.

16) c.

34.52%.

17) d.

39.63%.

18) e.

38.45%.

19) 305. The following information was obtained from the manager of a city water department for predicting the consumption of water (in gallons) from the size of household: Household Water Size (x) 2 7 9 4 12 6 9 3 3 2 1)

Here:

Σx = 57

Used (y) 650 1200 1300 430 1400 900 1800 640 793 925

Σy = 10,038

2

Σx = 433 Σy2 = 11,641,474 Σxy = 67,669

2) 3)

The equation of the least squares regression for water consumption on household size is given by:

4)

a.

ˆ = 97053.7 + 96.692 X Y

5)

b.

ˆ = 999.220 + 0.803 X Y

6)

c.

ˆ = -1.0028 + 0.0067 X Y

209

7)

d.

ˆ = 452.66 + 96.692 X Y

8)

e.

ˆ = 1003.8 - 96.692 X. Y

9)

306. There is an approximate linear relationship between the height of females and their age (from 5 to 18 years) described by:

10) Height = 50.3 + 6.01(age), Where height is measured in cm and age in years. 11) Which of the following is not correct? 12) a. The estimated slope is 6.01which imply that children increase by about 6 cm for each year they grow older. 13) b.

The estimated height of a child who is 10 years old is about 110 cm.

14) c. The estimated intercept is 50.3 cm, which implies that children reach this height when they are 50.3/6.01=8.4 years old. 15) d. The average height of children when they are 5 years old is about 50% of the average height when they are 18 years old. 16) e. My niece is about 8 years old and is about 115 cm tall. She is taller than average. 17) Based on the following information answer the questions no. 307 - 309. X Y

6 6 6 6 6 62 70 66 68 67 69 71 53748 6 6 6 6 6 66 68 65 71 67 68 70 86859

1)

307. Which of the following represent the best-fit line of Y on X?

2)

a.

ˆ = 1.479 + 1.086 X. Y

3)

b.

ˆ = 1.679 + 1.036 X. Y

4)

c.

ˆ = 1.779 + 1.096 X. Y

5)

d.

ˆ = 1.179 + 1.026 X. Y

6)

e.

ˆ = 1.479 + 1.036 X. Y

7)

308. What is the coefficient of correlation between average receivables balance and bad debts?

8)

a.

0.70268.

9)

b.

0.6965.

10) c.

0.6856.

11) d.

0.6725.

12) e.

0.7125.

13) 309. The percentage of variations in Y that is explained by the variations X is? 14) a.

70.98%.

15) b.

70.27%.

16) c.

69.25%.

17) d.

71.56%.

18) e. 69.56%. 19) Based on the following information answer the questions no. 310 - 311. 20) 21) 22) 23)

n = 10; Σ2x = 250 ; Σ y = 310; 2 Σ(x − 35) = 162; Σ(y − 31) = 222 Σ(x − 35)(y − 31) = 92

310. What is the covariance between X and Y for the above data?

210

24) a. 9.5. 25) b. 9.3. 26) c. 9.2. 27) d. 9.4. 28) e. 9.6. 29) 311. What is the Correlation Coefficient from the above data? 30) a. 0.39. 31) b. 0.43. 32) c. 0.49. 33) d. 0.52. 34) e. 0.53. 35) 312. For a given data set of 10 observations of two variables X and Y, the covariance between X and Y is -121.5 and the standard deviation of the variable X is 9.0. The sums of the observations of X and Y in the given data set is 55 and 33 respectively. If a regression line is plotted for estimating the value of Y for a given value of X, then what is the estimated value of Y for X equal to 5? 36) a. 2.15 37) b. 3.25 38) c. 4.05 39) d. 5.15 40) e. 6.25. 41) 313. The following is the matrix of correlation coefficients between three variables Y, X1 and X2. X2 Y X1 Y 1 0.7 0.5 X1 0.7 1 0.6 X2 0.5 0.6 1

7)

1) How much variation in Y is explained by X1 if X2 is kept constant? 2) a. 33 % 3) b. 58 % 4) c. 61 % 5) d. 72 % 6) e. 83 %. 314. The city council of Goa has gathered data on the number of minor traffic accidents and number of youth soccer games that occur in town over a weekend. He feels that there is a close relationship between the number of accidents and the number of games during weekends. No. of Soccer games 20 30 10 12 15 25 34 No. of Minor accidents 6 9 4 5 7 8 9

1)

Assuming that there is a linear relation between these two, what would be your prediction about the number of minor accidents that will occur on a weekend during which 33 soccer games take place in Goa? 2) a. 6.25. 3) b. 7.35. 4) c. 8.65. 5) d. 9.25. 6) e. 9.85.

7)

315. Rola Cola is studying the effect of its latest advertising campaign. People chosen at random were called and asked how many cans of Rola Cola they had bought in the past week and how many Rola Cola advertisements they had seen on television in the past week.

211

No. of advertisements seen No. of cans of Rola Cola purchased 1)

7)

3 10

7 17

4 9

2 4

0 7

4 6

1 3

What is the coefficient of correlation for the above sample of the number of advertisement seen and the number of cans purchased in the past week? 2)

a.

0.6548

3)

b.

0.7458

4)

c.

0.8001

5)

d.

0.9134

6)

e.

0.9868.

316. Raj Aryan, the auditor of a state public school system, has reviewed the inventory records to determine if the current inventory holdings of textbooks are typical. The following inventory amounts are from the previous five years: Year 1999 2000 2001 2002 2003 Inventory (in thousand rupees) 4,620 4,910 5,490 5,730 5,990

1) If the time variable has been coded as x =X - X , then which of the following is the linear trend equation for estimating inventory level for a given year in thousand rupees? ˆ = 4, 332 + 356 X Y 2) a. ˆ = 5, 348 + 356 X Y 3) b. ˆ = 53, 48, 000 + 356 X Y 4) c. ˆ = −7, 07, 008 + 356 X Y 5) d. ˆ = -7, 07, 008 + 2001 X Y 6) e. . 7)

7)

317. Two variables, X and Y, are related by a regression equation such that X is the independent variable and Y is the dependent variable. The following details are available:

Coefficient of correlation between X and Y =

0.80

Standard deviation of X

=

5

Standard deviation of Y

=

8

1)

The slope of the regression equation is

2)

a.

0.80

3)

b.

1.28

4)

c.

6.4

5)

d.

32

6)

e.

64.

318. The following details are available with regard to a simple regression relationship:

Total sum of squares = 15730 Error sum of squares = 1530 1) What percentage of the variations in the dependent variable is explained by the regression relationship? 2)

a.

0.

3)

b.

9.73%.

4)

c.

0.8%.

5)

d.

90.3%.

6)

e.

100%. 212

7)

7)

319. The following details are available with regard to a regression relationship: Sum of the observations of the independent variable = 120 Sum of the observations of the dependent variable = 425 Number of data points = 10 Slope of the regression line = 2.5 1) What is the estimated value of the dependent variable if the independent variable is equal to 10? 2) a. 2.5. 3) b. 12.5. 4) c. 25. 5) d. 37.5. 6) e. 42.5. 320. The following data relate to the age of a sample of ten employees in an organization and the number of days they reported sick during a period of six months:

Age (in years) Number of sick days

21 2

31 3

33 1

36 4

41 5

47 7

53 6

56 5

59 9

63 8

1) What is the coefficient of correlation between the age of the employees and the number of sick days?

7)

9)

2)

a.

0.163.

3)

b.

0.683.

4)

c.

0.871.

5)

d.

0.187.

6) e. 0.759. 321. A simple regression equation is developed which relates the variables X and Y ; X is the independent variable and Y is the dependent variable. The following details are available: Y

112

100

88

110

1)

∑ XY

= 21040

2)

Number of data points = 10

3)

What is the standard error of estimate of the regression equation?

4)

a.

2.88.

5)

b.

12.37.

6)

c.

13.83.

7)

d.

98.74.

∑X

110

130

150

∑X

= 140

160

140

200

2

= 2528

8) e. 151.96. 322. A simple regression equation is developed which relates the variables X and Y ; X is the independent variable and Y is the dependent variable. The following details are available: Y

31

40

30

34

20

1)

∑ XY

2)

Number of data points = 6

3)

What percentage of the variations in Y does the regression line, not explain?

4)

a.

3.2%.

5)

b.

17.4%.

6)

c.

20%.

7)

d.

82.6%.

= 1000

∑ X = 30 ∑ X

25

2

= 200

213

9)

8) e. 90%. 323. Ten salesmen in a firm were put to a competency test. The test scores obtained by the salesmen and the monthly sales made by them are given below: Test score Monthly sales (Rs. ’000)

30 25

60 50

40 35

50 45

70 55

40 30

80 60

30 30

50 40

50 30

1) A regression equation has to be developed for estimating the amount of monthly sales from the test scores. On the basis of this regression equation what is the estimated monthly sales for a salesman who scores 90 in the competency test?

7)

2)

a.

Rs.687.5.

3)

b.

Rs.5,625.

4)

c.

Rs.50,625.

5)

d.

Rs.61,875.

6)

e.

Rs.67,500.

324. For a simple regression equation the following results were obtained:

8)

ˆ If X = 5, Y = 31

9)

ˆ If X = 10, Y = 47

ˆ is the estimated value of the dependent variable Y. 10) Where X is the independent variable and Y

11) ∑ X = 60 12) Number of data points = 6 13) What is the mean of the observed values of the dependent variable, Y? 14) a.

15.

15) b.

32.

16) c.

47.

17) d.

18.2.

18) e.

25.

19) 325. The production (in thousand quintals) of a sugar factory during seven years is given below: Year Production ( '000 quintals)

1996 75

1997 85

1998 87

1999 80

2000 89

2001 95

2002 90

1) What is the estimated production of sugar for the year 2003 on the basis of a linear estimating equation that describes the trend in the sugar production by the factory ? 2)

a.

85857 quintals.

3)

b.

95429 quintals.

214

4)

c.

88250 quintals.

5)

d.

89857 quintals.

6)

e.

487904 quintals.

7)

8) Multiple Regressions 9)

326. For the following data:

Y X1 X2

7.88 7.43 8.38 7.42 7.97 7.49 8.84 8.29 3.00 2.00 4.00 2.00 3.00 2.00 5.00 4.00 2.00 1.00 3.00 1.00 2.00 2.00 3.00 2.00

1)

What is the multiple regression plane?

2)

a.

ˆ Y

= 7.2563 + 0.4133 X1 + 0.0759 X2.

3)

b.

ˆ Y

= 6.5191 + 0.5127 X1 + 0.0759 X2.

4)

c.

ˆ Y

= 6.5191 + 0.4133 X1 + 0.0759 X2.

5)

d.

ˆ Y

= 8.5191 + 0.5133 X1 + 0.0859 X2.

6) 7)

ˆ = 7.5191 + 0.4225 X + 0.0767 X Y e. 1 2. 327. Fit a least squares parabola of the form, Y = a + bX + cX2 to the following data?

Y X

2.4 0.0

2.1 1.0

1)

a.

Y = 2.5094 - 1.9998X + 0.7333X2.

2)

b.

Y = 2.6094 - 1.9998X + 0.7333X2.

3)

c.

Y = 2.5094 - 1.9998X + 0.6222X2.

4)

d.

Y = 3.1094 - 1.5258X + 0.8666X2.

5)

e.

Y = 2.1099 - 1.8888X + 0.5233X2.

3.2 2.0

5.6 3.0

9.3 4.0

14.6 21.9 15.0 16.0

6)


7)

A firm wants to obtain a regression equation relating monthly sales (in thousands of units) with the number of advertisements in a month in a national newspaper and the monthly advertisement cost (in thousands of rupees, for ads in the newspaper). The computer output obtained by regressing sales on the two explanatory variables using the data for the past one year is given below. Dependent Variable: Sales Total Sales (thousands of rupees) Analysis of Variance Source DF Sum of Mean F Value Prob > |F| Squares Square Regression 2 309.99 (3) (5) 0.006 Error (1) (2) (4) Total 11 453.19 Root MSE = (6) R2 = (7) 1)

The Regression Equation is Sales = 6.58 + 0.62 ADs + 2.14 cost Variable Intercept Ads Cost

1)

Parameter Estimate 6.584 0.625 2.139

Standard Error 8.542 1.120 1.470

T for H: Parameter Prob > |T| =0 0.77 0.461 (8) 0.591 (9) 0.180

328. What will come in place of (1), (2) and (3) in the above table?

215

2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)

a. 9,143.20 and 154.95. b. 8,147.25 and 155.92. c. 7,148.36 and 165.63. d. 6, 151.25 and 166.63. e. 10, 152.63 and 168.54. 329. What will come in place of (4), (5) and (6) in the above table? a. 16.71, 9.7392, and 3.9887. b. 15.91, 9.7392, and 3.9887. c. 18.68, 12.7392, and 3.9887. d. 15.91, 9.7392, and 4.8256. e. 16.91, 12.7392, and 4.9887. 330. What will come in place of (7), (8) and (9) in the above table? a. 0.584, 0.558 and 1.455. b. 0.128, 0.598 and 1.455. c. 0.684, 0.558 and 1.455. d. 0.584, 0.658 and 1.255. e. 0.524, 0.458 and 1.455. Based on the following information answer the questions no. 331 - 333.

20) One of the phases in the selection process for recruitment of salespersons by a company is an aptitude test. The aptitude test has four sections: Creativity, English language, abstract thinking and quantitative ability. The sales manager has collected sales growth data for 25 of the sales persons who were hired, along with scores from the four sections of the aptitude test. The manager obtained the following computer output by regressing sales growth on scores from the four sections of the aptitude test. Dependent Variable: Growth Analysis of Variance Source DF

F

Prob > F

(5)

0.00

1)

Regression Error Total Root MSE = (6) The Regression Equation is,

2)

Growth = 70.1 + 0.422 Creat + 0.271 QUANT + 0.745 ABST + 0.420 English Variable

1) 2) 3) 4) 5) 6)

331. a. b. c. d. e.

(1) 20 24

Sum of Squares Mean Square 1050.78 (3) 1183.88 (4) (2) R - Square = (7)

Source

Parameter Estimate DF

Intercept Creat Quant Abst English

70.06600 0.42160 0.27140 0.74504 0.41955

Standard Error SUM OF SQUARES 2.13000 0.17192 0.21840 0.28982 0.06871

T for H: Parameter Prob > |T| =0 MEAN SQUARE F (8) (9) (10) (11)

0.000 0.024 0.228 0.018 0.000

What will come in place of (1), (2) and (3) in the above table? 9,1143.20 and 254.95. 8,1147.25 and 255.92. 7,1148.36 and 265.63. 6, 1151.25 and 266.63. 4, 1134.66 and 262.695.

216

7)

332. What will come in place of (4), (5) and (6) in the above table?

8)

a.

4.194, 62.6359, and 2.0479.

9)

b.

4.256, 62.6359, and 2.0479.

10) c.

4.194, 62.7392, and 3.9887.

11) d.

5.91, 69.7392, and 4.8256.

12) e.

6.91, 12.7392, and 4.9887.

13) 333. What will come in place of (7), (8) and (9) in the above table? 14) a.

0.984, 2.4523 and 1.455.

15) b.

0.9281, 2.8225 and 1.455.

16) c.

0.984, 2.3265 and 1.455.

17) d.

0.9261, 2.4523 and 1.2427.

18) e

0.924 , 2.458 and 1.455.

19) 334. The Tasty-Smack hamburger chain has significantly increased its investment in inventory over the last 6 years. The information is printed below: Year Inventory (100,000)

1977 1978 1979 1980 1981 1982 4 4.5 6 8 8.5 10

1)

What is the linear estimating equation that best describes these data?

2)

a.

Y = 5.8333 + 0.6286X.

3)

b.

Y = 6.8333 + 0.5286X.

4)

c.

Y = 6.8333 + 0.6286X.

5)

d.

Y = 6.7225 + 0.6125X.

6)

e.

Y = 7.8325 + 0.5295X.

7)

335. In the above question no. 334, What is the second-degree estimating equation that best describes these data?

8)

a.

Y = 5.781 + 0.6286X + 0.0045X2.

9)

b.

Y = 6.781 + 0.6001X + 0.0045X2.

10) c.

Y = 6.781 + 0.6286X + 0.0025X2.

11) d.

Y = 6.781 + 0.6286X + 0.0045X2.

12)

Y = 7.126 + 0.52286X + 0.0235X2.

e.

13) 336. From the above question no. 335, Estimate the 1984 inventory level from the second degree equation?

14) a.

$11.203 lakh.

217

15) b.

$10.803 lakh.

16) c.

$13.125 lakh.

17) d.

$14.803 lakh.

18) e.

$12.803 lakh.

19) 337. A multiple regression plane passes through the points (2, 3, -1), (1, 5, 3) and (1, -1, 1). 20) What is the equation of the multiple regression plane? 21) a. 22) b. 23) c. 24) d. 25) e.

Y=

13 10 1 − X1 + X 2 3 3 3 .

Y=

14 11 1 − X1 + X 2 3 3 3 .

Y=

14 10 2 + X1 + X 2 3 3 3 .

Y=

17 13 1 − X1 + X 2 3 3 3 .

Y=

14 10 1 − X1 + X 2 3 3 3 .

26) 338. In the above equation 337 let X1 = - 5 and X2 = 1. What is the range of values taken by the dependent variable if X1 and X2 vary by ± 10% from the above given values? 27) a.

3.4.

28) b.

3.10.

29) c.

3.95.

30) d.

4.25.

31) e.

3.80.

32) 339. A multiple regression equation has to be developed between the variables Y, X1 and X2. X1 and X2 are the independent variables, and Y is the dependent variable. The following details are available: 33) Number of data points = 10

7)

∑ Y = 272

∑ X1Y = 12005 ∑ X 2 Y = 4013 ∑ X1X 2 = 6485

∑ X1 = 441

∑ X2

= 147

2

2

∑ X1 = 19461 ∑ X 2 = 2173

1)

On the basis of the multiple regression relationship what is the value of Y, if X1 = 40 and X2 = 10?

2)

a.

-546.38.

3)

b.

35.772.

4)

c.

-93.756.

5)

d.

19.722.

6)

e.

541.566.

340. A multiple regression relationship contains two independent variables. The standard error of estimate is 4.8. Error sum of squares = 576.

8)

What is the number of data points?

218

9)

a.

24.

10) b.

25.

11) c.

26.

12) d.

27.

13) e.

28.

219

Index Numbers 1)

341. The sales manager of Alpha Company is examining the commission rate employed for the last 3 years. Below are the commission earnings of the company’s top five sales persons. (Commission earnings in rupees.)

2001 48,500 41,900 38,750 36,300 33,850

A B C D E

2002 55,100 46,200 43,500 45,400 38,300

2003 63,800 60,150 46,700 39,900 50,200

1)

Using 2001 as the base period, express the commission earnings in 2002 and 2003 in terms of the unweighted aggregates index.

2)

a.

114.6513 and 130.8329

3)

b.

112.2635 and 128.563

4)

c.

113.5261 and 132.6523

5)

d.

115.2631 and 134.652

6)

e.

111.263 and 131.26.

7)


8)

Telly Vision is a player in the television industry. The selling prices and the sales figures for the past 3 years are as follows. Model Vision 2,000 Emperor Prince Crown Pearl

Selling Price (in Rs.) 2001 2002 2003 17,000 17,900 18,600 13,100 13,600 13,700 11,200 11,400 11,900 8,500 8,700 8,800 5,500 5,600 5,800

Units Sold 2001 2002 11,200 12,000 8,000 9,600 9,600 10,400 11,200 10,400 12,000 14,400

2003 12,000 11,200 12,000 12,000 18,000

1) Using 2001 as the base year calculate the Laspeyres, Paasche’s and Fisher’s price indices for the year 2002 and 2003. 2) 342. Using 2001 as the base year, What is the Laspeyres price indices for the year 2002 and 2003? 3) a. 102.536 and 105.4548. 4) b. 103.4472 and 106.4548. 5) c. 103.4472 and 107.965. 6) d. 105.236 and 108.56. 7) e. 101.263 and 105.236. 8) 343. Using 2001 as the base year, what is the Paasche’s a price indices for the year 2002 and 2003? 9) a. 104.536 and 105.4548. 10) b. 101.4472 and 106.4548. 11) c. 103.4472 and 107.965. 12) d. 102.236 and 108.56. 13) e. 103.4536 and 106.3137. 14) 344. Using 2001 as the base year, what is the Fisher’s price indices for the year 2002 and 2003? 15) a. 102.536 and 105.4548. 16) b. 101.4472 and 106.4548. 17) c. 103.4504 and 106.3842. 18) d. 102.236 and 107.56.

220

19) e.

102.4536 and 107.3137.

20) 345. What is the unweighted average of relative’s price index for the years 2002 and 2003 with 2001 as the base year? 21) a.

101.536 and 105.4548.

22) b

102.4472 and 106.4548.

23) c.

103.2504 and 106.3842.

24) d.

103.0135 and 105.8452.

25) e.

102.4536 and 107.3137.

26) 346. What is the weighted average of relatives price index taking 2002 as base year and 2002 year values as weights for year 2001 and 2003 (Use the respective years as the base years). 27) a.

95.536 and 105.4548.

28) b.

94.4472 and 106.4548.

29) c.

93.2504 and 106.3842.

30) d.

96.6616 and 102.9101.

31) e.

99.4536 and 107.3137.

32) 347. What is the Laspeyres, quantity indices for the years 2002 and 2003 using 2001 as the base year for the above mentioned data?

33) a.

108.8523 and 121.6697.

34) b.

107.2563 and 120.2361.

35) c.

106.256 and 121.2563.

36) d.

109.856 and 119.2335.

37) e.

110.2789 and 122.5465.

38) 348. What is the Paasche’s quantity index for the years 2002 and 2003 using 2001 as the base year for the above mentioned data?

39) a.

107.89 and 123.563.

40) b.

108.59 and 121.5085.

41) c.

106.23 and 119.263.

42) d.

105.63 and 122.253.

43) e.

108.256 and 124.53.

44) 349. What is the unweighted average of relatives quantity indices for the years 2002 and 2003 with 2001 as the base for the above data? 45) a.

109.66 and 125.8571.

46) b.

108.63 and 124.5632.

47) c.

111.25 and 125.2365.

48) d.

107.25 and 122.5632.

49) e.

106.52 and 127.8652.

50) 350. What is the weighted average of relatives quantity indices with 2001 as the base year, and values as weights for year 2002 and 2003? 51) a.

108.8523 and 121.6697.

52) b.

107.5623 and 119.4562.

53) c.

106.5632 and 121.526.

221

54) d.

108.6356 and 117.256.

55) e.

107.9856 and 118.2635.

56) 351. Refer to the above calculate the value indices for the year 2002 with 2001 as the base and for the year 2003 with 2002 as the base. From these indices calculate the average growth rate of sales for the period 2001-03. 57) a.

14.65

58) b.

15.21

59) c.

16.09

60) d.

15.50

61) e.

16.95.

62) 352. In January, 2003, a factory paid out a total of Rs.2,40,000 to 120 employees on the payroll. In July, 2003 the factory had 30 more employees on the payroll and paid out Rs.36,000 more than in January. 63)

Using January, 2003 as the base, find the employment index number (quantity relative) for July.

64) a.

120%

65) b.

121%

66) c.

122%

67) d.

123%

68) e.

125%.

69) 353. In the above question no. 352 Using January, 2003 as the base find the labor expense index (value index) for July. 70) a.

112%

71) b.

113%

72) c.

114%

73) d.

115%

74) e.

116%.

75) 354. Consider the following data Commodities 2001 Price A 4 B 7 C 7 D 3

Quantity 16 10 25 18

2002 Price 16 17 10 6

Quantity 13 14 32 22

1)

What is the Laspeyres, Paasche’s price indices for year 2002? Use 2001 as the base year.

2)

a.

144.3526 and 142.7273

3)

b.

142.5623 and 140.3025

4)

c.

143.25 and 141.2362

5)

d.

145.23 and 143.2562

6)

e.

146.2325 and 141.2365.

7)

355. For the above question no. 354 What is the Marshall-Edge worth’s Index?

8)

a.

141.233.

9)

b.

142.632.

10) c.

143.462.

11) d.

144.632.

222

12) e.

145.265.

13) 356. What is the Marshall-Edge worth price indices given that Laspeyres Index and Paasche’s Index are 144.3526 and 142.7273? 14) a. 140.2533. 15) b. 141.6325. 16) c. 142.6325. 17) d. 143.5376. 18) e. 144.3225. 19) 357. The following table contains information from the raw material purchase records of a tire manufacturer for the years 2001, 2002 and 2003. Material Average Annual Purchase Value of Purchase Price/Ton (Thousands) 2001 2002 2003 2003 Butadiene Rs.17 Rs.15 Rs.11 Rs.50.0 Styrene 85 89 95 210.0 Rayon cord 348 358 331 1,640 Carbon black 62 58 67 630.0 Sodium Pyrophosphate 49 567 67 90.00 1) 2) 3) 4) 5) 6) 7)

1)

7)

What is the weighted average of relative price index for each of those 3 years, using 2003 for weighting and for the base year? a. 100.28,101.14 and 100. b. 100.78,101.54 and 100. c. 100.98,101.01 and 100. d. 100.52,101.25 and 100. e. 100.89,101.64 and 100. 358. This information describes the unit sales of a bicycle shop for 3 years: Number Sold Price Model 2001 2002 2003 2001 Sport 45 48 56 $ 89 Touring 64 67 71 104 Cross-country 28 35 27 138 Sprint 21 16 28 245 Calculate the weighted average of relative quantity indices, using the prices and quantities from 2001 to compute the value weights, with 2001 as the base year. 2) a. Index for 2002 - 102.63 and Index for 2003-119.70 3) b. Index for 2002 - 102.23 and Index for 2003-114.25 4) c. Index for 2002 - 103.52 and Index for 2003-117.63 5) d. Index for 2002 - 105.03 and Index for 2003-119.90 6) e. Index for 2002 - 101.63 and Index for 2003-116.70. 359. Harry Wadia, a purchasing agent, has compiled the price information presented below. Using 2000 as the base period, calculate the unweighted aggregates price index for 2001, 2002 and 2003. Material 2000 2001 2002 2003 Aluminum $ 0.96 $ 0.99 $1.03 $1.06 Steel 11.48 11.54 11.55 11.59 Brass tubing 10.21 10.25 10.26 10.31 Copper wire 10.06 10.08 10.07 10.09

1)

a.

104.23, 107.4 and 112.5

2)

b.

105.5, 107.4 and 112.5

223

3)

c.

105.5, 106.25 and 112.5

4)

d.

105.5, 107.4 and 112.63

5) 6)

e. 104.25,108.12 and 113.21. 360. From the following data taken from the official records and dealer’s registers, construct index numbers of wholesale prices for the year 2002-03 with base 2001-02 using Laspeyres method. Commodity Rice Wheat Jowar Bajra Barley Grain Sugarcane Cotton

Production 2001-02 (in tonnes) 2,342 2,676 2,604 2,292 2,266 2,380 2,526 2,165

Price per ton 2001-02 (Rs.) 6.50 4.50 3.90 3.70 3.75 4.20 4.20 14.30

Production 2002-03 (in tonnes) 3,061 1,072 750 360 230 440 1,183 255

Price per ton 2002-03 (Rs.) 19.3 16.5 16.2 16.4 16.2 15.8 11.2 22.5

1)

a.

1552.63

2)

b.

156.75

3)

c.

157.85

4)

d.

160.32

5)

e.

159.56.

6)

361. In the above question no.360, what is the index numbers using Paasche method?

7)

a.

162.1572.

8)

b.

159.1254.

9)

c.

158.6325.

10) d.

161.2563.

11) e.

163.5241.

12) 362. For the table given in question no.360, what is the index number using Marshall-Edge worth method? 13) a.

157.95.

14) b.

159.90.

15) c.

158.65.

16) d.

155.20.

17) e.

154.23.

18) 363. You are given the following price indices for items A and B. Year

Price relatives for A

2001 (base year) 100 2002 120 2003 135 1)

Unweighted average of relatives price index for A and B 100 130 150

What weighted aggregates price index for A and B giving a weightage to A which is three times the weightage you give to B with base year 2001?

224

2)

a.

2002 - 125 and 2003 -142.5.

3)

b.

2002 - 122 and 2003 -145.

4)

c.

2002 - 127 and 2003 -145.

5)

d.

2002 - 128 and 2003 -144.

6)

e.

2002 - 127.5 and 2003 -142.5.


1) A company produces three products, viz. erasers, pencil sharpeners and ballpoint pens. The table below gives prices and units sold (Qty.) in each of the years 2001, 2002 and 2003. Product Erasers Pencil sharpeners Ballpoint pens

2001 Price Re.0.50 Rs.2.00 Rs.3.00

Qty 30 25 20

2002 Price Rs.0.75 Rs.4.00 Rs.3.50

Qty. 30 20 25

2003 Price Re.1.00 Rs.3.00 Rs.3.50

Qty. 35 25 20

Using 1987 as the base, find 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25)

364. What is the unweighted aggregates price index and quantity index for 2003? a. 105. b. 104.63. c. 105.632. d. 106.67. e. 107.25. 365. What is the weighted aggregates index for 2003, by using both Laspeyres method and the Paasche method? a. 140 and 141.18. b. 135 and 142. c. 138 and 141.18. d. 140 and 142.26. e. 139 and 142.63. 366. What is the unweighted average of relative price index and unweighted average of relative quantity index for 2003? a. 102.56. b. 104.32. c. 106.53. d. 105.56. e. 104.92. 367. The Laspeyres and Fisher’s Ideal Index for a quantity are 103.28 and 108.91 respectively. What is the Paasche’s Index from this data? a. 112.63. b. 113.45. c. 114.85. d. 116.45. e. 115.23. 368. The movements of a particular index are given in the following table: Year 1999 2000

Index movement (%) 4.3 6.7

225

1)

2001 - 4.8 2002 2.1 2003 - 4.6 Using 1998 as the base year, what is the value of index in 2000 and 2003?

2) 3) 4) 5) 6)

a. b. c. d. e.

7)


111.29 and 103.19 110.26 and 103.19 112.25 and 103.19 113.63 and 102.45 114.63 and 104.63. Commodity A B C D

2001 Price 40 20 30 10

Quantity 10 5 6 9

2002 Price 50 30 40 20

Quantity 7 8 10 10

1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)

369. What is the Marshall-Edgeworth Index for the year 2002? a. 135.26. b. 140.37. c. 139.23. d. 142.25. e. 143.35. 370. What is the Laspeyre’s Index for the year 2002? a. 132.25. b. 135.26. c. 140.65. d. 138.96. e. 139.25. 371. Following are the price and quantities of 4 commodities used as a basket of goods to measure index. Base Year Current Year Commodity Price (P0) Rs. Quantity (Q0) Price (P1) Quantity Q1) kg. Rs. kg. A 5 13 8 12 B 7 8 12 9 C 8 6 11 10 D 3 16 4 25

1)

Which of the following is the price index for the current year using Marshall-Edgeworth Method? a. 149.25. b. 149.99. c. 150.95. d. 150.30. e. 151.05. 372. The following data is available for a basket of commodities:

2) 3) 4) 5) 6) 7)

Item

Price 2001

2002

Average quantity consumed by a family in 2001 2002

226

Bread (Nos.) Milk (liters) Eggs (dozen) Mutton (kg.) 1) 2) 3) 4) 5) 6)

6.00 10.00 12.00 80.00

8.00 15.00 18.00 120.00

360 540 240 100

500 600 300 80

What is the Fisher’s ideal price index from the above data? a. 145.25. b. 147.71. c. 147.05. d. 148.32. e. 148.95. 373. The sales manager of Direct Sales Company is examining the commission paid to its salespersons over the last 3 years. Given below are the commission earnings of the company’s top five salespersons. (Commission earnings in rupees) Salesperson 2000 2001 2002 A 48,500 55,100 63,800 B 41,900 46,200 60,150 C 38,750 43,500 46,700 D 36,300 45,400 39,900 E 33,850 38,300 50,200

1) 2)

You are required to express the commission earnings of the salespersons in 2001 and 2002 in terms of unweighted aggregates index using 2000 as the base period. a. 114.65 and 130.83 b. 114.96 and 130.83

3)

c.

114.65 and 131.05

4)

d.

115.25 and 131.05

5)

e.

115.95 and 132.025.

6)

374. The following table produces the prices and quantities of some fruits in the years 2001 and 2002:

Fruit

1)

2001 2002 Price (Rs. per Quantity Price (Rs. per Quantity (kgs) kg) (kgs) kg) Mangoes 15 30 25 26 Oranges 45 40 65 30 Apples 60 25 65 25 Papaya 08 50 12 45 Grapes 18 30 25 28 What is the price index according to Paasche’s method taking 2001 as the base year?

2)

a.

133.16.

3)

b.

135.02.

4)

c.

135.96.

5)

d.

134.96.

6)

e.

133.98.


1)

ABC Inc. sell three types of product P, Q and R. Listed below is the quantity sold and the price for 1998 and 2000. 1998 Item

Quantity

Price

2000 Quantity

Price

227

P Q R

sold 600 3,000 10,000

Rs.1.00 Rs.10.00 Rs.3.00

sold 200 5,000 9,000

Rs.5.00 Rs.8.00 Rs.4.00

1)

375. What is the simple price index for Q with 1998 as the base period?

2)

a.

80.00.

3)

b.

82.25.

4)

c.

86.25.

5)

d.

92.56.

6)

e.

78.50.

7)

376. What is the aggregate price index for the group of items, using 1998 as the base period?

8)

a.

120.

9)

b.

119.25.

10) c.

121.43.

11) d.

122.45.

12) e.

123.

13) 377. What is the Laspeyres’ price index for 2000, with 1998 as the base period?

14) a.

105.26.

15) b.

108.36.

16) c.

110.56.

17) d.

111.25.

18) e.

113.63.

19) 378. What is the Paasche’s price index for 2000, with 1998 as the base period? 20) a.

102.23.

21) b.

97.94.

22) c.

98.25.

23) d.

101.63.

24) e.

99.74.

25) 379. Compute a value index for 2000, with 1998 as the base period? 26) a.

125.

27) b.

126.03.

28) c.

127.06.

29) d.

128.36.

30) e.

130.

31) 380. A wholesaler is comparing the change in prices of the commodities he trades in. The following information is given by him: Commodity

Year

228

Rice Wheat Salt Sugar Pulses

2001 Price (Rs. /kg) 9.00 6.75 2.50 10.50 20.00

2002 Price (Rs./kg) 12.00 8.50 4.50 15.00 30.00

Quantity (kg.) 500 800 400 7.50 400

1)

What is the weighted average of relatives price index for the year 2002 using 2002 for weighting and 2001 as the base year?

2)

a.

140.

3)

b.

142.34.

4)

c.

141.92.

5)

d.

144.23.

6)

e.

143.62.

7)

381. Find the sample aggregative index number from the following data:

Commodity Rice Sugar Oil Wheat Pulse

Base Price 140 100 400 125 160

Current Price 180 300 550 150 200

1)

a.

145

2)

b.

148

3)

c.

149.2

4)

d.

151

5)

e.

152.

6)

382. What is the weighted aggregative method, the index number of the following data?

Commodity Rice Oil Sugar Wheat Fish 1) 2) 3) 4) 5) 6)

Base Price 140 400 100 125 200

Current Price 180 550 250 150 300

Weight 10 7 6 8 4

a. 142. b. 143. c. 144.7. d. 145. e. 149. 383. Prepare Laspeyre’s price index numbers for 2003 with 2001 as base year from the following data: Commodity A B C D

Unit Kg Quintal Dozen Kg

2001 Quantity 5 7 6 2

Price (Rs.) 2.00 2.50 3.00 1.00

2003 Quantity 7 10 6 9

Price (Rs.) 4.50 3.20 4.50 1.80

229

1) 2) 3) 4) 5) 6)

a. b. c. d. e. 384.

159 158 160 161 162. What is the Paasche’s price index numbers for 2003 with 2001 as base year for the above data?

7)

a.

158.

8)

b.

159.

9)

c.

165.

10) d.

160.

11) e.

162.

12) 385. What is the fisher’s price index numbers for 2003 with 2001 as base year for the above data?

13) a.

158.5.

14) b.

160.5.

15) c.

161.25.

16) d.

159.85.

17) e.

158.75.

18) 386. Calculate the price index number for 2003 with 2000 as base year by the aggregative method, using base year quantities as weights and Commodity

1)

a.

A B C D 139.7

2)

b.

137.2

3)

c.

138.45

4)

d.

139.2

5)

e.

141.

2000 Quantity (000 tons) 350 200 140 80

Price per ton (Rs) 100 130 50 125

2003 Quantity (000 tons) 400 180 200 100

Price Per ton (Rs.) 120 200 110 140

387. In the above question no.386, what is the given year quantities as weights, from the following data? 7) a. 140. 8) b. 141. 9) c. 139.7. 10) d. 142.5. 11) e. 146. 12) Based on the following information answer the questions no. 388-389.

6)

Commodity A B

Unit Kg Quintal

2002 Quantity 5 7

Price (Rs.) 2.00 2.50

2004 Quantity 7 10

Price (Rs.) 4.50 3.20

230

C D 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 392.

1) 2) 3) 4) 5) 6) 7)

1)

Dozen Kg

6 2

3.00 1.00

6 9

4.50 1.80

388. What is the Laspyere’s price index numbers for 2004 with 2002 as base year for the above data? a. 160. b. 161. c. 159. d. 157. e. 158. 389. Prepare Paasche’s price index numbers for 2004 with 2002 as base year for the above data? a. 162. b. 165. c. 166. d. 168. e. 163.50. 390. What is fisher’s index numbers for 2004 with 2002 as base year from the following data? Pasche’s Index = 145 Laspyere’s Index = 148 a. 146.50. b. 147.62. c. 148.25. d. 149.62. e. 150.00. 391. What is the Marshall-Edgeworth’s Index for the following data ? ∑ p1q 0 = 825, ∑ p1q1 = 875, ∑ p 0 q 0 = 780, ∑ p0 q1 = 800 a. 105. b. 108.60. c. 107.60. d. 109.25. e. 110.65.

2001 2004 Commodity Price Qty. Price Qty. A 6 50 10 56 B 2 100 2 120 C 4 60 6 60 D 10 30 12 24 E 8 40 12 36 What is the Fisher ideal index for the above data? a. 138. b. 139.85. c. 140.5. d. 141.25. e. 143. 393. Find index numbers for the years 2003 by the chain base method, with base-year 2000 from the following table: Year Link index a. 115

2000 100

2001 110

2002 95.5

2003 109.5

231

2) 3) 4) 5) 6)

b. c. d. e. 394.

116 117 114 118. From the following data calculate the weighted average of relative index

Commodity Q0 P0 P1 A 200 0.91 1.19 B 300 0.79 0.99 C 100 3.92 4.50 Total 1) a. 120.5. 2) b. 121.65. 3) c. 122.00. 4) d. 123.65. 5) e. 124. 6) 395. What is the value of Fisher’s index, Laspeyre’s index and Paasche’s index is 121.6697 and 121.5085? 7) a. 99.99. 8) b. 100.01. 9) c. 115.75. 10) d. 121.5891. 11) e. 125.6361. 12) 396. For the data given below simple price weighted index will be Price/share at the beginning of the period Stock A 35.00 Stock B 47.00

Price/share at the end of the period 42.00 56.00

Number of shares outstanding 1,20,000 2,50,000

1)

a.

117.57

2)

b.

118.87

3)

c.

119.51

4)

d.

121.47

5)

e.

123.97.

6)

397. If the prices and the quantities consumed for a basket of commodities in two time periods are such that ∑ P1Q0 = 1280, ∑ P0 Q0 = 1140, ∑ P1Q1 = 1680 and ∑ P0 Q1 = 1500, then what is the Fisher’s

ideal price index? 7)

a.

105.12.

8)

b.

108.14.

9)

c.

110.26.

10) d.

112.14.

11) e. 116.16. 12) 398. A national shopping survey was conducted to study the average weekly buying habits of a typical Indian family in 2001 and 2002. The data collected are as follows: Items Cheese (8 grams)

2001 Unit Price (in Rs.) 11.9

Quantity 2

2002 Unit price (in Rs.) 20.9

Quantity 1

232

Bread (1 loaf) Eggs (1 dozen) Milk (1 liter)

7)

7.9 8.4 13.6

3 2 2

10.9 13.5 23.9

3 1 2

1)

What is the Marshall-Edgeworth price index for the year 2002 using 2001 as the base period?

2)

a.

162.38.

3)

b.

164.17.

4)

c.

166.64.

5)

d.

168.55.

6) e. 171.34. 399. The prices and quantities of some commodities consumed by a family in the years 1998 and 2002 are given below: Commodity

Tea Sugar Milk Powder Rice Flour

Year 1998 Price (Rs./kg) 600 14.50 90 13.50 12.00

Quantity (kgs) 10 150 15 500 300

2002 Price (Rs./kg)

Quantity (kgs)

800 15.00 120 15.50 13.50

8 120 12 480 250

1) What is the price index by Laspeyre’s method for the basket of commodities consumed by the family, for the year 2002, considering 1998 as the base year?

7)

2)

a.

83.33.

3)

b.

118.50.

4)

c.

102.92.

5)

d.

112.07.

6)

e.

120.00.

400. The following data are collected from a wholesaler of commodities:

Commodity

Rice Wheat Salt Sugar Pulses

Year 1997 Price (Rs./kg.) 11.00 8.50 3.50 12.50 24.50

2002 Price (Rs./kg) 14.00 11.50 5.00 14.00 29.50

1999 Price (Rs./kg) 13.00 9.75 4.25 13.00 27.00

Quantity (kg.) 2500 1000 500 750 1200

1) What is the weighted average of relative price index for the year 2002, using the year 1999 for weighting and the year 1997 for the base year? 2)

a.

123.33.

3)

b.

124.27.

4)

c.

127.57.

5)

d.

90.54.

6)

e.

111.67.

233

7)

401. The following data are collected from a fruit merchant:

Commodity

Mangoes Bananas Oranges Pineapple Apple

Year 1996 Price Rs.10 per kg. Rs.8 per doz. Rs.30 per kg. Rs.8 per piece Rs.40 per kg.

Quantity 500 kgs. 600 doz. 350 kgs. 450 pieces 300 kgs.

2002 Price Rs.15 per kg. Rs.12 per doz. Rs.50 per kg. Rs.12 per piece Rs.70 per kg.

Quantity 400 kgs. 500 doz. 300 kgs. 400 pieces 200 kgs.

1) What is the price index by Marshall-Edgeworth method for the fruits traded by the merchant, for the year 2002, considering 1996 as the base year? 2)

a.

60.38.

3)

b.

81.82.

4)

c.

162.87.

5)

d.

165.63.

6)

e.

122.22.

Time Series Analysis 1)

Based on the following information answer the questions no. 402-404. Clean environ Ltd. was able to sell the following number of devices during the past seven years.

Years 1992 1993 1994 1995 1996 1997 1998 Number of devices sold (in hundreds) 11 13 14 16 18 20 23 These devices reduce the amount of pollutants released by the tannery and other industries. With stricter enforcement of laws, the sales are expected to pick up. 1) 3)

402. For this data, which of the following represent a Linear estimating equation? ˆ = 16.43 + 1.93x. Y a. ˆ = 16.43 + 1.63x. Y b.

4)

c.

5)

d.

2)

6) 7) 8) 9) 10) 11) 12)

ˆ Y ˆ Y

= 15.84 + 1.93x. = 17.53 + 1.23x.

ˆ Y

e. = 16.23 + 1.53x. 403. Which of the following represent the Curvilinear equation from the above data? ˆ = 14.93 + 1.93x - 0.12x2. Y a. ˆ = 15.93 + 1.23x - 0.12x2. Y b. ˆ = 15.93 + 1.93x - 0.45x2. Y c. ˆ = 17.02 + 1.87x - 0.18x2. Y d. ˆ = 15.93 + 1.93x - 0.12x2. Y e.

13) 404. Calculate the number of devices that will be sold in 1999 by using both the estimating equations. Which of the two estimates is more accurate? 14) a.

2173.

15) b.

2251.

16) c.

2150.

17) d.

2190.

18) e.

2105. 234

19) 405. The level of the working capital needed by a small firm during the last six years is shown below. Year 1993 1994 1995 1996 1997 1998 Working Capital (in Rs. lakh) 5.10 5.85 6.35 6.65 7.10 7.35 1) 2) 3)

For this data, which of the following represent at a linear trend? ˆ = 6.98 + 0.2186x. Y a. ˆ = 6.4 + 0.2186x. Y b.

4)

c.

5)

d.

ˆ = 6.4 + 0.2925x. Y ˆ = 6.25 + 0.2225x. Y

6)

e.

ˆ = 6.55 + 0.2045x. Y

7)

406. From the above question no. 405 and estimate the level of working capital required during the years 2000 and 2001 respectively?

8)

a.

Rs.8.367 lakh and Rs.8.58 lakh. respectively.

9)

b.


10) c.


11) d.


12) e.

Rs.8.102 lakh and Rs.8.40lakh. respectively.

13) 407. As a finance manager you are responsible to raise funds from different sources so that the cost of capital is minimum possible. The first step you would take is to ascertain the amount of funds that you need in the current financial year. Towards this end you collected the following information and obtained an equation for best linear fit as Year Amount required (in lakh)

1992 1993 1994 1995 1996 1997 1998 65 71 79 82 90 102.50 105.00

1)

Calculate the Percent of Trend for this data and state in which year there is largest fluctuation from the trend line?

2)

a.

1993.

3)

b.

1994.

4)

c.

1995.

5)

d.

1996.

6)

e.

1997.

7)

408. In the above question no.407. Calculate the Relative Cyclical Residual Trend and state in which year there is largest fluctuation from the trend line?

8)

a.

1994.

9)

b.

1995.

10) c.

1996.

11) d.

1997.

12) e.

1998.

13) 409. The exchange rates (Rs./Dollar) for the last two years are given below. Month Exchange rate (Rs./Dollar) 1997-98 1998-99 April 35.80 39.63 May 35.79 40.45 June 35.80 42.18 235

July August September October November December January February March 1)

35.72 35.88 36.40 36.80 37.18 39.09 39.21 38.83 39.49

42.44 42.69 42.46 42.28 42.33 42.52 42.46

Calculate the quarterly moving average of exchange rates from June, 1998 to December, 1998 and state which of the following statement is correct? 2)

i.

Quarterly moving average for the August 98 is 42.44

3)

ii.

Quarterly moving average for the September 98 is 42.53

4)

iii.

Quarterly moving average for the October 98 is 42.47

5)

iv.

6)

a.

Only (i), (ii), (iii).

7)

b.

Only (i), (ii), and (iv).

8)

c.


9)

d.


Quarterly moving average for the November 98 is 42.38.

10) e. Only (i) and (ii). 11) 410. Following are the price and quantities of 4 commodities used as a basket of goods to measure index. Commodity A B C D

Base Year Price (P0) Rs. 5 7 8 3

Quantity (Q0) kg 13 8 6 16

Current Year Price (P1) Quantity (Q1) Rs. kg 8 12 12 9 11 10 4 25

1)

What is the price index for the current year using Marshall-Edge worth method?

2)

a.

149.20.

3)

b.

150.30.

4)

c.

152.65.

5)

d.

148.65.

6) 7)

e. 153.52. 411. The following data pertains to the estimated demand and actual demand for a product for the last 8 periods. Period 1 2 3 4 5 6 7 8

Estimated Demand (’000 units) 714 546 568 739 577 662 788 1097

Actual Demand (’000 units) 750 526 652 667 682 859 1022 981

You are required to identify the cyclical variation in demand using percent of trend measure and state

236

1) 2) 3) 4) 5) 6)

7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)

in which period fluctuation is maximum? a. 3. b. 4. c. 5. d. 6. e. 7. 412. In a study of sales company has obtained the following least square trend equation: Y = 16+2x (orgin year 1977, X units = 1year, Y= total no. of unit sold). The company has physical facilities to provide only 30units in a year and it believes that at least for the next decade the trend will continue before. By what year the company’s expected sales have equaled its present capacity? a. 2003. b. 2004. c. 2005. d. 2006. e. 2007. 413. In the above question no. 412. What is the estimated sales for the year 2001? a. 22 units. b. 23 units. c. 24 units. d. 25 units. e. 26 units. 414. Suppose we have a series of quarterly production figure (I thousands) in an industry for the years 1990 to 1996, and the equation of the linear trend fitted to the annual data is

19) Xt = 107. 2 +2093t 20) Where t = year 1993 and Xt = annual production in time t. Estimate the annual production for 1991. 21) a.

101.34

22) b.

102

23) c.

102.56

24) d.

102.96

25) e.

103.01.

415.

Period

1)

Estimated Expenditure

Actual expenditure (’000 units) Y

ˆ (’000 units) Y 1995 250 200 1996 265 245 1997 295 280 1998 345 325 1999 415 420 2000 450 425 2001 465 440 2002 480 470 You are required to identify the cyclical variation in demand using percent of trend measure and state in which period fluctuation is maximum?

2)

a.

1995.

3)

b.

1996.

4)

c.

1998.

237

5)

d.

2000.

6)

e.

2001.

7)

416. Mr. Prasad, the accounts manager of Xcel Ltd., has compiled the following data regarding the level of accounts receivable over a period of five years in Rs. Thousands.

1)

Year Quarter I Quarter II 1994 102 120 1995 110 126 1996 111 128 1997 115 135 1998 122 144 For this data, help him calculate the Modified mean correct?

2)

i.

Modified mean for 1st quarter is 106.76

3)

ii.

Modified mean for 2nd quarter is 122.39

4)

iii.

Modified mean for 3rd quarter is 91.72

5)

iv.

Modified mean for 4th quarter is 82.52.

6)

a.

Only (i), (ii) and (iv).

7)

b.


8)

c.


9)

d.


10) e.

Quarter III Quarter IV 90 78 95 83 97 86 103 91 110 98 and state which of the following statement is

Only (i) and (ii).

11) 417. In the above question calculate the seasonal index and state which of the following statement is correct? Seasonal Index for 1st quarter is 106.76 12) i. 13) ii. Seasonal Index for 2nd quarter is 122.45 14) iii. Seasonal Index for 3rd quarter is 91.76 15) iv. Seasonal Index for 4th quarter is 78.98 16) a. Only (i), (ii) and (iv). 17) b. Only (i), (ii) and (iii). 18) c. Only (ii), (iii) and (iv). 19) d. Only (i), (iii) and (iv). 20) e. Only (i) and (ii). 21) 418. If we depersonalizing the data, and fit a linear equation, which of the following is represent the equation for the trend? ˆ = 102.841 + 0.64x. Y 22) a. 23) b. 24) c. 25) d.

ˆ = 101.841 + 0.84x. Y ˆ = 101.841 + 0.64x. Y ˆ = 103.241 + 0.94x. Y ˆ = 104.841 + 0.74x. Y

26) e. 27) 419. The production in (thousands) in a Shoe factory during 1998-2004 has been as follows: Period 1998 1999 2000

Production 87 98 104

238

2001 2002 2003 2004 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)

95 101 108 100

Which of the following represent a straight line that best represent the data? a. Yt = 99 + 2X. b. Yt = 91 + 2.2X. c. Yt = 93 + 2.5X. d. Yt = 95 + 2.6X. e. Yt = 98 + 3X. 420. What is the monthly increase in production? a. 166.67. b. 167.68. c. 165.63. d. 162.53. e. 160.25. 421. A trend equation using monthly sales data for the years 1999 through 2003 is Y’ = 2.50 + 4.56t (January, 1999 = 1). What is the unadjusted forecast for April 2004? a. 291. b. 292.5. c. 293.6. d. 294.34. e. 296.52.

19) 422. The April seasonal index is 75. What is the seasonally adjusted forecast for April 2004? 20) a.

214.25.

21) b.

218.35.

22) c.

220.755.

23) d.

225.63.

24) e.

205.63

ˆ 25) 423. The estimation equation for a time series is given by Y = 60 + 1.5x , where x is the coded time such that x = X - X . The mean year is given as 1998. What would be the relative cyclical residual for the year 2003, if the observed value for that year were 70?

26) a.

−3.70.

27) b.

3.57.

28) c.

3.70.

29) d.

96.4.

30) e.

103.7.

31) 424. The Gulmarg Winter Sports resort has just recently tabulated its data on the number of customers (in thousands) it has had during each season of the last 5 years. Spring Summer Fall Winter 239

1999 2000 2001 2002 2003 1)

200 175 225 200 175

300 250 300 350 300

125 150 200 225 200

325 375 450 375 350

Which of the following is the correct combination of seasonal index for the quarters Spring, Summer, Fall and Winter respectively? (Use four quarter moving average) 2)

a.

45.36, 65.58, 51.36, 95.64.

3)

b.

48.75, 75.00, 45.00, 93.75.

4)

c.

64.56, 105.45, 71.25, 126.56.

5)

d.

73.37, 113.34, 66.02, 147.27.

6)

e.

84.26, 125.64, 75.84, 156.56.

425. The linear trend estimating equation for a time series is given below:

1)

ˆ = 139 + 7.5x Y

2)

where x = Year - 1998

3)

ˆ is the estimated value of the dependent variable Y. Y

4)

The observed value of Y for the year 2000 is 160.

5)

What is the relative cyclical residual for the year 2000?

6)

a.

-3.9.

7)

b.

10.8.

8)

c.

3.9.

9)

d.

96.25.

10) e.

103.9.

240

Quality Control 1)

426. Mr. Mathur is responsible to overlook the emergency services at Medicare Hospital. He receives many complaints about the time taken by the ambulance in reaching the necessary location. He decides to check whether the complaints are genuine and whether the process is In-control. Each day he takes 9 calls and he repeats this for 21 days. The data collected is as follows.

1)

Mon Tue Wed Thu 11.6 17.4 14.8 13.8 R 14.1 19.1 22.9 18.0 Second week x 9.5 12.7 17.7 16.3 R 12.6 17.0 12.0 15.1 Third week 11.4 16.0 11.0 13.3 x R 12.1 21.1 13.5 20.3 What is the upper and the lower control limits of the test?

2)

a.

8.725 and 21.075.

3)

b.

7.725 and 20.075.

4)

c.

6.725 and 19.075.

5)

d.

5.725 and 18.075.

6)

e.

4.725 and 17.075.

7)

427. Bolt India Ltd. is manufacturing 20,000 bolts of a diameter of 1.05 cms per day. The standard deviation of the diameter is known to be 0.005 cm. Mr. Sanjeev Isser, the quality control engineer, takes a sample of 25 bolts per hour to monitor the diameter of bolts to ensure that the process is in control. The samples of last 5 hrs. observed by him at 9:00 AM on a particular day have indicated the following dimensions.

First week

x

Time 9:00 AM 8:00 AM 7:00 AM 6:00 AM 5:00 AM

Fri 13.9 14.6 10.5 22.1 19.3 16.8

Sat 22.7 23.7 22.5 24.1 21.5 20.7

Sun 16.6 21.0 12.6 21.3 17.9 23.2

Sample diameter (cm) 1.0524 1.0475 1.0489 1.0513 1.0524

1)

With the help of x chart, examine whether the above process is in control or not and state which of the following statement is correct? i. Upper control limit is 1.053

2)

ii.

Lower control limit is 1.047

3)

iii.

There is no significant pattern in these observations, and the process seems to be in control

4)

iv.

There is significant pattern in these observations, and the process is not in control.

5)

a.

Only (i) above.

6)

b.

Only (i) and (ii) above.

7)

c.

Only (i), (ii), and (iii) above.

8)

d.

Only (i), (ii) and (iv) above.

9)

e.

Only (ii) and (iv) above.

10) 428. The manager of a food chain outlet wants to ensure that the variability in the service time by their bearers is in control. He does a sampling of the average time taken by the bearers in serving the customers in the outlet. He finds that the mean of the sample range to be 1.5 minutes. If he has collected five samples each of ten observations to conduct the test then What is the upper control limit of the R chart? (Given that d2 =3.078 and d3 = 0.797 for sample size of 10)

241

11) a.

0.335.

12) b.

1.005.

13) c.

1.500.

14) d.

2.005.

15) e. 2.665. 16) 429. Sportsline Bicycle Parts manufactures precision ball bearings for wheel hubs, bottom brackets, headsets, and pedals. Ashok is responsible for quality control at Sportsline. He has been checking the output of the 5 mm bearings used in front wheel hubs. For each of last 6 hours, he has sampled 5 bearings, with the following results: Hour 1 2 3 4 5 6 1)

Bearing Diameter (mm) 5.03 5.06 4.97 4.94 5.02 4.98 4.92 4.93 5.01 4.99 5.00 4.95

4.86 5.09 4.94 4.90 4.93 5.10

4.90 4.78 4.95 4.92 5.06 4.85

4.95 4.88 4.80 4.96 5.01 4.91

Ashok wishes to control the variation in the bearing diameter by plotting a x chart. Which of the following is true for the given sample? i.

The grand mean of the bearing diameter is 4.953 mm.

ii.

The grand mean of the bearing diameter is 5.001 mm.

iii.

The mean of sample range is 0.195 mm.

iv.

The mean of sample range is 0.201 mm.

v.

The Lower Control Limit of the x chart will be 4.841 mm.

vi.

The Lower control Limit of the x chart will be 4.891 mm.

vii. The Upper Control Limit of the x chart will be 5.066 mm. viii. The Upper Control Limit of the x chart will be 5.112 mm. ix.

On the basis of the control chart it can be concluded that the process is out of control.

x.

On the basis of the control chart it can be concluded that the process is in control.

1)

a.

Only (i), (iii), (v), (vii) and (x).

2)

b.

Only (ii), (iv), (vi), (viii) and (ix).

3)

c.

Only (i), (iv), (vi), (viii) and (x).

4)

d.

Only (ii), (iii), (v), (vii) and (ix).

5)

e.

Only (i), (iii), (vi), (viii) and (x).

6)

242

Chi-Square Test Analysis of Variance 1)

430. Sales manager of a leading newspaper which brings out its editions in two languages wanted to know its newspaper readership profile. He mused whether the educational qualifications of a person determined the reading habits.

He conducted survey and prepared the findings in the form of the table which is shown below.

2)

Frequency of Readership Never Sometimes Morning or Evening Both Editions Total

Level of Educational Achievement Post College High School High School Graduates Students Students Drop Outs 10 17 11 21 12 23 8 5 35 38 16 7 28 19 6 13 85 97 41 46

Total 159 148 1 96 166 269

1)

What is the Chi-square Statistic for the above data?

2)

a.

29.65.

3)

b.

32.88.

4)

c.

33.65.

5)

d.

34.25.

6)

e.

35.0.

7)

431. In the above question no. 430. Do you think that at a significance level of 10%, the frequency of the newspaper readership differs according to the educational qualification of the readers?

8)

a.

2 Yes and the χ Statistic is 14.684.

9)

b.

Yes and the degree of freedom for the test is 9.

10) c.

2 Yes and the χ Statistic is 18.254.

11) d.

Both a and c above.

12) e. a and b above. 13) Based on the following information answer the questions no. 432-434. 14) A retailing chain in order to improve sales has adopted four promotion policies at five of its stores. The data collected is shown below. Free sample One pack gift Discount on cost price Refund by mail if unsatisfied 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11)

432. a. b. c. d. e. 433. a. b. c. d.

Store 1 78 94 73 79

Store 2 87 91 78 83

Store 3 81 87 69 78

Store 4 89 90 83 69

Store 5 85 88 76 81

What will be the grand mean for the above data? 81.95. 86. 85.26. 82.5. 80.54. What is the first estimation of population variance using between-column variance methods? 204.05. 203.25. 201.25. 205. 243

12) e. 206.63. 13) 434. In the above question, what is the first estimation of population variance using within-column variance methods? 14) a.

18.

15) b.

19.

16) c.

20.

17) d.

21.

18) e.

22.

19) 435. Maya Investment Corporation is planning a new venture. It wants to analyze the risk of the project through simulation. The analyst of the company has identified annual demand, sales price and variable costs as the factors based on which the simulation can be carried out. The probability distributions of the three factors are as follows: Annual demand Units 20,000 25,000 30,000 35,000

Sales price Prob Rs./unit Probability abili ty 0.05 50 0.2 0.10 70 0.6 0.20 90 0.2 0.30 40,000

Variable costs Rs. per unit Probability 17.50 20.00 25.00

0.25 0.50 0.25

244

Part II: Solutions Probability Distribution and Decision Theory 1.

(a) The expected value or mean of the random variable X is given by E(X) = ΣXP(X) = (0) (0.1) + (1) (0.15) + (2) (0.25) + (3) (0.3) + (4) (0.1) + (5) (0.07) + (6) (0.03) = 2.48 2 The variance of X is given by V(X) = ΣP(X − X) X P(X) X−X P(X).(X − X) 2 (X − X) 2

0 1 2 3 4 5 6 Total

0.1 0.15 0.25 0.30 0.10 0.07 0.03

-2.48 -1.48 -0.48 0.52 1.52 2.52 3.52

6.1504 2.1904 0.2304 0.2704 2.3104 6.3504 12.3904

0.61504 0.32856 0.0576 0.08112 0.23104 0.444528 0.371712 2.1296

V(X) = 2.1296. 2.

(d) P(1 ≤ X ≤ 5) = P(1) + P(2) + P(3) + P(4) + P(5) = 0.15 + 0.25 + 0.30 + 0.10 + 0.07 = 0.87.

3.

(c) P(2 ≤ X)

4.

= P(2) + P(3) + P(4) + P(5) + P(6) = 0.25 + 0.30 + 0.10 + 0.07 + 0.03 = 0.75.

(c) P(X < 4)

5.

= P(0) + P(1) + P(2) + P(3) = 0.1 + 0.15 + 0.25 + 0.30 = 0.8. (b) If the event of a candidate getting the job could be called a success then we have p = 0.12, q = 0.88 and n = 15 The probability of r successes in n trials is given by

⎛n⎞ ⎜⎜ ⎟⎟ r P(r) = ⎝ ⎠ pr qn - r

6.

⎛15 ⎞ 15! ⎜⎜ ⎟⎟ 5 5!10! P(5) = ⎝ ⎠ (0.12)5 (0.88)10 = (0.12)5 (0.88)10 = 0.0208. (a) P(Less than 4) = P (0) + P (1) + P (2) + P(3) ⎛15 ⎞ ⎜⎜ ⎟⎟ 1 15 = (0.88) + ⎝ ⎠ (0.12) (0.88)14 +

⎛15 ⎞ ⎜⎜ ⎟⎟ ⎝ 2 ⎠ (0. 12)2 (0.88)13 +

⎛15 ⎞ ⎜⎜ ⎟⎟ ⎝ 3 ⎠ (0.12)3 (0.88)12

= 0.147 + 0.3006 + 0.287 + 0.1696 = 0.9042. 7.

(b) If the event of answering a question correctly is taken as a success, then we have p = 0.25, q = 0.75, and n = 25.

The mean is given by np = (25)(0.25) = 6.25. Therefore, the expected number of correct answers is 6.25.

245

8.

(a) The probability of answering exactly r = 5 questions out of n = 25 questions correctly is given by

⎛n⎞ ⎜⎜ ⎟⎟ r P(r) = ⎝ ⎠ pr qn - r = 9.

⎛ 25 ⎞ ⎜⎜ ⎟⎟ ⎝ 5 ⎠ (0.25)5 (0.75)20 = 0.1645.

(d) We can take the event of a customer making a purchase as a success.

Then we have p = 0.3, q = 0.7, and n = 15. The probability of r successes in n trials is given by ⎛n⎞ ⎜⎜ ⎟⎟ r P(r) = ⎝ ⎠ pr qn - r

P(at least 1) = 1 - P(0) ⎛15 ⎞ ⎜⎜ ⎟⎟ 0 P(0) = ⎝ ⎠ (0.3)0 (0.7)15 = (0.7)15 = 0.0047

P(at least 1) = 1 - P(0) = 1 - 0.0047 = 0.9953. 10. (c) Let the event of a person owning a home be taken as a success. Then p = 0.30 and q = 0.70. ⎛7⎞ ⎜⎜ ⎟⎟ 2 P(2 home owners) = ⎝ ⎠ (0.3)2 (0.7)5 7! 2!5!

(0.3)2 (0.7)5 = 0.3177. 11. (a) Note: The sample size is small but σ is known. Hence, we use Z-test (and not t-test). =

Given: µ0 = 16; σ = 0.5; n = 25 x = 16.35; x = 15.85 Z=

Test Statistic is

x − µ0 σ/ n

=

16.35 − 16 0.5 25 = 3.5.

12. (b) Let X denotes the random variable marks got by students.

The mark of 82 can have any value from 81.5 to 82.5. Therefore, the value of Z=

X − µ 81.5 − 82 − 0.5 = = = − 0.17 σ 3 3

Z=

X − µ 82.5 − 82 0.5 = = = 0.17 σ 3 3

P (81.5 < X < 82.5) = P(-0.17 < Z < 0.17) = 2(0.0675) = 0.135. 13. (a) Here probability of an application being approved = p = 0.8

Mean µ

= np = (0.8) (1460)

= 1168

2

Variance σ = npq = (1460) (0.8) (0.2) = 233.6 Standard deviation =

npq

=

233.6 = 15.284

The number of applications approved in a year can be said to follow a normal distribution with mean 1168 and standard deviation 15.284. Let X be the random variable, denoting the number of applications approved in a year.

246

X = 1180 implies

Z=

X − µ 1180 −1168 = ~ σ 15.284 = 0.7851 − 0.79.

P(X ≤ 1180) = P(Z ≤ 0.79) = 0.5 + 0.2852 = 0.7852. 14. (c) P(X ≥ 1180) = P(Z ≥ 0.79) = 0.5 - 0.2852 = 0.2148. 15. (d) Production Decision

Conditional Profit States of Demand (in thousands)

50 400 330 260 190 120

50 60 70 80 90

60 400 480 410 340 270

70 400 480 560 490 420

80 400 480 560 640 570

90 400 480 560 640 720

Expected Profit for each Decision Alternative Decision Alternative Expected Profit (in thousands) 50 (400) (0.1) + (400) (0.12) + (400) (0.15) + (400) (0.33) + (400) (0.3) = 400 60 (330) (0.1) + (480) (0.12) + (480) (0.15) + (480) (0.33) + (480) (0.3) = 465 70 (260) (0.1) + (410) (0.12) + (560) (0.15) + (560) (0.33) + (560) (0.3) = 512 80 (190) (0.1) + (340) (0.12) + (490) (0.15) + (640) (0.33) + (640) (0.3) = 536.5 90 (120) (0.1) + (270) (0.12) + (420) (0.15) + (570) (0.33) + (720) (0.3) = 511.5.

We observe that expected profit is maximized when the publishing house prints 80,000 calendars. Expected profit under certainty (perfect information) is = (400) (0.1) + (480) (0.12) + (560) (0.15) + (640) (0.33) + (720) (0.3) = 40 + 57.6 + 84 + 211.2 + 216 = 608.8. Therefore, expected value of perfect information = 608.8 - 536.5 = 72.3 = Rs.72,300. 16. (a)

Conditional Profit Table Stock Possible Demand Decision 20 21 22 20 449 449 449 21 447.45 471.45 471.45 22 445.90 469.90 493.90 23 444.35 468.35 492.35 24 442.80 466.80 490.80

Stock Decision 20 21 22 23 24

23 449 471.45 493.90 516.35 514.80

24 449 471.45 493.90 516.35 538.80

Expected Profit Table Expected Profit (449) (0.23) + (449) (0.21) + (449) (0.22) + (449) (0.16) + (449) (0.18) = 449 (447.45) (0.23) + (471.45) (0.21) + (471.5) (0.22) + (471.5) (0.16) + (471.5) (0.18) = 465.93 (445.9) (0.23) + (469.9) (0.21) + (493.9) (0.22) + (493.9) (0.16) + (493.9) (0.18) = 477.82 (444.35) (0.23) + (468.35) (0.21) + (492.35) (0.22) + (516.35) (0.16) + (516.35) (0.18) = 484.43 (442.8) (0.23) + (466.8) (0.21) + (490.8) (0.22) + (514.8)

247

(0.16) + (538.8) (0.18) = 487.20 The expected profit is the highest, when stocking decision is 24. The students should therefore order 2400 t-shirts to maximize their profits. 17. (b) There is a fleet of N(= 24) trucks of which M(= 4) emit excessive amounts of pollutants and the remaining N - M (= 24 - 4 = 20) do not emit excessive amounts of pollutants. In this problem, we are interested in the probability that x (= 0), that is none of the trucks out of chosen n (= 6) emit ⎛M⎞ ⎜⎜ ⎟⎟ x pollutants. But we are not replacing them back. Therefore, there are ⎝ ⎠ ways of choosing x trucks ⎛N − M⎞ ⎜⎜ ⎟ n − x ⎟⎠ out of M trucks and ⎝ ways of choosing n - x (= 6) trucks out of N - M (= 20) trucks. Hence, M N − M ⎛ ⎞ ⎛ ⎞ ⎛ N⎞ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ x n−x ⎠ n there are ⎝ ⎠ ⎝ ways of choosing x out of n - x trucks. Also, there are ⎝ ⎠ ways of choosing n out of N trucks. Since all events are equally likely, the probability of choosing x trucks out of n trucks, is given by

P(X = x) =

⎛M⎞⎛ N − M⎞ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎝ x ⎠⎝ n − x ⎠ ⎛ N⎞ ⎜⎜ ⎟⎟ ⎝n⎠

P(X = 0) =

⎛ 4 ⎞ ⎛ 20 ⎞ ⎜ ⎟⎜ ⎟ ⎝ 0 ⎠ ⎝ 6 ⎠ = 0.2880. ⎛ 24 ⎞ ⎜ ⎟ ⎝6⎠

18. (a) X

6 9 11 12 15 17 Total

Y 11 14 15 19 23 25

P(X, Y) XP(X, Y) 0.03 0.18 0.20 1.80 0.07 0.77 0.50 6.00 0.10 1.50 0.10 1.70 11.95

YP(X, Y) 0.33 2.80 1.05 9.50 2.30 2.50 18.48

(X − X)

(Y − Y)

P(X, Y) (X − X)2

P(X, Y) (Y − Y)2

P(X, Y) (X − X) (Y − Y)

-5.95 -2.95 -0.95 0.05 3.05 5.05 Total

-7.48 -4.48 -3.48 0.52 4.52 6.52

1.0621 1.7405 0.0632 0.0012 0.9302 2.5502 6.3474

1.6785 4.0141 0.8477 0.1352 2.043 4.2510 12.9695

1.3352 2.6432 0.2314 0.0130 1.3786 3.2926 8.8940

E(X) = X = ΣXP(X, Y)

= 11.95

E(Y) = Y = ΣYP(X, Y)

= 18.48

Var(X)

= ΣP(X - X )2

= 6.3474

248

Var(Y)

= ΣP(Y - Y )2

= 12.9695.

19. (e) Cov (X, Y) = ΣP(X - X )(Y - Y ) = 8.8940. 20. (c)

E(Z)

= E(4X - 3Y) = 4E (X) - 3E(Y) = 4 (11.95) - 3 (18.48) = - 7.64

Var(Z)

= Var (4X - 3Y) = (4)2 (6.3474) + (3)2 (12.9695) - 2(4)(3)(8.894) = 4.8279

[Note: The expected value of X and Y, E(X), E(Y) are denoted by X and Y ]. 21. (b) Wildcat Probability P(oil) =0.10 P(gas) =0.20 P(dry) = 0.70

i.

Profit/(Loss) Rs.77,00,000 Rs.10,00,000 (Rs.12,00,000)

Drilling Probability P(oil) = 0.20 P(gas) = 0.05 P(dry) = 0.75

Deeper Profit/(Loss) Rs.2,40,00,000 Rs.15,00,000 (Rs.10,00,000)

Refurbishing Probability P(improve) = 0.60 P(no change) = 0.30 P(worse) = 0.10

Profit/(Loss) Rs.4,00,000 Rs.4,00,000 (Rs.8,00,000)

Expected profit for the ‘wildcat’ decision =

(0.1) (77,00,000) + (0.2) (10,00,000) + (0.7) (-12,00,000)

= 7,70,000 + 2,00,000 - 8,40,000 = Rs.1,30,000 ii.

Expected profit for the ‘drilling deeper’ decision = (0.2) (2,40,00,000) + (0.05) (15,00,000) + (0.75) (-10,00,000) = 4,80,000 + 75,000 - 7,50,000 = 41,25,000

iii.

Expected profit for the refurbishing decision (not considering municipal bonds investment) = (0.6)(4,00,000)+ (0.3) (-4,00,000) + (0.10) (-8,00,000) = 40,000. Total expected profit for the ‘wildcat’ decision = Rs.1,30,000 Total expected profit for ‘drilling deeper’ decision = 41,25,000 + 10,000 = Rs.41,35,000 Total expected profit for refurbishing decision = 40,000 + 40,000 = Rs.80,000

Total expected profit if Rs.12,00,000 is invested in bonds = Rs.60,000 Therefore, the alternative ‘drilling deeper’ gives the maximum expected annual profit. 22. (b) 1 1 1 6 6 + = 3 P(1 or 2) =

249

1 P(3 or 4) = 6 + 1 P(5 or 6) = 6 +

1 6 = 1 6 =

1 3 1 3

1 Therefore, the probability that any of the three alternatives is marked in a question is 3 . 1 The probability of marking a correct answer is 3 . 1 2 Therefore, p = 3 , q = 3

⎛ 8 ⎞ ⎛ 1 ⎞4 ⎛ 2 ⎞4 8! 16 70 x16 P(n = 4) = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ . = = 0.1707. 4 3 3 4!4! 81x 81 81x 81 ⎝ ⎠⎝ ⎠ ⎝ ⎠ =

23. (b) Here the probability of selecting a defective can be called a success. Therefore, p = 0.1, q = 0.9 Here, n = 5 ⎛5⎞ 5! ⎜⎜ ⎟⎟ 2 2! 3! 2 3 x (0.1)2 (0.9)3 = 0.0729. P(2 successes) = ⎝ ⎠ (0.1) (0.9) = 24. (b) Conditional Profit Table Requirement of annual Alternatives regarding availability of mechanic hours mechanic hours 10,000 12,000 14,000 16,000 hrs. hrs. hrs. hrs. 10,000 hrs. 66,400 47,680 28,960 10,240 12,000 hrs. 66,400 79,680 60,960 42,240 14,000 hrs. 66,400 79,680 92,960 74,240 16,000 hrs. 66,400 79,680 92,960 1,06,240 Notes to the table:

1.

The hourly profit is Rs.(16.00 - 9.00) = Rs.7.00

2.

It has been assumed that every mechanic gets a 2 week paid vacation i.e., a sum of Rs.720 [= Rs.9.00/hour x 40 hours per week x 2 weeks] as leave wages. The leave wages have been deducted from the overall profits.

3.

The number of mechanics to be employed have been calculated as follows: Annual requirement (in hours) 10,000

Number of mechanics 10,000 =5 2,000

12,000

12,000 =6 2,000

14,000

14,000 =7 2,000

16,000

16,000 =8 2,000

Expected Profit Table

Availability of

Expected Profit

250

Mechanic hours 10,000 12,000 14,000 16,000

[66,400 x 0.2 + 66,400 x 0.3 + 66,400 x 0.4 + 66,400 x 0.1] = 66,400 [47,680 x 0.2 + 79,680 x 0.3 + 79,680 x 0.4 + 79,680 x 0.1] = 73,280 [28,960 x 0.2 + 60,960 x 0.3 + 92,960 x 0.4 + 92,960 x 0.1] = 70,560 [10,240 x 0.2 + 42,240 x 0.3 + 74,240 x 0.4 + 1,06,240 x 0.1] = 55,040

As the availability of 12,000 mechanic hours results in maximum expected profits, the optimum course of action would be to employ 6 mechanics. 25. (d) Conditional Profit Table under Certainty Requirement of Availability of mechanic hours mechanic hours 10,000 hrs. 12,000 hrs. 10,000 hrs. 66,400 12,000 hrs. 79,680 14,000 hrs. 16,000 hrs. -

14,000 hrs. 16,000 hrs. 92,960 1,06,240

Expected profit under certainty = Rs. [66,400 x 0.2 + 79,680 x 0.3 + 92,960 x 0.4 + 1, 06,240 x 0.1] = Rs.84, 992 Cost of perfect information = Rs.[84,992 - 73,280] = Rs.11,712.

26. (a) Minimum probability that will justify stocking an additional rim =

7.30 0.80 + 7.3

= 0.90

Let Q be the quantity of rims that need to be stocked at which the minimum required probability is obtained. Given that the pattern of sales follows normal distribution with mean of 120 and standard deviation of 28, the position of Q in the normal curve will be as follows:

The shaded area of the curve denotes the required probability. Q lies to the left of the mean covering an area of 0.40 between the point and the mean. From the tables, we find that a point with an area of 0.3997 between the point and the mean is 1.28 standard deviations from the mean. Therefore, the value of Q will be = Mean - 1.28 standard deviations = 120 - 1.28 x 28 = 84 rims. 27. (a)

(1) (2) (3) (4) (5) (6)

Production Level (in lbs) Expected Revenue (Rs.) Fixed Cost (Rs.) Variable Cost (Rs.) Total Cost (Rs.) = (3) + (4) Profit (Rs.) = (2) - (5)

45,000 50,000 67,500 75,000 30,000 30,000 22,500 25,000 52,500 55,000 15,000 20,000

Pay-off table Production Expected Sales 45,000 50,000 Alternative (in lbs) Probability 0.40 0.35 45,000 15,000 15,000 50,000 12,500 20,000 55,000 10,000 17,500 Total cost of production to meet average sales = Rs.55,000.

55,000 82,500 30,000 27,500 57,500 25,000 55,000 0.25 15,000 20,000 25,000

Expected Profit 15,000 17,000 16,375

251

From the pay-off table we find that Keystone should produce 50,000. 28. (d) Expected value of perfect information = 15,000 (0.4) + 20,000 (0.35) + 25,000 (0.25) - 17,000 = Rs.2,250. 29. (e) Action: Anjali can hire 8, 9, 10 or 11 MBAs Event: Number of jobs awarded An incomplete job results in a loss of Rs.10,000 Event Action

b.

8 9 10 11 Probability Expected profit from hiring i.

Conditional Profit Table 24 27 30 2,40,000 2,10,000 1,80,000 1,80,000 2,70,000 2,40,000 1,20,000 2,10,000 3,00,000 60,000 1,50,000 2,40,000 0.3 0.2 0.4

33 1,50,000 2,10,000 2,70,000 3,30,000 0.1

8 MBAs 2,40,000 x 0.3 + 2,10,000 x 0.2 + 1,80,000 x 0.4 + 1,50,000 x 0.1 = 2,01,000

ii.

9 MBAs 1,80,000 x 0.3 + 2,70,000 x 0.2 + 2,40,000 x 0.4 + 2,10,000 x 0.1 = 2,25,000

iii.

10 MBAs 1,20,000 x 0.3 + 2,10,000 x 0.2 + 3,00,000 x 0.4 + 2,70,000 x 0.1 = 2,25,000

iv.

11 MBAs 60,000 x 0.3 + 1,50,000 x 0.2 + 2,40,000 x 0.4 + 3,30,000 x 0.1

= 1,77,000

Maximum profit is attained by hiring 9 or 10 MBAs. 30. (a) Expected profit under certainty

= 2,40,000 x 0.3 + 2,70,000 x 0.2 + 3,00,000 x 0.4 + 3,30,000 x 0.1 = 2,79,000 EVPI = 2,79,000 - 2,25,000 = 54,000. 31. (b) Let ‘C’ be the height in inches of the doorway required to be erected. Then, we need to find C such that P (X ≤ C) = 0.95, where X is the random variable denoting the height of a person.

Since X ~ N (60, 5) P (X ≤ C) = 0.95 ⇒ P

(Z <

C − 60 ) = 0.95 5

The value of the standard normal variable with 0.95 probabilities is approximately 1.64 C − 60 5 = 1.64 or C = 68.2. Hence, 32. (a) Net sales of ABC = X1 ~ N (3.7, 0.4) Net sales of XYZ = X2 ~ N (4.0, 0.8) P [ABC’s sales > XYZ’s sales] = P [X1 > X2] = P [X1 - X2 > 0]

⎡ ⎤ (0.3) ⎡ (X − X 2 ) − Mean (X1 − X 2 ) 0 − Mean (X1 − X 2 ) ⎤ Z > ⎢ ⎥ P⎢ 1 > ⎥ 0.16 + 0.64 ⎥⎦ ⎢ SD (X1 − X 2 ) SD (X1 − X 2 ) ⎦ = ⎣ = P⎣ ⎡ 0.3 ⎤ ⎢Z > ⎥ variance ((x1 ) + (x 2 )) 0.8 ⎦ =P ⎣ (Since SD (X1 - X2) = = P [Z > 0.3354] = 0.5 - P [0 < Z < 0.3354] = 0.5 - 0.1331 = 0.3669.

252

33. (b) If XYZ’s sales are Rs.4.5 crore P [ABC’s < XYZ’s sales] = P [X1 < 4.5] ⎡ X − Mean (X1 ) 4.5 − 3.7 ⎤ P⎢ 1 < ⎥ SD (X1 ) 0.4 ⎦ = ⎣ 0.8 ⎤ ⎡ ⎢ Z < 0.4 ⎥ ⎦ = P [ Z < 2] = 0.5 - P [0 > Z > 2] = 0.5 + 0.4772 = 0.9772. =P ⎣

34. (a) Let X = Cost of a flat in Hyderabad

Now µ = 4,00,000; σ = 30,000 P(X ≥ 4,15,000) ⎛ 4,15,000 − 4,00,000 ⎞ P ⎜⎜ Z ≥ ⎟⎟ 30,000 ⎠ = P (Z ≥ 0.5) = 0.3085. = ⎝

35. (e) Let X and Y denote the (Random Variables) returns on securities X and Y respectively. E(X) = E(P1) = 20% V(X) = V(P1) = 16% E(Y) = E(P3) = 40% V(Y) = V(P3) = 0.25%

Expected returns on P4 E(P4) = E(0.25X + 0.75Y) = 35% V(P4) = (0.25)2V(X) + (0.75)2V(Y) + (2)(0.25) ⎛1⎞ ⎜ ⎟ ρ XY ρ 2 = 1 + 0.140625 + 0.75 XY (0.75)(4) ⎝ ⎠

V(P4) = 1.140625 + 0.75

ρ XY

V(P2) = V(0.5X + 0.5Y) = 0.25V(X) + 0.25V(Y) + (2) (0.5) (0.5) ⎛1⎞ ⎜ ⎟ ρ XY 2 (4) ⎝ ⎠

25

= 4 + 0.0625 +

ρ XY ρ XY

= 20.9375

Therefore, V(P4) = 1.140625 + (0.75) (20.9375) = 16.84375(%)2 and E(P4)

= 35%.

36. (b) Selling price per shirt = Rs.32.50

Cost price per patch

= Rs.7.50

Cost price per shirt

= Rs.15.00

Therefore, cost price per shirt along with color transfer = 15.00 + 7.50 = Rs.22.50.

253

It is known that unsold white cotton T-shirts will be purchased by a local merchant. Since salvage value is not given, it is assumed that they will be purchased back at cost price itself. Also, for patches there will be no market after the celebration. Mean for the shirts Standard deviation MP = 32.50 - 22.50

= 200 = 34 = 10.00

ML = Cost price of the patch = 7.50 Therefore P* =

ML MP + ML

=

7.50 = 0.428 10 + 7.50

The area from the mean to the point Q is 0.071. This area is located between the mean and a point 0.18 standard deviation to the right of the mean. It is given that 1 standard deviation is 34 units. ~ So 0.18 times 34 = 6.12 − 6 Since point Q is 6 units to the right of the mean it must be 200 + 6 = 206 units. The committee must buy 206 patches. 37. (b)

Vol. of business (Rs. cr.) No. of persons required Expected profit from business under various volumes (Rs. lakh)

5 8 10 15 20 32 40 60 50 80 100 150

Therefore, the company can hire either 20, 32, 40 or 60 people. The conditional profit table is prepared as follows: (Rs. lakh) Vol. of business (in crore) No. of people 5 hired 20 50 32 32** 40 20 60 -10 Probability 0.5

8

10

15

35* 80 68 38 0.35

25.00 70.00 100.00 70.00 0.10

0.00 45.00 75.00 150.00 0.05

* Profit when volume of business is Rs.8 cr. but only 20 people are hired = Profit on Rs.5 cr. business less loss on Rs.3 cr. business refused = 5 x 10 - 3 x 5 = Rs.35 lakh. ** Profit when volume of business is Rs.5 cr. but 32 people are hired = Profit on Rs.5 cr. business less payment towards additional staff = 5 x 10 - 12 x 1.5 = Rs.32 lakh. Now expected net profit from hiring:

254

i.

20 professionals = 50 x 0.5 + 35 x 0.35 + 25 x 0.1 + 0 x 0.05 = 25 + 12.25 + 2.5 + 0 = Rs.39.75 lakh

ii.

32 professionals = 32 x 0.5 + 80 x 0.35 + 70 x 0.1 + 45 x 0.05 = 16 + 28 + 7 + 2.25 = Rs.53.25 lakh

iii.

40 professionals = 20 x 0.5 + 68 x 0.35 + 100 x 0.1 + 75 x 0.05 = 10 + 23.8 + 10 + 3.75 = Rs.47.55 lakh

iv.

60 professionals = -10 x 0.5 + 38 x 0.35 + 70 x 0.10 + 150 x 0.05 = -5 + 13.3 + 7 + 7.5 = Rs.22.8 lakh

The company’s profits are maximum when the strength of professional staff is 32. Therefore, it should hire 32 people. 38. (a) The expected value of perfect information

=

50 x 0.5 + 80 x 0.35 + 100 x 0.1 + 150 x 0.05 - 53.25

=

25 + 28 + 10 + 7.5 - 53.25 = Rs.17.25 lakh.

39. (b) Assume that the returns on various accounts are normally distributed with a mean µ = 22% and σ. As all returns lie between 20% - 24%, those will be the outer limits (3σ limits) of the normal distribution.

∴φ0 µ + 3σ = ∴φ0 22 + 3σ = σ = 0.666

24 24

Let p1 indicate the probability of the return being between 21% to 23%. Let p2 indicate the probability of striking a deal. Given that the deal is struck the probability of returns falling between 21% and 23% (p1) is shown in the shaded region.

x −µ We know Z = σ

⇒

23− 22 Z = 0.666

The shaded region in normal curve = 0.4332 [from Normal Table] (one side only) Therefore, the probability = 0.4332 x 2 = 0.8664 The probability of striking a deal with the customer p2 = 0.20 Therefore, the probability of striking a deal and the returns falling between 21% and 23% =

p1 x p2 = 0.8664 x 0.2

255

=

0.1733 = 17.33%.

40. (a)

Marginal profit of product A = 50 - 40 = Rs.10 Marginal loss of product A = Rs.40 The probability that will justify the production of an additional unit can be calculated as follows: P =

ML 40 = = 0.80. MP + ML 40 + 10

41. (d) We are required to find the sample size so that the mean annual earnings of the CFAs should fall within plus or minus Rs.2,000 at a confidence level of 95%.

The Z-value for 95% level of confidence is 1.96. ∴φ0 1.96

σx

=

Rs.2,000

2,000 1.96 = Rs.1,020.408 = ∴φ2 σ The standard error x is also given by the following formula σ σx = n σx

Rs.

5, 000

⇒

n 1,020.408 = 5, 000 1, 020.408 n =

⇒ ∴ταβn = 24.01 or 25 ∴φ0 Τηε σαµπλε µαψ χονταιν 25 ΧΦΑσ ατ ρανδοµ.

42. (a)

µ

=

Rs.6,000

σ

=

Rs.3,000

We have to find out how many observations will fall in the following 3 categories: i.

Less than 5,000 (Probability P1)

ii.

Between 5,000 and 10,000 (Probability P2)

iii.

More than 10,000 category (Probability P3)

256

i.

Z1 =

X1 − µ 5, 000 − 6, 000 = = − 0.333 σ 3, 000

∴φ0 Ρεθυιρεδ προπορτιον Π1 = 0.5 − 0.1293 (φροµ νορµαλ ταβλεσ) ∴φ0 Π1 = 0.3707 X2 − µ

ιι.

iii.

Ζ2

=

σ

10, 000 − 6, 000 4, 000 = = 1.33 3, 000 3, 000 =

P2

= 0.5 + 0.4082 [From normal tables] - P1 = 0.5375

P3

= 1 - P1 - P2 = 0.0918

∴β Expected frequencies will be 37, 54, 9.

2 27 43. (e) The probability of projects giving abnormal returns = =p 2 25 27 27 Hence, the probability of projects not giving abnormal returns = 1 = =q

If n = number of projects = 6 The probability function f(x) of ‘x’ success in ‘n’ trials for the binomial distribution is x

⎛ 2 ⎞ ⎛ 25 ⎞ ⎜ ⎟ ⎜ ⎟ 27 27 f(x) = nCx px qn-x = 6Cx ⎝ ⎠ ⎝ ⎠

n −x

The probability of 2 projects succeeding out of 6 projects: 6

f(2) =

2

4

2 4 ⎛ 2 ⎞ ⎛ 25 ⎞ ⎛ 2 ⎞ ⎛ 25 ⎞ C2 ⎜ ⎟ ⎜ ⎟ 15 x ⎜ ⎟ ⎜ ⎟ = 0.06. ⎝ 27 ⎠ ⎝ 27 ⎠ = ⎝ 27 ⎠ ⎝ 27 ⎠

44. (d) The probability of none of the projects succeeding: 0

⎛ 2 ⎞ ⎛ 25 ⎞ C0 ⎜ ⎟ ⎜ ⎟ ⎝ 27 ⎠ ⎝ 27 ⎠ f(0) = 6

6−0

6 ⎛ 25 ⎞ ⎜ ⎟ = 0.63. = 1 x 1 x ⎝ 27 ⎠

45. (e)

H0: µ = 21.5% H1: µ < 21.5% Given, σ = 9.72% For the given sample, n = 4 and x = 9.15% Standard error of mean,

σx =

σ n

=

9.72 4

= 4.86

From the standard normal distribution table, at 5% significance level, Z < - 1.64 Critical region: Z < -1.64

257

For the given sample, Z =

X − µ 9.15 − 21.5 = = − 2.54 < − 1.64 σx 4.86

Since the Z-value for the observed sample falls in the critical region the null hypothesis (H0) can be rejected and it can be concluded that the long-term average returns from stock markets have declined in recent years.

46. (a) The problem can be solved by using the binomial distribution. According to the binomial distribution formula: Probability of ‘r’ successes in ‘n’ trials, P(r) = nCr pr qn-r Where,

p q n r

= = = =

a.

1 = 0.25 Probability of success (i.e. probability of a customer to buy something) = 4 Probability of failure (i.e. probability of a customer not buying anything)= 1 - p = 0.75 Number of customers visiting the site = 10 Number of successes desired (in this problem success means a customer buys something)

Probability that at least 6 customers will buy something = P(r ≥ 6) =

P(r = 6) + P(r = 7) + P(r = 8) + P(r = 9) + P(r = 10)

=

10

=

0.01622 + 0.00309 + 0.00039 + 0.00003 + 0.000001 = 0.019731 = 1.97% (approx).

C6 (0.25)6 (0.75)4 + 10C7 (0.25)7 (0.75)3 + 10C8 (0.25)8 (0.75)2 + 10C9 (0.25)9 (0.75)1 + C10 (0.25)10 (0.75)0

10

47. (e) r=0

∴ταβProbability that no customer will buy anything = nCr pr qn-r = 10C0 (0.25)0 (0.75)10-0 = 1 x 1 x (0.75)10 = 0.0563 i.e. 5.63%. 48. (c) Probability of getting a call for buying or selling (p) = 80% = 0.80. Probability of not getting a call for buying or selling (q) = 1 - 0.80 = 0.20. n = 10 10! 5!5!

P (r = 5) = nCr pr qn- r = (0.80)5 10-5 (0.20) = 0.0264 = 2.64%. 49. (c)

258

P(r > 7) = P(r = 8) + P(r = 9) + P(r = 10) P(r = 8) =

10

C8 (0.80)8 (0.20)10-8

P(r = 9) =

10

P(r = 10) =

10

9

C9 (0.80) (0.20) 10

= 0.302

10-9

C10 (0.80) (0.20) Total

= 0.268

10-10

= 0.107 = 0.677

∴φ0 Π (ρ > 7) = 0.677 = 67.7%. 50. (χ)

Ρεγρετ Ταβλεσ (Ρυπεεσ) Πεσσιµιστιχ 5,000 − 5,000 5,000 − (−2,000) 5,000 − (−5,000)

Πορτφολιο Α Β Χ

Οπτιµιστιχ 65,000 − 50,000 65,000 − 40,000 65,000 − 65,000

Νορµαλ 35,000 − 25,000 35,000 − 35,000 35,000 − 30,000

Πορτφολιο

Οπτιµιστιχ

Α Β Χ

15,000 25,000 0

Νορµαλ Πεσσιµιστι χ 10,000 0 0 7,000 5,000 10,000

(Ρυπεεσ) Μαξιµυµ Ρε γρετ 15,000 25,000 10,000

Αχχορδινγ το τηε ρεγρετ χριτεριον τηε αλτερνατιϖε ωηιχη γιϖεσ τηε µινιµυµ οφ µαξιµυµ ρε γρετσ σηουλδ βε σελεχτεδ. Φροµ τηε αβοϖε ταβλε, ιτ χαν βε σεεν τηατ τηε πορτφολιο Χ γιϖε σ τηε µινιµυµ αµονγστ τηε µαξιµυµ ρεγρετσ. (Ρσ.10,000 ισ τηε µινιµυµ αµονγστ τηε µαξιµυµ ρεγρετσ). Σο τηε ινϖεστορ ωιλλ σελεχτ πορτφολιο Χ ιφ ηε υσεσ τηε ρεγρετ χριτεριον.

51. (α) (Ρυπεεσ) Πορτφολιο Οπτιµιστιχ Νορµαλ Πεσσιµισ τιχ Α 50,000 25,000 5,000 Β 40,000 35,000 −2,000 Χ 65,000 30,000 −5,000

Μινιµυµ Προφ ιτ 5,000 −2,000 −5,000

Αχχορδινγ το τηε µαξιµιν χριτεριον τηε αλτερνατιϖε ωηιχη γιϖεσ τηε µαξιµυµ οφ µινιµυµ π ροφιτσ σηουλδ βε σελεχτεδ. Φροµ τηε αβοϖε ταβλε, ιτ χαν βε σεεν τηατ τηε πορτφολιο Α γιϖ εσ τηε µαξιµυµ οφ µινιµυµ προφιτσ (Ρσ.5,000 ισ τηε µαξιµυµ αµονγστ τηε µινιµυµ προφιτσ). Σο τηε ινϖεστορ ωιλλ σελεχτ πορτφολιο Α, ιφ ηε υσεσ τηε µαξιµιν χριτεριον. 52. (α) Γιϖεν :ν = 4 π = 0.45 θ = 0.55 Π(ρ ≥ 3) = P(r = 4) + P(r = 3) According to the binominal theorem, P (r = 4) = nCr pr qn-r =

4! 4!(4 − 4)!

(0.45)4 x 1

= 1 x (0.45 )4 = 0.0410

259

P(r = 3) = 4C3 (0.45)3 0.55 =

4! 3!(4 − 3)!

(0.45)3 0.55 = 0.2005

P(r ≥ 3) = 0.0410 + 0.2005 = 0.2415. 53. (e) P(r = 0) = 4C0 (0.45)0 0.554-0

=

4! 0!(4 − 0)!

(0.45)0 (0.55)4 = 0.0915

The probability that Johnson Foods will not secure any of the supply contracts is 9.15%. 54. (a) The binomial distribution should be used. According to the binomial distribution formula, probability of ‘r’ success in ‘n’ trials,

P(r) = nCr pr qn-r p = Probability of being defective = 5% = 0.05 q = Probability of not being defective = 1- p = 1 - 0.05 = 0.95 n = Number of trials = 10 r = Number of successes i.e. defective jobs in this case = 3 Probability of maximum 3 out of 10 jobs being defective is P(≤ 1) = P(0) + P(1) 10C0 (0.05)0 (0.95)10 + 10C1 (0.05)1 (0.95)9 = 0.5987 + 0.3151 = 0.9131. 55. (b) Marginal Profit (MP) = 8 - 6.50 = 1.50

Marginal Loss (ML) = 6.50 The minimum required probability of selling one additional packet in order to justify its stocking can be obtained as follows: Expected (MP) = Expected (ML) P* MP = (1 - P*) ML 6.5 ML + 6.5 1.5 = 0.8125. P* = MP+ML =

56. (c) The stock of packets can be increased as long as the probability of selling the additional packet is greater than or equal to the minimum required probability of selling an additional packet, which is this case is 0.8125. From above, it can be seen that the cumulative probability becomes less than the minimum required probability of selling an additional packet at a demand level of 203 packets and it continues to decrease at every demand level thereafter. So the optimum number of packets (i.e. which maximizes the expected profit in a day) is 202 packets. 3

2

⎛1⎞ ⎛1⎞ 5C3 ⎜ ⎟ ⎜ ⎟ = 0.3125. ⎝2⎠ ⎝2⎠ 57. (a) b/p(X=3) = 58. (c)

p = 0.02 q = 0.98 n = 6 r = 0 P(r) = nCr. qn-r. pr P(0) = 6C0. 0.986. 0.020

260

= 1.0.8858. 1 = 0.8858. 59. (a)

p = 0.55 q = 0.45 n = 12 “at least two” ⇒ r = 2 or 3 …… or 12. P( at least two ) = P(2) + P(3).…. + P(12) = 1- [ P(0) + P(1) ] P(0) =

12

C0 . 0.4512. 0.550

= 1.0.000 069. 1 = 0.000 069 P(1) =

12

C1 . 0.4511. 0.551

= 12.0.000153.0.55 = 0.001011 P(0) + P(1) = 0.001080 1 - 0.001 080 = 0.998920. 60. (a) The probability function f(x) of ‘x’ success in ‘n’ trials for the binomial distribution is x

⎛ 2 ⎞ ⎛ 25 ⎞ ⎜ ⎟ ⎜ ⎟ 27 27 f(x) = nCx px qn-x = 6Cx ⎝ ⎠ ⎝ ⎠ 10

f(7) = =

C0 ( 0.3)

0

n−x

( 0.7 )

10

1 x 1 X0.0282 = 0.0282.

61. (b) The probability function f(x) of ‘x’ success in ‘n’ trials for the binomial distribution is f(x) = px qn-x 8! 2!6! P(1) = 0.52.0.56 = 28 x 0.25 x (0.015625) = 0.1094.

n

Cx

62. (c) Marginal profit of product A = 120 - 90 = Rs.30

Marginal loss of product A = Rs.90 The probability that will justify the production of an additional unit can be calculated as follows: P =

ML 90 = = 0.75. MP + ML 30 + 90

63. (b) P (168
261

67. (a)

E( X ) = 0 x 0.16 + 0.5 x 0.4 + 1 x 0.33 + 1.5 x 0.1 + 2 x 0.01 = 0.7 E( X ) = 0 x 0.16 + 0.52 x0.4 + 12 x 0.33 +1.52 x 0.1+22 x 0.01 = 0.695 2

⎡ E(X) ⎤ ⎦ = 0.695 - 0.49 = 0.205. ? = Var ( X ) = E( X ) - ⎣ 20 − 24 ⎡ 28 − 24 ⎤ = 0.8 and = −0.8⎥ ⎢ 5 5 ⎦ of mean 68. (e) X lies within ± 0.8 times standard deviation ⎣ So the probability is 2 × 0.2881 = 0.5762= 57.6%. 2X

2

npq 60 x 0.7 x (1 − 0.7) 69. (e) Standard deviation of binomial distribution = = = 4.01. 70. (d) Expected return on the portfolio = wAkA + wBkB = (2/5)(15) + (3/5)(25) = 21%. 71. (a) The binomial distribution should be used. According to the binomial distribution formula, probability of ‘r’ success in ‘n’ trials, P(r) = nCr pr qn-r p = Probability of being defective = 5% = 0.05 q = Probability of not being defective = 1- p = 1 - 0.05 = 0.95 n = Number of trials = 10 r = Number of successes i.e., defective jobs in this case = 3 Probability of maximum 3 out of 10 jobs being defective is

P(≤ 3) = P(0) + P(1) + P(2) + P(3) 10 = C0 (0.05)0 (0.95)10 + 10C1 (0.05)1 (0.95)9 + 10C2 (0.05)2 (0.95)8 + 10C3 (0.05)3 (0.95)7 = 1 × 1 × 0.5987 + 10 × 0.05 × 0.6302 + 45 × 0.0025 × 0.6634 + 120 × 0.000125 × 0.6983 = 0.5987 + 0.3151 + 0.0746 + 0.0105 = 0.9989 i.e. 99.89%. 72. (a) The expected value or mean of the random variable X is given by E(X) = ΣXP(X) = (0)(0.05) + (1)(0.1) + (2)(0.15) + (3) (0.2) + (4)(0.25) + (5)(0.10) + (6)(0.15) = 3.40 2 The variance of X is given by V(X) = ΣP(X − X) X P(X) X−X (X − X) 2 0 0.05 -2.4 5.76 1 0.10 -1.4 1.96 2 0.15 -0.4 0.16 3 0.20 0.6 0.36 4 0.25 1.6 2.56 5 0.10 2.6 6.76 6 0.15 3.6 12.96 V(X) = 3.84.

73. (a) P(1 ≤ X ≤ = 5) = 74. (c) P(2 ≤ X) 75. (d) P(X < 4)

P(X − X) 2 0.288 0.196 0.024 0.072 0.640 0.676 1.944

P(1) + P(2) + P(3) + P(4) + P(5) 0.1 + 0.15 + 0.20 + 0.25 + 0.10 = 0.80.

= =

P(2) + P(3) + P(4) + P(5) + P(6) 0.15 + 0.20 + 0.25 + 0.10 + 0.15 = 0.85.

= =

P(0) + P(1) + P(2) + P(3) 0.05 + 0.1+ 0.15 + 0.20 = 0.50.

76. (b)

262

Prob. of a ‘6’ = p = 1/6 No. of tosses = n = 1200 1 ∴φ0 µ = np = 1200 x 6 = 200

σ=

npq

1200 x

=

1 5 x 6 6 = 12.91

To find P(190 ≤ x ≤ 210) (x = Binomial) = P(189.5 ≤ x ≤ 210.5) (x = Normal) 210.5 − 200 ⎞ ⎛ 189.5 − 200 ⎜ 12.91 ≤ Z ≤ 12.91 ⎟ ⎠ = P⎝ = P (-0.81 ≤ Z ≤ 0.81) = 2.ϕ (0.81) = 2 x 0.2910 = 0.582. 77. (a) E(x) = 0.1 x (15) + 0.6 x (13) + 0.3 x (7) = 11.4. 78. (c) σ 2 = Var(x) = 0.1× (15 -11.4) 2 + 0.6× (13 -11.4) 2 + 0.3× (7 -11.4) 2 = 8.64 σ = Var(x) = 8.64

= 2.94.

79. (c)

Let the probability of the girl child be taken as p. Then p = 0.5; q = 0.5; and n = 4

The probability of r successes is given by

⎛n⎞ ⎜⎜ ⎟⎟ r P(r) = ⎝ ⎠ pr qn - r

The probability that a family of four will consist of 2 boys and 2 girls is given by ⎛ 4⎞ ⎜⎜ ⎟⎟ 2 P(2) = ⎝ ⎠ (0.5)2 (0.5)2 = 0.375

Number of families expected to have two boys and two girls = (0.375) (800) = 300. 80. (a)

Given µ = 50,000 and σ = 10,000 P(x > 55,000) 55, 000 − 50, 000 ⎞ ⎛ ⎜Z> ⎟ 10, 000 ⎠ = ⎝ = P(Z > 0.5) = 0.5 - 0.1915 = 0.3085 = 30.85%. 81. (b) P(45,000 < x < 60,000)

=P

⎛ 45,000 − 50,000 60, 000 − 50, 000 ⎞
= P (- 0.5 < x < 1) = 0.1915 + 0.3413 = 0.5328 = 53.28%. 82. (c)

263

Given, m = 16 Using Normal approx. µ = 16 and σ = 16 = 4 To find (P (x > 23) (Poisson Variate) = P(x > 23.5) (Normal Variate) 23.5 −16 ⎞ ⎛ ⎜Z> ⎟ 4 ⎝ ⎠ =P = P(Z > 1.875) = 0.5 - 0.4696 = 0.0304. 83. (b) Maximax criterion: The maximax criterion is applied when the decision-maker is optimistic. When this criterion is applied the decision-maker selects that alternative which provides the maximum pay-off amongst the maximum pay-offs of all the alternatives. In this case, the maximum amongst the maximum pay-offs is provided by the alternative ‘Build’ (Rs.30 crore). So the alternative ‘Build’ will be selected by the decision-maker under the maximax criterion. Alternative Demand Low Moderate High Maximum of each alternative Expand 10 15 25 25 Build 5 18 30 30 Sub-contract 8 10 20 20 84. (a) Maximin criterion: The maximin criterion is applied when the decision-maker is pessimistic. When this criterion is applied the decision-maker selects that alternative which provides the maximum pay-off amongst the minimum pay-offs of all the alternatives. In this case, the maximum amongst the minimum pay-offs is provided by the alternative ‘Expand’ (Rs.10 crore). So the alternative ‘Expand’ will be selected by the decision-maker under the maximin criterion. Alternative Demand Low Moderate High Minimum of each alternative Expand 10 15 25 10 Build 5 18 30 5 Sub-contract 8 10 20 8 85. (a) Marginal Profit (MP) = 100 - 80 = 20

Marginal Loss (ML) = 80 The minimum required probability of selling one additional packet in order to justify its stocking can be obtained as follows: Expected (MP) = Expected (ML) P* MP = (1 - P*) ML ML 80 = P* = ML+MP 80 + 20 = 0.80. 86. (b) No. of Packets

100 101 102 103 104 105

Probability 0.06 0.1 0.15 0.3 0.2 0.1

Cumulative Probability 1 0.94 0.84 0.69 0.39 0.19

264

106 Total

0.09 1

0.09

The stock of packets can be increased as long as the probability of selling the additional packet is greater than or equal to the minimum required probability of selling an additional packet, which is this case is 0.80. From above, it can be seen that the cumulative probability becomes less than the minimum required probability of selling an additional packet at a demand level of 103 packets and it continues to decrease at every demand level thereafter. So the optimum number of packets (i.e., which maximizes the expected profit in a day) is 102 packets. 87. (e) Marginal Profit (MP) = 16 - 10 = Rs.6

Marginal Loss (ML) = Rs.10 Minimum required probability of selling one or more additional pieces to justify their preparation, ML 10 = P* = ML+MP 10 + 6 = 0.625.

88. (a) The number of pieces prepared must be such that the probability of selling three or more number of pieces (i.e. cumulative probability) is at least equal to the minimum required probability (p*) of 0.625.

From the above table it can be seen that For 153 pieces, cumulative probability = 0.70 > p* For 154 pieces, cumulative probability = 0.40 > p* Hence, the optimal number of pieces to be prepared is 153. 89. (c) The expected price of share = (0.50) (40) + (0.20) (25) + (0.30) (30) = Rs.34. 90. (d) The bionomial distribution formula has to be used.

n = 3, r = 2, p = 0.5, q = 0.5 ∴φ0 Π(2) = νΧρ πρ θν− ρ = 3Χ2 0.52 0.53−2 = 37.5%. 91.

(β) Ιν α Βερνουλλι εξπεριµεντ τηε προβαβιλιτψ οφ ρ συχχεσσεσ ιν ν τριαλσ ισ γι ϖεν βψ νΧρ πρ θν− ρ Γιϖεν: Προβαβιλιτψ οφ συχχεσσ, π = 0.20 Νο. οφ συχχεσσεσ, ρ = 1 Νυµβερ οφ τριαλσ ν = 7 ∴φ0 Προβαβιλιτψ οφ φαιλυρε, θ = 1 − π = 0.80 ∴φ0 Π(2) = 7Χ1 0.21 0.87−1 = 0.367.

92. (ε)

Τηε εξπεχτεδ νυµβερ οφ µοϖιεγοερσ = 500 × 0.15 + 550 × 0.20 + 600 × 0.30 + 650 × 0.35 = 592.5 ≈ 593 (approx).

93. (b) Given that p = 0.60 , q = 0.40, n = 4, n r n −r P(r) = Cr p q 4 3 4 −3 P (3) = C3 × (0.60) × (0.40) = 0.3456.

94. (d) Probability that the normal variable with mean 12 and standard deviation 5 will take a value in the interval 5.6 and 21.8 is the area under the standard normal curve between the standard normal values 5.6 − 12 21.8 − 12 5 5 and .

The area between z = −1.28 and z = 1.96 is 0.3997 + 0.4750 = 0.8747.

265

Therefore the required probability is 87. 47 %. 95. (a) The probability that a randomly selected person is pro NDA is 0.55. Then the probability that he/she does not support NDA is 1- 0.55 = 0.45

Therefore, we can assume that it follows a binomial distribution with probability of success p = 0.55 and probability of failure q = 0.45. The probability that atleast eight out of ten persons chosen are pro NDA is P(8) + P(9) + P(10) =

10

C8 × (0.55)8 × (0.45) 2 + 10 C9 × (0.55)9 × (0.45)1 + 10 C10 × (0.55)10 × (0.45)0

= 0.0763 + 0.0207 + 0.0025 = 0.0995. 96. (d) According to Hurwicz criterion, the decision is made based on the weighted profit, which is calculated as,

Weighted profit = α (Maximum profit for alternative) + (1- α) (Minimum profit for alternative) Hence, Weighted profit for Portfolio A = 0.7 × 30,000 + 0.3 × 2000 = Rs.21,600 Weighted profit for Portfolio B = 0.7 × 40,000 + 0.3 × (-15,000) = Rs.23,500 Weighted profit for Portfolio C = 0.7 × 50,000 + 0.3 × (-30,000) = Rs.26,000 Weighted profit for Portfolio D = 0.7 × 40,000 + 0.3 × 5,000 = Rs.29,500 Weighted profit for Portfolio E = 0.7 × 20,000 + 0.3 × (-5,000) = Rs.12,500 Therefore according to Hurwicz criterion, the analyst should invest in portfolio D. As portfolio D has the highest weighted profit. 97. (c) The marginal profit per hour = Rs.16 − Rs.9 = Rs.7

The marginal loss per hour = Rs.9 The conditional profit (in rupees) for different levels of demand is tabulated below: Available Demand (hours) Expected (hours) Profit (Rs.) 10,000 12,000 14,000 16,000 (0.2) (0.3) (0.4) (0.1) 10,000 70,000 70,000 70,000 70,000 70,000 12,000 52,000 84,000 84,000 84,000 77,600 14,000 34,000 66,000 98,000 98,000 75,600 16,000 16,000 48,000 80,000 112,000 60,800 The manager expects to make maximum profit of Rs.77,600 if he has 12,000 mechanic hours available. The number of hours a mechanic will work in a year = (52 weeks - 2 weeks) × 40 hours = 2,000 hours. Therefore, the number of mechanics required for 12,000 mechanic hours = 12,000 ÷ 2,000 = 6 mechanics. Expected profit under certainty = 70,000×0.2 + 84,000×0.3 + 98,000×0.4 + 112,000×0.1 = Rs.89,600 Therefore, Expected Value of Perfect Information (EVPI) = Rs 89,600 - Rs 77,600 = Rs.12,000. 98. (e)

E(Z) = E(2X + 5Y)

= 2E(X) + 5E(Y) =

2 ⎡⎣(10 × 0.20 ) + (15 × 0.40 ) + ( 20 × 0.30 ) + ( 25 × 0.10 ) ⎤⎦ + 5 ⎡⎣( 8 × 0.25 ) + (12 × 0.55 ) + (16 × 0.20 ) ⎤⎦

= 33 + 59 = 92 .

266

99. (e) n = 15

p = 0.40 q = 1- 0.40 = 0.60 P (r ≥ 4) = 1- P (r ≤ 3) ⎡ P ( 0 ) + P (1) + P ( 2 ) + P ( 3) ⎤⎦ = 1- ⎣

= =

1− ⎡

⎢⎣

15

0 15 15 1 14 15 2 13 15 3 12 C0 ( 0.40 ) . ( 0.60 ) + C1 ( 0.40 ) ( 0.60 ) + C 2 ( 0.40 ) ( 0.60 ) + C3 ( 0.40 ) ( 0.60 ) ⎤ ⎥⎦

1 − [ 0.0004702 + 0.004702 + 0.021942 + 0.063388]

= 0.9095.

1 100. (a) p = 4 1 3 = ∴φ0 θ = 1− π = 1 − 4 4

Variance = npq

∴φ0

( 3)

2

⎛ 1 ⎞⎛ 3 ⎞ 3 n ⎜ ⎟⎜ ⎟ = n ⎝ 4 ⎠ ⎝ 4 ⎠ or 3 = 16 or n = 16 8

16

∴φ0 Π(ρ = 8) = 101. (c)

x = 70

x = 82

⇒z=

⇒z=

8

⎛1⎞ ⎛3⎞ C8 ⎜ ⎟ ⎜ ⎟ = 0.0197. ⎝4⎠ ⎝4⎠

70 − 76 −6 −3 = = = −0.75 8 8 4

82 − 76 = 0.75 8

from the tables =

P ( −0.75 < x < 0 ) = 0.2734

P ( 0 < x < 0.75 ) = 0.2734

∴φ0

P ( −0.75 < x < 0.75 ) = 2 × 0.2734 = 0.5468

∴φ0 Τηε νυµβερ οφ στυδεντσ σχορινγ βετωεεν 70 ανδ 82, ισ = 0.5468 × 600 = 328.08 ≅ 328. Minimum required probability of selling an additional unit of the product which justifies ML = p* ML MP + stocking that unit =

102. (c)

Marginal Profit, MP = Rs.(210 - 120) = Rs.90 Marginal Loss, ML = Rs.(120 - 30)

∴φ0

p* =

= Rs.90

90 90 = = 0.50. 90 + 90 180

267

268

Statistical Inferences 103. (b) The sampling distribution of means of random samples of size 4 drawn from the given population σ 9 σx = = = 4.5 kg. n 4 will have a mean µ = 70 and standard deviation

If the mean weight of the four persons exceeds 75, the maximum load of 300 will be exceeded. Then, P( X > 75) is the required probability. Z= We have,

X − µ 75 − 70 = = 1.11 σx 4.5

P( X > 75) = P(Z > 1.11) = 0.5 - 0.3665= 0.1335. 104. (a) Here the person’s weight has to exceed 75 kg. So that, he and his luggage combined exceed the 300 kg mark.

Let the random variable X denote the weight of the person. Then,

Z=

X − µ 75 − 70 5 = = = 0.56 σ 9 9

P(X > 75) = P(Z > 0.56) = 0.5 - 0.2123 = 0.2877. 105. (b) Since the sample size is large, the sampling distribution of means can be said to follow a normal distribution. Mean of the sampling distribution of mean S = µ = Population mean. σ

Standard deviation of the sampling distribution of means =

n

=

15 100

=1.5

We have to find P( X > µ + 3 and X < µ - 3). We have Z=

µ + 3−µ =2 1.5

Z=

µ −3−µ =−2 1.5

P( X > µ + 3 and X < µ - 3) = P(Z > 2 and Z < - 2) = (0.5 - 0.4772) + (0.5 - 0.4772) = 0.0456. 106. (a) Mean of the sampling distribution of means = µ = 200 ml. σ

Standard deviation of the sampling distribution of means =

n

=

15

15 = 2.5 ml 36 6 =

We have to find P( X > 204)

=

X − µ 204 − 200 4 = = =1.6 σx 2.5 2.5

P( X > 204) = P (Z > 1.6) = 0.5 - 0.4452 = 0.0548. 107. (e) If

σX

is the standard deviation of the sampling distribution, then the limits within which the

269

sample mean should fall are given by µ ± Z 0.95 here. σ We also have Z x = 0.2

σX

where, Z depends on the required probability which is

When the required probability is 0.95, Z = 1.96. σX

Therefore, 1.96

= 0.2 or

σX

0.2 = 1.96

σ

σx =

n (σ is the standard deviation of the population)

We have σ = 1 1 σ 1.96 0.2 = = = 9.8or n = 96.04 −% 97. n= σ x 1.96 0.2

108. (b) The confidence intervals are given by Given X = 4.6 and σˆ = 0.49 σ

X

σˆ

=

=

n

0.49 6

=

0.49 2.4495

X±tσ X

= 0.2

The appropriate t value = 2.02 The confidence intervals are given by: =

4.6 ± (2.02)(0.2) = 4.196 to 5.004.

109. (d) The null and alternative hypothesis for part (a) are H0: µ = 8

H1: µ ≠ 8 Standard error of the mean is σx =

σ n

=

14 49

=

14 =2 7

For a two tailed test at 5% level of significance, the Z value is 1.96. µH 0 ± Z σ X

= 8 ± (1.96)(2) = 8 ± 3.92

The acceptance region falls between 4.08 and 11.92. 110. (a) The null and alternative hypothesis for part (b) are H0: µ = 8

H1: µ > 8 For this one tailed test at 0.05 level of significance the appropriate Z value is 1.65. µ+Z

σX

= 8 + 1.65(2) = 8 + 3.3 = 11.3.

Since the sample mean, X = 10.6 is within the acceptance region we accept the null hypothesis. 111. (c) The null and alternate hypothesis for the given problem are H0: µ = 4700

H1: µ > 4700

270

The standard error of the mean is

σX =

σ n

=

1460 100

=

1460 = 146 10

The appropriate Z value for this one tailed test at 1% level of significance is 2.33. µ + ZσX

= 4700 + (2.33) (146)

= 4700 + 340.18 = 5040.18. Since the sample mean falls in the acceptance region we do not reject the null hypothesis. The new process is not better than the old process at this level of significance. 112. (d) The given sample has a mean X = 39 and a standard deviation S = 1.5811 The null and alternative hypothesis are

H0: µ = 40 H1: µ ≠ 40 σˆ

Standard error of the sampling distribution of the mean = n

=

1.5811 5

= 0.7071

The appropriate t value for this two tailed test at 5% level of significance and four degrees of freedom is 2.776. µH 0 ± t σ X

40 ± (2.776)(0.7071) = 40 ± 1.9629. The acceptance region lies in between 38.0371 to 41.9629. Since the sample mean lies in the acceptance region we accept the null hypothesis. The manufacturer’s claim is justified. 113. (d) Here the null and alternative hypothesis are H0: µ = 40

H1: µ < 40 The appropriate t value for this one tailed test at 4 degrees and 5% level of significance is 2.132. µH 0 − t σ X

= 40 - (2.132)(0.7071) = 40 - 1.5075 = 38.49 Since sample mean X = 39 > 38.49, we accept H0. The number of matches is not significantly less than 40. σX

114. (a)

1

− X2

=

σ12 σ12 + n1 n 2

(2000) 2 (1500) 2 + 50 40 = 369.1206. = 115. (a) The null and alternative hypothesis for this problem are

H0: µ1 - µ2 = 0 H1: µ1 - µ2 ≠ 0. σ12 σ12 + n1 n 2

(2000) 2 (1500) 2 + 1 50 40 = 369.1206. = The Z value for this two tailed test at 1% level of significance is 2.58 σX

− X2

=

271

(µ1 - µ2) ± Z

σX

1

−X

2

= 0 ± (369.1206) 2.58 = ± 952.3311

Here, (X1 − X 2 ) = 1500 > 952.3311. Therefore, we reject H0 and conclude that there is significant difference in the quality of the two tyres. σˆ X

1 − X2

=

σˆ 12 σˆ 22 (90)2 (120)2 + = + n1 n 2 100 75

116. (c) = 16.5227. 117. (a) The null and alternative hypothesis for this problem are

H0: µ1 - µ2 = 0 H1: µ1 - µ2 ≠ 0. σˆ X

1 − X2

=

σˆ 12 σˆ 22 (90)2 (120)2 + = + n1 n 2 100 75

= 16.5227 The appropriate Z value for this two-tailed test at 5% level of significance is 1.96. (µ1 - µ2) ± Z

σˆ X

1− X2

= 0 ± (1.96)(16.5227) = ± 32.3845

X −X

Here 2 1 = 1230 - 1190 = 40 > 32.3845. Since the difference between the sample means is greater than 32.3845, we reject the null hypothesis and conclude that there is a significant difference in the lifetimes of the two brands of light bulbs. 118. (e) The appropriate Z Value for this two tailed test at 1% level of significance is 2.58. (µ1 - µ2) ± Z = ± 42.6286

σ X2 − X1

= 0 ± (2.58)(16.5227)

X 2 − X1 =

40 < 42.6286. Since the difference between the sample means is less than 42.6286 we accept the null hypothesis and conclude that there is no significant difference in the lifetimes of the two brands of bulbs. 119. (c) Let p denote the proportion of the equipment that conforms to specifications. The null and alternative hypothesis for this problem are H0: p = 0.95 H1: p < 0.95 σp =

pq (0.95) (0.05) = = 0.0154 n 200

The appropriate Z value for this single tailed test at 1% level of significance is 2.33. p H0 − Zσ p

= 0.95 - (2.33) (0.0154) = 0.9141 Sample proportion p = 0.91 < 0.9141 Therefore, we reject the hypothesis at 1% level of significance and conclude that the manufacturer’s claim is not justified. 120. (b) Let p1 and p2 be the proportions of defective bolts produced by machines A and B respectively. The null and alternative hypothesis are H0: p1 - p2 = 0

H1: p1 - p2 ≠ 0. pˆ

=

n1 p1 + n 2 p 2 n1 + n 2

272

p1 =

19 = 0.095 200

p2 =

5 = 0.05 100

=

pˆ

(200) (0.095) + (100) (0.05) 200 + 100

= 0.08

qˆ = 1 − qˆ = 1 − 0.08 = 0.92 pˆqˆ pˆqˆ + n1 n 2

pˆ p

−p

=

1 ⎞ ⎛ 1 + (0.08) (0.92) ⎜ ⎟ ⎝ 100 200 ⎠ = 0.0332

1

= 2

The appropriate Z value for this two-tailed test at 5% level of significance is 1.96. The upper and lower limits of the acceptance region are 0 ± (1.96) (0.0332) p −p 1 2

= ± 0.0651

= 0.095 - 0.05 = 0.045.

Since the difference in the sample proportions lies within the acceptance region we accept the null hypothesis and conclude that the two machines are not showing different qualities of performance. 121. (a) The confidence intervals are given by p ± Z σp

Where p is the population proportion and distribution of proportion.

σp

=

pq n is the standard error of the sampling

For a 99% confidence interval, Z = 2.58 Here, Z 0.01 = 2.58

pq n

⇒

σp

0.01 2.58 =

= 0.01 pq n 2

⎛ 2.58 ⎞ ⎜ ⎟ 0.01 ⎠ pq ⇒ n= ⎝ The maximum value of pq is attained when, p = 0.5, q = 0.5. 2

⎛ 2.58 ⎞ 2 ⎜ ⎟ (0.5) 0.01 ⎠ ⎝ Therefore, we have n = = 16641. 122. (a)

µH0 =

14,500 Hypothesized value of the population mean.

σ n x

2,100 Population standard deviation. 25 Sample size.

= =

= 13,000 Sample mean. At a 0.01 significance level, we have to find out whether the average life of the press is too less. H0: µ = 14,500 (Null hypothesis: The average life of the press is 14,500 hrs.) H1: µ < 14,500 (Alternative hypothesis: The average life is less than 14,500 hrs.)

273

α: 0.01 level of significance for testing the hypothesis. σx =

σ

=

2,100

= 420 n 25 Standard error: Hinton press wishes to know whether the average life of the press is less than 14,500 hours. This is a left-tailed test. Since the level of significance is 0.01, the acceptance region consists of 49 percent on the left side of the distributions plus the entire right side (50 percent) for a total area of 99 percent. By referring to the tables we can determine that the appropriate Z value for 99 percent of the area under curve is 2.33. The acceptance region’s lower limit is µH - 2.33 σ x = 14,500 - 2.33 (420) = 0

13,521.

The lower limit x L = 13,521 hours. We can see that the sample mean lies outside the acceptance region. Therefore, H0 is rejected. It can be concluded that the average life of the press is significantly less. 123. (c) H0: p = 0.05 (Null hypothesis: The proportion that buys extra-spicy ketchup has not increased.) H1: p > 0.05 (Alternative hypothesis: Proportion after 2 years is higher than 0.05.) α pH0 qH0 n p

= = = =

2% Level of significance 0.05 Population proportion that buys 0.95 Population proportion that does 6,000 Sample size

not buy

= 0.05583 Sample proportion. S.E of the proportion 0.05 x 0.95 = 0.00281 6, 000

σp =

It wants to know whether there is an increased interest. It is a right-tailed test. The appropriate Z value for 0.48 of the area under the curve is 2.05. The acceptance region’s upper limit is σ p + 2.05 p = 0.05 + 2.05 (0.00281) = 0.05578 The sample proportion 0.05583 lies outside the acceptance region. So H0 is rejected. The company can conclude that there is an increased interest in the extra-spicy flavor.

124. (d)

µH0 n x

= 70 Hypothesized value of the population mean = 22 Sample size = 83 Sample mean

σˆ = s = 12.5 Sample standard deviation

H0: µ

= 70 (Null hypothesis: The population mean is 70)

H1: µ > 70

(Alternative hypothesis: The mean is greater than 70)

α = 2.5% Level of significance σˆ

12.5

ˆ n = 22 = 2.665 S.E σ x = It has to be tested whether the mean is greater than 70. So, it is right-tailed test. The appropriate degrees of freedom is 21 that is 22 - 1. If we have the t-distribution for one-tailed test we need to determine the area located in only one tail. So to find the appropriate t value for a one-tailed test at a significance level of 0.025 with 21 degrees of freedom, we would look in the Tables under the 0.05

274

column opposite the 21 degrees of freedom which is 2.08. The upper limit of the acceptance region is ˆ µH + 2.08 σ x = 70 + 2.08 (2.665) = 75.54 0

The sample mean 83 lies outside the acceptance region. So, H0 is rejected. 125. (d) H0 = µ = 10 (Null hypothesis: The population mean is 10 pounds)

H1: µ ≠ 10 (Alternative hypothesis: The mean weight is not 10 pounds) α = 0.01 Level of significance n

= 18 Sample size x = 12.4 ← Sample mean

σˆ = S = 2.7 ← Sample standard deviation σˆ

S.E σˆ x

=

2.7

n =

18 = 0.6364

Since we are to find out whether the true mean weight is larger or smaller than the hypothesized weight, a two-tailed test is used. The appropriate t value for two-tailed test at a significance level of 0.01 with 17 degrees of freedom is 2.898. The limits of the acceptance region are: xL xu

=

µH0 - 2.898 σˆ x

=

10 - 2.898 (0.6364) = 8.156

=

µH0 - 2.898 σˆ x

= 10 + 2.898 (0.6364) = 11.84 The sample mean 12.4 does not lie within the acceptance region. So, H0 is rejected. There is significant evidence to doubt the validity of the claimed 10 pound value. 126. (a)

H0: µ1 = µ0 (Null hypothesis: There is no difference between the mean) H1: µ1 ≠ µ2 (Alternative hypothesis: A difference exists)

α = 1% x1 = 86; σˆ 1 = 6 ˆ n2 = 75; x 2 = 82; σ 2 = 9 Standard error of the difference between the two means

n1 = 60;

σˆ 12

σˆ 22

62 92 + n1 n 2 1 2 = 60 75 = 1.296. 127. (d) At the significance level of 1%, the acceptance region contains two equal areas of 0.495 each. From the tables, we can determine the appropriate Z value for 0.495 of the area under the curve to be 2.57. σˆ

X

−X

=

+

The hypothesized difference between the two population means is zero. The limits of the acceptance region are Upper limit

= 0 + 2.57(1.296) = 3.331

275

Lower limit

= 0 - 2.57 (1.296) = - 3.331

The difference between the two sample means 86 - 82 = 4 lies outside the acceptance region. So, H0 is rejected. They appear to come from different populations. 128. (a) p 1

q

1

n1 p q

= 0.52 Sample proportion of working mothers in section I = 0.48 Proportion of not working = 200 Sample size

2

= 0.4

Sample proportion of working mothers in section II

2

= 0.6

Sample proportion of not working

n2

= 150 Sample size

H0:p1 = p2

(Null hypothesis: There is no difference between the proportion of working mothers)

H1: p1 ≠ p2

(Alternative hypothesis: There is a difference between them)

α

Level of significance

pˆ

pˆ =

= 4%

=

n1 p1 + n 2 p 2 n1 + n 2

200 (0.52) + 150(0.4) 200 + 150

164 = 350 = 0.4686.

129. (e) pˆ = 0.4686 qˆ = 1 - pˆ = 0.5314 qˆ p − p 1 2=

= qˆ p − p 1 2

pˆ qˆ pˆ qˆ + n1 n2

0.0012 + 0.0017

= 0.0539.

130. (a) Since we want to know whether there is a difference between the two proportions, this is a twotailed test. The approximate Z value for 0.48 of the area under the curve is 2.05.

The limits of the acceptance region are Upper limit = 0 + 2.05 (0.0539) = 0.1105 Lower limit = 0 - 2.05 (0.0539) = - 0.1105 The difference between the proportion (0.52 - 0.4) = 0.12 lies outside the acceptance region. So H0 is rejected. There is a significant difference in the proportion of working mothers in the two areas of the city. 131. (c)

276

p

1

q

1

p q

= 0.68 Sample proportion of reduction of first system. = 0.32 Proportion of not reducing.

2

= 0.76 Sample proportion of reduction of second system.

2

= 0.24 Proportion of non-reduction.

n1 = 200; n2 = 250 H0: p1 = p 2 (Null hypothesis: There is no difference between the two systems) H1: p1 < p 2 (Alternative hypothesis: There is a difference) α = 2% Level of significance. S.E n1 p1 + n 2 p 2 pˆ =

=

n1 + n 2 200 (0.68) + 250(0.76) 200 + 250

pˆ = 0.7244.

132. (e) qˆ

= 0.2756 pˆ qˆ

σˆ p1 − p 2 =

n1

+

pˆ qˆ n2

0.00099 + 0.00079 = = 0.00178 = 0.04219. 133. (a) Since the management wants to know whether the expensive system is more effective than the inexpensive one, the appropriate test is a one-tailed test. The appropriate Z value for 0.49 under the curve is 2.33. The lower limit of the acceptance region is: (σˆ ) 0 - 2.33 p1 − p 2 = 0 - 2.33 (0.04219) = - 0.0983 The difference between the sample proportion (0.68 - 0.76) = -0.08 lies within the acceptance region. So, H0 is accepted. There is no significant difference in reduction of emission of pollutants between the two systems. And, the management can install the less expensive system.

134. (a) Participant Before After 1 38 45

Difference X (Difference)2 = (x2) +7 49

277

2 3 4 5 6 7 8 9 10 Total

11 34 25 17 38 12 27 32 29

24 41 39 30 44 30 39 40 41

+13 +7 +14 +13 +6 +18 +12 +8 +12 110

169 49 196 169 36 324 144 64 144 1344

110 ∑x x = n = 10 = +11 2

σˆ = S

∑ x2 n x − n −1 n −1

=

=

1344 10(11) 2 − 9 9 = 3.859.

135. (e)

H0: µ1 - µ2 = 8 (Null hypothesis: The difference in increase is 8.) H1: µ1 - µ2 > 8 (Alternative hypothesis: The difference is more than 8.) α = 2.5% Level of significance σˆ 3.859 ˆσ x = n = 10 = 1.22 S.E Since the club wants to prove that the increase is more than 8, one-tailed test is used. The appropriate t value at a 2.5% significance level with 9 degrees of freedom is the t value at 0.05 significance level with 9 degrees of freedom which is 2.262. The upper limit of the acceptance region is = 8 + 2.262 (1.22) = 10.76 Since the mean difference 11 lies outside the acceptance region, H0 is rejected. The club’s claim is supported. 136. (a) H0: µ H1: µ σ x N S.E σx

= < = = = =

40,000 (Null hypothesis: The population mile is 40,000) 40,000 (Alternative hypothesis: The mean mileage is less than 40,000) 7,600 miles 38,500 miles 64 sample size σ

7, 600

n = = x−µ σx

64 = 950 miles 38,500 − 40,000 950 = - 1.58 = From the tables, we see that the probability that Z is not less than 1.58 is 0.5 - 0.4429 = 0.0571. 137. (c) Test whether there is any evidence that broker X is more reliable, at 5% level of significance. pA: True proportion of bad deliveries from X pB: True proportion of bad deliveries from Y

Z

278

12 p nA = 250, A = 250 = 0.048 20 p nB = 300, B = 300 = 0.067 H0: pA = pB H1: pA < pB α = 5% n A p A + n B p B 250 x 0.048 + 300 x 0.067 32 nA + nB pˆ 250 + 300 = = = 550 = 0.058. 138. (a) pˆ = 0.058 qˆ = 1 - pˆ = 0.942 pˆ qˆ pˆqˆ + n1 n 2

σˆ p A − p B =

=

(0.058) (0.942) (0.058) (0.942) + 250 300

0.000218 + 0.000182

=

139. (d) Sample mean = Sample variance = s = i. H0: µ = 10 H1: µ > 10 ii.

= 0.02 i.e. p A − p B ~ N(0, 0.02).

10.875 4.125 2.031

x −µH

X = 10.875, The test statistic = t = distribution. 2.031 σˆ σˆ x = n = 8 = 0.718066

σˆ x

0

is a t8

10.875 −10 t observed = 0.718066 = 1.218549 = 1.22 iii. For a right tailed test at α = 5%, the critical point is 1.895.

Since the observed t value falls in the acceptance region, H0 is accepted. 140. (a) n = 1,000

Let x = number of persons having banking accounts = 480 + 320 = 800 800 p Hence, = sample proportion of persons having any banking a/c = 1000 = 0.8

Since n > 30 we can use the normal distribution. 95% confidence interval for p is given by p ± 1.96 σ p pq

=

p ± 1.96

n

= 0.8 ± 1.96

(0.8) (0.2) 1000 = 0.8 ± 1.96

⎛ 0.16 ⎞ ⎜ ⎟ ⎝ 1000 ⎠

= 0.8 ± 1.96 (0.01264) = 0.8 ± 0.02479 = (0.7752, 0.8248) Hence, the 95% confidence interval for P is (77.52%, 82.48%).

279

141. (b) σ n

= = =

x

Rs.30 lakh 100 (sample size) Rs.5.04 crore (sample mean)

H0: µ = Rs.500 lakh H1: µ ≠ Rs.500 lakh α = 5% level of significance σ σx =

30

30 100 n = = 10 = Rs.3 lakh

The acceptance region is µ ± 1.96 σ x . That is, 500 ± 1.96 x 3 = (494.12, 505.88) Since sample mean of Rs.5.04 crore falls in this region, H0 is accepted. 142. (a)

X 70 65 112 58 25 147 95 68

Y 80 85 110 64 32 159 100 70

d 10 20 -2 6 7 12 5 2 60

d2 100 400 4 36 49 144 25 4 762

Let d = Y - X, where 60 8 = 7.5 d = 1 (∑ d i 2 − nd 2 ) n −1 = S.D. of (di) =

1 (762 − 8) (7.5) 2 7 = 6.67 (approx.)

6.67

Thus, S.D. of ( d ) =

6.67 8 = 2.83 = 2.3569.

143. (a)

H0: The increase in market price after declaration of dividends is not significant. H1: The increase is significant. If X & Y represent the average market price “before” and “after” declaration of dividends; then H0: µ1= µ2 H1: µ1 < µ2 That is, H0: µ1 - µ2 = 0 H1: µ1 - µ2 < 0 S.D. of ( d ) =2.3569 7.5 Hence, t = 2.3569 = 3.182 is the observed t value (with 7 degrees of freedom). 144. (c)

σ = Rs.30 lakh = Rs.0.3 crore 280

n = 100 x = 0.1008 x 100 = Rs.10.08 crore H0 : µ = Rs.10 crore H1 : µ ≠ Rs.10 crore α : 0.05 level of significance σ σx =

0.3

n = 100 = Rs.0.03 crore.

145. (a) Here, we have to compare a sample mean µ (nine months average returns from stock markets) with a population mean µ0 (10 years weekly returns from the stock markets)

The Hypothesis can be formulated as follows: H0: µ = µ0 (There is no significance difference between µ and µ0) H1: µ ≠ µ0 (There is significance difference between µ and µ0) µ = 0.32 S = 0.45 µ0 = 0.4 σ x = 0.35 0.35 σx =

39 = 0.056

To find out the critical value of sample mean the Z-value can be found out from standard normal distribution curve tables for a corresponding probability of 0.45. Z = 1.645 (from tables) The rejection region will lie below. µ 0 - 1.645 σ x = 0.40 - 1.645 x 0.056 = 0.308

As the sample mean is more than the critical value we accept H0, i.e., there is no significant difference between the average weekly returns of last nine months and last 10 years. 146. (a)

σ = 2.7 n = 81 Standard error of mean, σ x 2.7 σ n =

=

81 = 0.30.

147. (a)

H0: µ = 300 H1: µ < 300 α = 0.05 Critical region: Z < -1.64, Acceptance region: Z ≥ -1.64 For the observed data, Z

=

x−µ σx

298.5 − 300 0.30 = = -5.00 < -1.64

Since, for the observed data Z < -1.64 we reject H0 and accept H1. < 300.

281

148. (e)

H0: µ = 4.0 cm H1: µ ≠ 4.0 cm 4 σ = 4mm = 10 cm = 0.4 cm, n = 25, N = 10,000 25 n Sampling fraction, N = 10,000 = 0.0025 < 0.05

So, finite population correction factor need not be used for calculating the standard error of mean. Since, the standard deviation of the population is known the normal distribution will be used. σ

Standard error of mean (σ x ) =

n =

0.4 25 = 0.08.

149. (d)

Level of significance (α) = 1% = 0.01 (0.005 of the area on each of the tails). So, the null hypothesis will be accepted if the standardized values of the sample mean falls within 0.5 - 0.005 = 0.495 of the area in each of the tails. From the table, it can be seen that the boundaries of the acceptance region are Z = ± 2.57. Critical region on the standard normal distribution is Z < -2.57 or Z > 2.57. For the observed sample,

Z=

x−µ σx

4.4 − 4.0 = 0.08 = 5 > 2.57

Hence, the null hypothesis is rejected and the alternative hypothesis is accepted. So, it can be inferred that the mean diameter of the 10000 bolts bought by the company is different from 4.0 cms. So, the lot should not be accepted. 150. (b) Given:

n x

= 49 = Rs.48,000

s

= Rs.4,200

N

= 561

n 49 The population is finite and sampling fraction N = 561 = 0.0873 > 0.05. So the standard error of the sampling distribution of mean has to be calculated by using finite population correction factor along with the standard formula for standard error of mean. Further the standard deviation of population is not known. So the standard deviation of sample is to be used as an estimate of the population standard deviation. σˆ ∴ σˆ x =

n 4,200

ˆ or σ x =

49

N−n N −1

where, σˆ = s = 4,200

561 − 49 561−1

= 573.71.

282

151. (b) The sample size is more than 30. So the central limit theorem allows us to use the normal distribution. A 95% confidence interval for the estimate is required. This will include 47.5% of the area on either side of the mean of the sampling distribution. From the normal distribution table it can be seen that 0.475 of the area under the normal curve is located between the mean and a point 1.96 standard errors from the mean.

This means (2 x 0.475) = 0.95 of the area under the normal curve is located between plus and minus 1.96 standard errors from the mean. So the confidence limits are: ˆ Upper confidence limit: x + 1.96 σ x = 48,000 + 1.96 (573.71) = Rs.49,124.47 ˆ Lower confidence limit: x - 1.96 σ x = 48,000 - 1.96 (573.71) = Rs.46,875.52

At a confidence level of 95%, the average commission paid to the sales persons last year falls between Rs.49,124.47 and Rs.46,875.52. 152. (a) At a 5% level of significance you are required to find out whether the mean daily wages paid to the semi-skilled workers are same in both the industries.

Let the following notations be used: µ1 =

Population mean of daily wages paid in Industry 1

µ2 =

Population mean of daily wages paid in Industry 2

n1 =

Number of workers sampled in Industry 1 = 121 (given)

n2 =

Number of workers sampled in Industry 2 = 100 (given)

S1 =

Sample standard deviation of daily wages in Industry 1 = Rs.4.4/day (given)

S2 =

Sample standard deviation of daily wages in Industry 2 = Rs.5.2/day (given)

H0: µ1 = µ 2 i.e. µ1 - µ 2 = 0 There is no difference in the daily wages paid to the semi-skilled workers in the two industries. H1: µ1≠ µ2 i.e. µ1 - µ2 ≠ 0 -

A difference exists.

Since the population standard deviations are not known the sample standard deviations will be used as the estimates for the population standard deviations. ˆ ˆ ∴φ2 σ1 = S1 = 4.40; σ 2 = S2 = 5.20

Estimated standard error of the difference between the two means,

σˆ x1 − x 2 =

=

σˆ + σˆ 2 x1

2 x2

=

σˆ 12 σˆ 2 + 2 n1 n2

(4.40) 2 (5.20) 2 + 121 100

= 0.656 (approx). 153. (e) The 5% significance level implies a confidence level of 95% which means that the acceptance region lies between plus and minus 1.96 standard errors from the mean of the sampling distribution of the difference between the sample means.

The standardized value of the difference between sample means

283

Z

(x1 − x 2 ) − (µ1 − µ 2 ) H 0 σˆ x1 − x 2

=

(64 − 70) − 0 ~ 0.656 = -9.146 − -9.15 =

The acceptance region is -1.96 ≤ Z ≤ +1.96. Thus, the standardized difference between the samples means falls outside the acceptance region. So we reject the null hypothesis that there is no difference between the mean daily wages paid to the semi-skilled workers in the two industries. ∴φ0 Τηε µεαν δαιλψ ωαγεσ παιδ το τηε σεµι−σκιλλεδ ωορκερσ ισ διφφερεντ ιν τηε τωο ινδυσ τριεσ. 154. (β) Ηερε, π = 23/150 ανδ Π0 = 1/6 Η0: (Π=1/6) ανδ Η1 : ( Π ? 1/6) p − P0 P0 Q0 n

Τεστ Σπαστιχσ ιν Ζ =

=

23/150 − 1/6 1 5 x /150 6 6

= - 0.438. 155. (d)

N = 100; x = 1570; S = 120; µ 0 = 1600 Ho: ( µ = 1600) Test Statistic is x − µ 0 1570 − 1600 = = − 2.5 120 / 100 . H : ( µ ≠ 1600) Z = S / n 1

156. (a) Given, µ0 = 16; σ = 0.5; n = 25 X = 16.35; X − µ0

Test statistic is Z =

σ/n

=

16.35 − 16 0.5/ 25

= 3.5.

157. (c)

Given, n1 = 100 ; x1 = 200 σ1 = σ 2 = 11 n 2 = 150 ; x 2 = 220 H 0 : (µ1 = µ 2 ) H1 : (µ1 ≠ µ 2 ) Z=

−20 11 .01 + .0067

σ12 n1

Test statistic is =

x1 − x 2

=

+

σ 22 n2

=

200 − 220 112 112 + 100 150

−20

−20 = − 14. 11 .0167 11 x0.13

284

σ

158. (c) Standard error of mean =

n = 5.

159. (c) n1 = 1000, p1 = 450/1000 = 0.45

n2 = 800, p2 = 400/800

= 0.50

H0: ( P1 = P2) ; H1 : (P1
Test statistic is Z = Where, p =

n1p1 + n 2 p 2 n1 +n 2

n = 0.472 0.45 − 0.5 and Z =

0.472 x 0.528{1/1000 + 1/ 800}

= −2.11.

160. (a) n = 200; x = 0.824; S = 0.042

S/ n =

0.042

S.E =

200

=

0.0042 2

= 0.0042

2 2 = 0.0021 x 1.414 = .0029694 = 0.00297

Hence, the 95% confidence limits for µ are: µ

= x ± 1.96 (S.E)

µ

= 0.824 ± 1.96 (.00297) = 0.81818 and 082982, So, the lower limit is 0.81818.

161. (d) Here, n= 600

p = 45/600 and P0 = 0.05 H0: (P=0.05) and H1 : ( P >0.05) Test Statistics in p − P0

Z =

P0 (1 − P0 )/n

=

0.075 − 0.05 0.05 x 0.95 / 600

= 2.81.

162. (a)

n = 1600 ; x = 99 σ = 15 H0 : ( µ = 100) H : ( µ ≠ 100) 1

x − µo

=

99 − 100

Test statistic is Z = α / n 15 / 1600 = - 2.67 Z≥ The C.R. at 5% level is 1.96. The calculated value falls in the C.R. Hence, reject Ho. Hence, we conclude that this is not a sample from the given population. 163. (a) Here, we can notice that the problem relates to two sets of samples. Hence, paired t-test is not applicable in this case. It should be treated as a test of equality of Means.] H 0 : (µ1 = µ 2 ) ; H1 : (µ1 ≠ µ 2 )

285

x1 − x 2 s

Test Statistic t =

1 1 + n1 n 2

= −7

Hence, accept H0 there is no effect of training. (Computed: x1 = 8.2, x 2 = 8.9, S1 = 1.25; S2 = 2.68). 164. (b) z = (x - µ)/σ

z = (72 - 69)/4 z = 0.75

Obviously, the answer is a minority. The tables give 0.7734 1 - 0.7734 = 0.2266. 165. (b)

X1 = 65 X2 = 67 Z1 = (65 - 69) / 4 = -1 Z2 = (67 - 69) / 4 = - 0.5

Left tail area = 1 - 0.8413 = 0.1587 Right tail area = 0.6915 Total tail area = 0.8502 1 - 0.8502

= 0.1498.

166. (a)

Given, µ = 5.75cm; σ = 0.75cm; N = 1000 (i)

To find N x P(5 < x < 6.25)

286

6.25 − 5.75 ⎞ ⎛ 5 − 5.75 1000 x P ⎜ <Ζ < 0.75 ⎟⎠ ⎝ 0.75 = = 1000 x P(-1 < Z < 0.67) = 1000 (0.3413 + 0.2486) = 1000 x 0.5899 = 589.9 = 590. 167. (b) To find Z, such that the shaded area (i.e., to the left of Z1) is 0.20

Area to the right of -Z1 = Area to the left of Z1 (Since Normal Curve is uniform) i.e., Area from 0 to -Z1 = 0.3 But given area from 0 to 0.84 = 0.3 ∴φ0 −Ζ1 = 0.84 ⎛ x − 5.75 ⎞ ⎜ 0.75 ⎟ ⎠ = 0.84 where, x = height ι.ε., − ⎝

i.e., -(x - 5.75) = 0.84 x 0.75 = 0.63 i.e., x = 5.12cm. 168. (d)

N = 400; µ = 45; σ = 15 To find N x P(30 ≤ x ≤ 60) 60 − 45 ⎞ ⎛ 30 − 45 400 × P ⎜ ≤ Z≤ 15 ⎟⎠ ⎝ 15 = =

400 × P ( −1 ≤ Z ≤ 1)

= 400 = 400 x 2 x 0.3413 = 273.

[Note: Area between 0 and 1 is 0.3413]. 169. (a) Since the sampling is done without replacement, so we will use population correction factor, the ⎛ σ ⎞ N−n ⎜ ⎟ N −1 standard error of mean σ x = ⎝ n ⎠ ⎛ 12.5 ⎞ 205 − 49 ⎜ ⎟ 205 − 1 = 1.56156 ≈ 1.56. = ⎝ 49 ⎠

170. (a)

Given : n1 = 12 ; n2 = 10 x1 1, 050 ; x 2 = 980

H0 : (µ1 = µ 2 ) H1 : (µ1 ≠ µ 2 ) s=

(n1 − 1) S12 + (n 2 − 1) S22 11x 682 + 9 x 742 = n1 + n 2 − 2 12 +10 − 2

= 70.76.

x1 − x 2

171. ( b ) Test statistic is t =

1 1 s + n1 n 2

=

with n1 + n2 - 2 d.f.

1, 050 − 980 70.76

1 1 + 12 10 = 2.31.

172. (a) n1 = 8, 000 ; x1 = 168 ; S1 = 6.8 n 2 = 1, 600 ; x 2 = 172; S2 = 6.3

287

H 0 : (µ1 = µ 2 ) H1 : (µ1 = µ 2 )

Test Statistic is Z x1 − x 2 S12 S22 + n1 n 2

=

=

168 − 172 6.82 6.22 + 8, 000 1, 600

−2 x 89.44 = 15.44 = 11.58.

173. (b)

Given : n1 = 50 ; x1 = 1980 ; S1 = 80 n 2 = 70 ; x 2 = 2010 ; S2 = 60

To test whether difference is significant at 10% level. [Note : At 10% level the C.R. for two-tailed test is [Z] ≥ 1.64] H0 : ( (µ1 = µ 2 ) (µ ≠ µ2 ) H1: ( 1 Z=

x1 − x 2 S12

+

S22

1980 − 2010

=

−30 −30 30 802 602 = + =− = − 2.24 + 13.4 50 70 128 + 51.4 179.4

n1 n 2 Test statistic is The calculated value falls in the Critical Region Hence, reject Ho. Hence, we conclude that there is significant difference between the mean life of bulbs from the two manufactures. 174. (a) Given, p = 0.30; P0 = 0.22; n = 100 H0 : (P = 0.22) H1 ; (P > 0.22) p − Po 0.30 − 0.22 +0.08 Z= = Po Qo / n 0.22 x 0.88 / 100 = 0.044 = 1.818 Test Statistic is The Critical Region at 1% level is Z > 2.33 the calculated value does not fall in the Critical Region. Hence, accept Ho. Hence, conclude that the claim of the manufacturer is justified. 175. (a) H : (µ = 56) ; H1 : (µ ≠ 56) 0

n = 16 ; x = 53 ; ∑ (x − x) 2 = 160 µ0 = 56 S=

∑(x − x) 2 160 = = 3.162 n 16 x − µ0

Test Statistic is t = s / n − 1

=

53 − 56 3.162 / 15

=

−3 x 3.83 = − 3.63. 3.162

176. (a) The Critical Region at 5% level for t with d.f. 15 is [t] ≥ 2.23 The calculated value falls in the Critical Region. Hence reject Ho. Hence, the sample cannot be regarded as drawn from the population.

288

95% Confidence limits are: x ± C.R. x S.E.

⎛ 3.062 ⎞ ⎛ 3.062 ⎞ ± (2.131) ⎜ ⎜ ⎟ ⎟ n −1 ⎝ 3.83 ⎠ = 53 ± (2.131) ⎝ 15 ⎠ = 53 Where, = 53 ± (2.131) (0.8) = 53 ±1.7048 = 51.30 and 54.70. 177. (a) S.E. =

S

To find − Z1 and Z1 such that P(-Z ≤ Z ≤ Z1) = 0.50 i.e., Area from 0 to Z1 = 0.25 (uniformity) From the table Area from 0 to 0.675 = 0.25

∴φ0 Ζ1 = 0.675 x − 45 ιε 15 = 0.675 ie x = 45 + (15 x 0.675) = 45 + 10.125 = 55 Z1 = -0.675

x − 45 ie 15 = -0.675 ie x = 45 - (15 x 0.675) = 45 - 10.025 = 35

∴β0The middle 50% of students must get between 35 and 55 marks. 178. (a) Given: µ = 3 and σ = 4

To find the value of X such that P(3 ≤ X ≤ y) = 0.4772 (X = normal variate)

i.e.

y −3 ⎞ ⎛ 3−3 P⎜ ≤Z≤ 4 ⎟⎠ ⎝ 4 = 0.4772

i.e

y −3 ⎞ ⎛ P⎜0 ≤ Z ≤ 4 ⎟⎠ ⎝ = 0.4772

We are also given P(Z ≤ 2) = 0.9772 i.e P(0 ≤ Z ≤ 2) = 0.9772 - 0.5 = 0.4772 y−3 Comparing (1) and (2), 4 = 2 ⇒ y = 11.

179. (d) Given, N = 120; µ = 11.35; σ = 3.03

It x denotes the wages To find 100 x P(9 < x < 17) 17 −11.35 ⎞ ⎛ 9 −11.35
= 100 (0.2823 + 0.4686) (Pl. refer tables) = 75.09%. 180. (b) Given µ = 5.02; σ = 0.05

To find the percentages of defective bolts. i.e., to find

100 × {1 − P(4.96 ≤ z ≤ 5.08)}

289

⎧ 5.08 − 5.02 ⎞ ⎫ ⎛ 4.96 − 5.02 100 × ⎨1 − P ⎜ ≤z≤ ⎬ 0.05 0.05 ⎟⎠ ⎭ ⎝ ⎩ =

=

100 × {1 − P(−1.2 ≤ z ≤ 1.2)}

= 100 x {1 - 2 x 0.3849} = 100 {1 - 0.7698} = 23.02% = 23% (Approx). 181. (a) Since the sampling is done without replacement, so we will use population correction factor, ⎛ σ ⎞ N−n ⎜ ⎟ N −1 The standard error of mean σ x = ⎝ n ⎠ ⎛ 12.5 ⎞ 205 − 49 ⎜ ⎟ 205 − 1 = ⎝ 49 ⎠ = 1.56156 ≈ 1.56.

182. (d) The F ratio is given as

F ratio =

Estimate of population variance based on variance among the sample means Estimate of the population variance based on variance within the samples

3.362 = 1.124 = 2.991. 183. (c) The z value in the normal table for 89.9 percent confidence level is 1.64. We want our estimate to be within 0.02, so,

Z

σp

= 0.02

And z = 1.64 Then 1.64

σp

= 0.02

Substituting the value of 1.64

pq = 0.02 n or,

σp

in the equation,

pq n = 0.0122

36000 As the university feels that the proportion of students favoring the new system is 60000 = 0.60. We can take this value as the estimate of the population parameter.

Therefore,

0.6 × 0.4 0.6 × 0.4 = 0.0122 2 n or, n = (0.0122) = 1612.4697

Therefore a sample size of 1613 would be appropriate to estimate the proportion of the students favoring the new system in an interval of ± 0.02 at a confidence level of 89.9%. 184. (a) The sampling proportion is large as n/N = 50/700 = 0.07 (This is > 0.05)

Therefore we will use finite population correction factor to find standard error of mean,

σx =

σ n

1.2 N−n N − 1 = 50

700 − 50 700 − 1

= 0.1636

Since the sample size is more than 30, we use normal distribution to find the interval estimate. The confidence interval for the estimate is given as [X − z σ x , X + z σ x ] Therefore the interval estimate is 4.5 + 1.64 x 0.1636 and 4.5 - 1.64 x 0.1636 The interval estimate is 4.5 ± 0.2683. 185. (b) We want to test the hypothesis

290

H0 : µ = 44.95 ( Null Hypothesis: Average retail price has not changed) H1: µ < 44.95 (Alternative Hypothesis: Average retail price has reduced after sales promotion) σ

5.75

The standard error of mean = n =

25 = 1.15 X−µ

The standard normal value of the mean retail price of the sample = σ / n 42.95 − 44.95 −2 1.15 = = 1.15 = −1.739

Looking at the table for the critical value corresponding the area under the normal curve as 2% in the left tail i.e. -2.05. We find that the sample statistic falls in the acceptance region therefore we accept null hypothesis and at significance level of 2 % we can conclude that the price of the tennis balls has not reduced significantly after the trade promotions. 186. (d) Anand Electronics wants to open a store if more than 15 percent of the students own a stereo system costing Rs.5,000 or more. They conduct a survey and wants to test whether they should open a store in Varanasi or not. H 0 : p = 0.15 (Null hypothesis: The proportion of students owning a stereo costing Rs.5,000 or more is 15 percent) H1 : p > 0.15 (Alternative hypothesis: The proportion of students owning a stereo costing Rs.5,000 or more is more than 15 percent)

Therefore we will conduct a right-tailed test of hypothesis. The sample size is, n = 300 The sample proportion owning a stereo worth Rs.5,000 and more is, 57 = 0.19 p = 300 and q = 1 − 0.19 = 0.81

The standard error of proportion

σp

=

pq n =

0.19 × 0.81 300 = 0.02265 p − p H0

The standardized value of the sample proportion is z =

σp

0.19 − 0.15 = 0.02265 = 1.766

At a significance level of 5 percent the critical value is 1.64. Since the standardized value of the sample proportion exceeds 1.64, it falls in the rejection region, we can conclude at a significance level of 5 percent that the number of students owning stereos worth Rs.5,000 and more is more than 15 percent. Therefore they should open a store at Varanasi. 187. (b) We want to test on the basis of two samples, whether it is reasonable to conclude at significance level of 10 percent that the SEBI proposal will cause a greater reduction in EPS for high technology firms than the producers of consumer goods.

Therefore the Hypothesis are, H0: µ1 = µ 2 [Null Hypothesis: The reduction in the EPS of the high technology companies, with mean µ1 and the consumer goods companies, with mean µ2 are equal] H1: µ1 > µ 2 [Alternative Hypothesis: The reduction in the EPS in the high technology companies, with mean µ1 is more than the reduction in consumer goods companies with mean µ2] Since we are testing whether the mean reduction in the EPS is more in the high technology companies than the consumer goods companies, we will conduct a right-tailed test.

291

The sample statistics are as follows: n1 = 41 , x1 = 13.8, s1 = 18.9 and n2 = 35 , x 2 = 9.1, s2 = 8.7 Since both the samples have more than 30 observations we will use normal distribution to test the hypothesis. Estimated Standard error of the difference between two means, σˆ x1 − x 2

=

s12 s 2 2 + n1 n 2

=

18.92 8.7 2 + 41 35 = 3.2977 ≈ 3.298

The standardized value of the difference between sample means, (x1 − x 2 ) − (µ1 − µ 2 ) H0

z=

σˆ x1 − x 2

(13.8 − 9.1) − 0 3.298 = = 1.425

At a significance level of 10 percent the critical value is 1.28. We see that the standardized value of the difference between the sample means falls in the rejection region as 1.425 > 1.28. We can conclude at a significance level of 10 percent that the reduction in the EPS in the high technology company is more than the reduction in the EPS of the consumer goods company. 188. (a) Let X denote the price in Rs. and Y denote the quantity (in thousands) of toys that can be sold for the given price X.

Then,

∑X = 143, ∑Y= 1095, X = 23.83, Y = 182.5, n = 6 ∑XY = 25490, ∑X2 = 3463, ∑Y2 = 206575. n ∑ XY − ∑ X ∑ Y

6 × 25490 − 143 × 1095

2 2 The regression coefficient, b = n ∑ X − (∑ X) =

6 × 3463 − 1432

= -11.08

The Y intercept a = Y − b X = 182.5 - (-11.08) × 23.83 = 446.51257 ≈ 446.513 ˆ = 446.513 - 11.08 X Therefore, the estimating equation is Y

Estimated number of toys that can be sold at X= Rs 30,

ˆ = 446.513 - (11.08 × 30) = 114.113 ≈ 114 toys (approximately) Y Now the standard error of estimate is Se= =

∑ Y 2 − a ∑ Y − b ∑ XY n−2

206575 − 446.513 × 1095 − (−11.08)(25490) 6−2

= 4.256 Se

The standard error of the regression coefficient, Sb =

2

4.256 2

∑ X − nX =

3463 − 6 × 23.832

= 0.5698. 189. (b) We first calculate the between column variance as follows: n x x n ( x - x )2 6 58 57.93 0.0294 5 57 57.93 4.3245 4 59 57.93 4.5796 Total 8.9335

292

The estimate of the population variance based on the variance among the sample means 2

σˆ =

∑ n j (x j − x) 2 k −1

8.9335 = 3 − 1 = 4.46675 ≈ 4.467

Now we calculate variance within the column as follows: n x ∑(x − x) 2 System A System B System C

6 5 4

58 57 59

52 154 106

s2 10.4 38.5 35.33 Total

⎛ n −1 ⎞ 2 ⎜ ⎟s ⎝ nT − k ⎠ 4.3333 12.8333 8.8325 25.9991

The estimate of the population variance based on the variance within the samples is ⎛ nj −1 ⎞ 2 σˆ 2 = ∑ ⎜⎜ ⎟⎟ × s j − n k ⎝ T ⎠ = 25.9991 ≈ 26.0

Now calculating the F ratio as F ratio =

Estimate of the population variance based on the variance among the sample means Estimate of the population variance based on the variance within the samples

4.467 = F ratio = 26 0.172

Now looking at the critical value in the F distribution for a 5 percent significance level. The degrees of freedom in numerator is, (number of samples -1) = 3 - 1 = 2 The degrees of freedom in denominator is, (total sample size - number of samples) = 15 - 3 = 12 The F value from the table is F (2, 12, 5 %) = 3.89 (Critical value) We find that the F statistic falls in the acceptance region; therefore we can conclude at 5 percent level of significance that there is no significant difference in the time taken by the three systems of cleaning the oil spills. Sample standard deviation n

190. (a) Estimated standard error of mean =

Sample standard deviation, x=

s=

∑x

2

n −1

−

nx 2 n −1

∑ x = 160 = 10 n

16

1840 16 (10 ) − =4 15 15 2

∴φ0 σ =

4

∴φ0 Εστιµατεδ στανδαρδ ερρορ οφ µεαν = 191. (c)

16

=

4 = 1. 4

X = 31

s = 6.84 n = 60, N = 540

293

n 60 = = 0.11 > 0.05 Sampling fraction, N 540

∴φ0 Εστιµατεδ στανδαρδ ερρορ οφ µεαν,

σˆ X =

s n

N−n N −1

6.84 540 − 60 6.84 480 × 539 = 0.833 = 60 540 − 1 = 60

Since the sample size is greater than 30 the normal distribution will be used. Z-values for the upper and lower confidence limit are ±1.96

∴φ0 Υππερ χονφιδενχε λιµιτ = Lower confidence limit = p=

192. (b)

20 = 0.10, 200

X + 1.96 σˆ X = 31 + (1.96 × 0.833) = 32.63 years

X − 1.96σˆ X = 31 − (1.96 × 0.833) = 29.37 years.

q = 1 − p = 0.90

Estimated standard error or proportion,

σˆ p =

pq 0.10 × 0.90 = n 200 = 0.0212

Z - Values for the upper and lower confidence limit are ±2.32

∴φ0 Υππερ χονφιδενχε λιµιτ = Lower confidence limit =

p + ( 2.32 ×σˆ p ) = 0.10 + ( 2.32 × 0.0212 ) = 0.1492

p − ( 2.32 × σˆ p ) = 0.10 − ( 2.32 × 0.0212 ) = 0.0508.

193. (e) H 0 : µ = 6.70 H1 : µ > 6.70 α = 0.05

Estimated standard error of mean,

σˆ X =

s n

=

2.4 200

= 0.1697

Since the sample size is large (greater then 30), the appropriate distribution is the normal distribution. z=

Standardized value of sample mean,

x −µ σˆ X

7.25 − 6.70 = 3.24 = 0.1697

Critical Z - value = 1.64

294

Since the standardized sample statistic exceeds the critical value, the sample statistic falls in the rejection region. Hence, the null hypothesis is rejected at a significance level of 5 percent and it can be concluded that the mean number of hours of television viewing per household in the city is more than 6.70 per day. 194. (e) H 0 : p = 0.42 H1 : p < 0.42 α = 0.02 σp =

0.42 × (1 − 0.42 ) p HO q HO = = 0.0285 n 300

Sample proportion,

p=

120 = 0.40 300 Z=

p − p Ho σp

Standardized value of sample proportion,

=

0.40 − 0.42 0.0285

= - 0.702

Since the sample size is large we shall use the normal distribution. Critical Z - value = - 2.05

Thus, we can see that the sample statistic falls in the acceptance region. Hence, we accept the null hypothesis at a significance level of 2 percent. It can be concluded that the proportion of loans made to the women entrepreneurs has not reduced in the past five years. 195. (d) H 0 : µ = 14.50 H1 : µ > 14.50 α = 0.05

Estimated standard error of mean

σˆ X =

s n

=

1.25 16

= 0.3125

Since the population standard deviation is not known and the sample size is less than 30, the tdistribution will be used. Degrees of freedom = n - 1 = 16 - 1 = 15 Since this is a right tailed test with α = 0.05, the rejection area is 0.05 under the right tail. Hence we should look under the 0.10 column in the t-distribution (0.10 area in both tails combined).

∴φ0 Χριτιχαλ τ−ϖαλυε = 1.753 X − µ H0

Στανδαρδιζεδ σαµπλε στατιστιχ, τ =

σˆ X

=

15.00 − 14.50 = 1.60 0.3125

295

We find that the sample statistic falls in the acceptance region. Hence we accept the null hypothesis. It is concluded that the mean price of the rice is not more than Rs.14.50 per kg.

Linear Regression

196. (d) Coefficient of correlation, rXY =

Cov(x,y) σx σy

2.1

=

2.3 x 4.1

= 0.684.

197. (c) If y is dependent on variable x then we can write that the estimated value of y, yˆ = 3x + 5.

When x = 5 the estimated value of y would be 3 × 5 + 5 = 20. 198. (b) The standard error of estimate for the simple regression relationship ˆ ∑ (Y − Y)

2

n−2

= 178.56/9 = 4.45.

199. (a) Correlation coefficient is the square root of coefficient of determination. In this case, 36.0 = ± 0.60. As the slope of the regression equation is negative so the correlation coefficient is - 0.60. 200. (a)

∑X = 118, ∑Y=141, n = 10 ∑XY=1677, ∑X2 = 1404, X = 11.8, Y = 14.1

∑ XY − nXY 2

b = ∑ X − nX

2

=

1677 − 10 x 11.8 x 14.1 1404 − 10 x (11.8) 2

13.2 = 1.37 = 15106.2/116 = 130.22. = 11.6

201. (a)

Regression. Eqn. of y on x is: y = 2x

∴ byx = 2

Regression Eqn. of x on y is: 6x - y = 4 i.e. 6x = y + 4 ⎛1⎞ 4 ⎜ ⎟ 6 6 i.e x = ⎝ ⎠ y +

1 ⇒ bxy = 6

296

byx.bxy

∴r=

=

⎛1⎞ 2⎜ ⎟ ⎝6⎠ =

1 3 = 0.578.

202. (a) Let the Regression Eqn. of x on y be:

8x - 10y + 66 = 0 i.e. 8x = 10y - 66 ⎛ 10 ⎞ 66 10 ⎜ ⎟ 8 8 ⇒ ⎝ ⎠ ybxy = 8 x=

i.e.

Hence, Regression Eqn. of y on x is: 40x - 18y = 214 i.e. 18y = 40x - 214 ⎛ 40 ⎞ 40 ⎜ ⎟ 18 y = ⎝ ⎠ x - 18

i.e.

10 40 x >1 bxy.byx = r2 = 8 18

Since r2 cannot be greater than 1, our assumption is not correct. Hence, the Regression Eqn. of y on x is: 8x - 10y + 66 = 0 i.e. 10y = 8x + 66 8 66 8 4 x+ = 10 10 10 5 ⇒ byx = i.e. y = Regression Eqn. of x on y is: 40x - 18y = 214 18 9 40 20 ⇒ i.e. 40x = 18y + 214 byx = =

∴

9 4 r = bxy.byx = 20 x 5 2

i.e, r2 = 0.36 ⇒ r = 0.6.

203. (a) σy

bxy = r

σx

3 9 σ i.e. 20 = (0.6) y (Since variance of x = 9, σ x = 3

i.e.

σy

(0.6)(3)(20) 9 = = 4.

204. (a) Regression Eqn., of y on x is : y = x + 5 ⇒ byx = 1 Regression. Eqn., of x on y is: 16x - 9y = 94 i.e 16x = 9y + 94

297

⎛9⎞ 94 9 ⎜ ⎟ i.e. x = ⎝ 16 ⎠ y + 16 ⇒ bxy = 16 9 3 2 ∴ r = byx. bxy = 1 x 16 ⇒ r = 4 ⎛ bxy.σ y ⎞ σx ⎜ ⎟ r ⎠ σy ⇒ σx ⎝ = bxy = r. ⎛ (9 /16) (4) ⎞ ⎜ ⎟ (3 / 4) ⎠ = 3. = ⎝

205. (e) Cov(xy) 3 r = σ y . σ x ⇒ Cov (xy) = r. σ x . σ y = 4 .3.4 = 9.

206. (c) We know that the regression coefficient b=

Cov(X, Y) V(X)

−121.5 = −1.5 ∴φ0 β = 81

Given that, ∑X = 55 and ∑Y = 33 and the number of observations n = 10 Therefore,

X=

55 33 = 5.5 Y= = 3.3 10 10 and

Now, The Y-intercept, a = Y − bX

= 3.3 - (− 1.5 × 5.5) = 11.55 ˆ = 11.55 - 1.5 X The estimating equation is Y ˆ = 11.55 - 15 x 5 = 4.05. The estimate of Y for X = 5, Y ˆ = 11.55 - 1.5 × 5 = 4.05. X = 5, Y 207. (b) Given a probability distribution of paired data {X,Y}, we can compute the covariance using the formula Cov (X,Y) = ∑ [X − E (X)] [Y − E(Y)] P(X,Y) The calculations for the covariance are shown in the table below:

X

Y

40 25 -15 Total

35 25 20

P(X,Y) (p) 0.40 0.50 0.10

X×p (a) 16.0 12.5 -1.5 27

Y×p (b) 14.0 12.5 2.0 28.5

X−E(X) (c) 13 -2 -42

Y−E(Y) (d) 6.5 -3.5 -8.5

(c).(d).(p) 33.8 3.5 35.7 73.00

208. (b) Risk premium = β(Rm - Rf) Pro b.

kA

0.3 0 0.3 5 0.1

7 8 14 16

km (kA-

kA

)

(km-

km

)

(km-

km k

9 5 10 14

-3.2 -2.2 3.8 5.8

0.25 -3.75 1.25 5.25

)

(kA- A ) -0.8 8.25 4.75 30.45

(km-

km

) (kA-

(km-

km

)2 P

k

A ). P -0.24 2.8875 0.7125 6.09 9.45

0.01875 4.9218 0.2343 5.5125

298

5 0.2 0

kA = ∑ kA P km = ∑ km P = 10.20

βA =

10.68735

=8.75

Cov(k k m ) A Var(k m )

∑ P.(k A − k A )(k m − k m ) 9.45 = 0.884. P(k m − k m )2 ∑ = = 10.68735

Risk premium = 0.884 (8.75 - 6) = 2.43%. Hence, option (b) is the correct choice. Cov(k j k m )

ρ xm σ x σ m

Var(k m ) Var(k m ) 209. (c) β x = =

0.45 × 10 × 3.33 10 = 0.26. =

210. (a)

x = 6, y = 8

σx

= 5,

40 8 = 3 , r = 15

σy

σy

8 40 1 64 . = 15 x 3 x 5 = 45 (i) byx = r. 211. (b) Regression Eqn. of x on y is σx

σy

(y - y ) 5 8 40 i.e., x - 6 = 15 . 3 (y - 8) 1 i.e., x - 6 = 5 (y - 8) x -X = r

σx

i.e., 5x - 30 = y - 8 i.e., 5x - y = 22. 212. (b) Given, x = 100 to estimate y Regression Eqn. of y on x is σy σ y - y = r. x (x - x ) 64 i.e., y - 8 = 45 x 94 ⎛ 64 x 94 ⎞ ⎜ ⎟ i.e., y = 8 = ⎝ 45 ⎠ = Rs.141.70.

213. (b) X

Y

X−X

y-

Y

(X − X)2

(Y − Y)

2

(X − X)

299

(Y − Y)

1 3 4 6 8 9 11 14 Total 56 X

1 2 4 4 5 7 8 9 40

-6 -4 -3 -1 1 2 4 7

-4 -3 -1 -1 0 2 3 4

36 16 9 1 1 4 16 49 132

16 9 1 1 0 4 9 16 56

24 12 3 1 0 4 12 28 84

56 ΣX ΣY 40 = n = 8 = 7, Y = n = 8 = 5

CovXY =

Σ( X − X ) ( Y − Y ) n −1

84 = 7 = 12.

214. (b) Σ (X − X ) 2 n −1

S 2X

=

SX

= 4.3425 Σ (Y − Y ) 2 n −1

S 2Y

=

SY

= 2.8284

1 = 7 (132) = 18.8571

1 = 7 (56) = 8

Correlation coefficient, r = 215. (d) X 4 9 10 14 4 7 12 22 1 17 Total 100

Y 31 58 65 73 37 44 60 91 21 84 564

X2 16 81 100 196 16 49 144 484 1 289 1376

Cov XY SX SY

=

12 (4.3425) (2.8284)

Y2 961 3364 4225 5329 1369 1936 3600 8281 441 7056 36562

= 0.977.

XY 124 522 650 1022 148 308 720 2002 21 1428 6945

We need to fit a straight line of the form: ˆ = a + bX Y n ∑ XY − ( ∑ X)(∑ Y )

The slope, b =

2

n ∑ X − ( ∑ X)

2

(10) (6945) − (100)(564)

=

(10)(1376) − (100) 2

= 3.4707

564 100 a = Y - b X = 10 - (3.4707) 10 = 21.693.

216. (a)

300

Standard error of the estimate ∑ Y 2 − a ∑ Y − b ∑ XY n−2

Se =

36562 − (21.693)(564) − (3.4707) (6945) 10 − 2

=

= 5.2813.

217. (b) The coefficient of determination, r2, is given by

r2

=

a ∑ Y + b ∑ XY − nY 2 ∑ Y 2 − nY 2

(21.693)(564) + (3.4707)(6945) −10(56.4) 2 36562 −10(56.4) 2

=

= 0.9530

Therefore, 95.3% of the variation in the test scores is explained by the regression line. We could also say that 95.3% of the variation in the dependent variable, test scores is explained by the independent variable, number of hours studied. 218. (a) Regression eqn. of y on x is: σy σ y - y = r. x (x - x ) Cov(xy)

i.e., y - y =

σx 2

(x - x )

∑ xy ∑ x ∑ y − ∑y n n ⎛x − ∑x ⎞ − n ⎟ 2 ⎜ 2 n n ⎠ ∑x ⎛ ∑x ⎞ ⎝ −⎜ ⎟ n ⎝ n ⎠ i.e., y -

12815 ⎛ 424 ⎞⎛ 363 ⎞ −⎜ ⎟⎜ ⎟ 10 ⎝ 10 ⎠⎝ 10 ⎠ ⎛ x − 424 ⎞ ⎜ ⎟ 2 10 ⎠ ⎝ 21926 ⎛ 424 ⎞ ⎛ 363 ⎞ − ⎜ ⎟ ⎜ ⎟ 10 ⎝ 10 ⎠ i.e., y - ⎝ 10 ⎠ = 128150 − (424) (363)

i.e., y - 36.3 =

219260 − (424) 2

(x − 42.4)

− 25762 (x − 42.4) = 39484 = -0.65(x - 42.4)

i.e. y - 36.3 = -0.65x + 27.56 Hence, the Regression equation of y on x is y = - 0.65x + 63.86. 219. (b)

When x = 28, y = - 0.65(28) + 63.86 i.e. y = 45.66 Thus, if x = Rs.28,000, then y = 4,566 [Note: unit of x = ’000 and that of y = ’00] Estimate the level of defective parts delivered when inspection expenditure amounts to Rs.28,000.

301

220. (c) The total costs (Y) are related to units produced (X) by the following equation:

Y

= a + bX

Where, a = Fixed costs b = Variable cost per unit. Units Produced Total Costs (in lakh of (in lakh) X rupees) Y 5 140 7 155 9 170 11 180 14 200 17 230 20 240 22 260 24 275 28 310 Total 157 2160 n ∑ XY − (∑ X)(∑ Y) b=

X2

XY

25 49 81 121 196 289 400 484 576 784 3005

700 1085 1530 1980 2800 3910 4800 5720 6600 8680 37805

n ∑ X 2 − (∑ X) 2

10(37805) − (157)(2160) 10(3005) − (157)

b =

2

38930 = 5401 = 7.2079

⎛ 157 ⎞ 2160 ⎜ ⎟ ~ 10 ⎠ a = Y - b X = 10 - 7.2079 ⎝ = 102.83597 − 102.836 ˆ = 102.836 + 7.2079 X We now have the relation Y

Therefore, Fixed costs = Rs.102.836 lakh, and Variable cost per unit = Rs.7.2079. 221. (e) Market Efficient XY (X) (Y)

15 13 14 -9 12 9 Total 54

X2

X- X

180 225 6 117 169 4 -154 196 5 -72 81 -18 132 144 3 36 81 0 239 896 ∑(X −X )(Y− Y ) −58 n −1 CovXY = = 5 = -11.6 S2X

S2Y

12 9 -11 8 11 4 33

(Y - Y ) (X - X ) (Y Y) 6.5 39 3.5 14 -16.5 - 82.5 2.5 - 45.0 5.5 16.5 -1.5 0 -58

=

=

∑( X − X ) 2 n −1

1 = 5 (410) = 82

∑( Y − Y ) 2 n −1

1 = 5 (365.50) = 73.1

(X − X ) 2

(Y − Y ) 2

36 16 25 324 9 0 410

42.25 12.25 272.25 6.25 30.25 2.25 365.50

For the characteristic line Y = a + bX, the slope b and intercept a are given by 302

n ∑ XY − (∑ X) (∑ Y) 2

n ∑ X − (∑ X)

b=

2

6(239) − (54) (33) 6(896) − (54)

=

2

=

1434 − 1782 5376 − 2916

− 348 = 2460 = - 0.1415

33 54 6 - (- 0.1415) 6 = 5.5 + 1.2735 = 6.7735 a= Y -bX = ˆ = 6.7735 - 0.1415 X. The characteristic line is given by Y

222. (b) From the characteristic line equation, we can see that the beta is - 0.1415. 223. (a) Systematic risk = β2 x Variance of market returns = (-0.1415)2 (82) = 1.6418. Cov XY

224. (c) Correlation coefficient, r =

S 2X

S 2Y

− 11.6

=

(82) (73.1)

= -0.1498

2

Coefficient of determination r = 0.02244. 225. (c) 80 Σx x = n = 20 = 4

y

40 Σy = n = 20 = 2

Σxy ⎛ Σx ⎞ ⎛ Σy ⎞ 480 −⎜ ⎟⎜ ⎟ n n n ⎝ ⎠ ⎝ ⎠ = 20 - (4)(2) = 24 - 8 = 16 Cov(xy) = 2

σx

2

Σx 2 ⎛ Σx ⎞ 1680 2 −⎜ (4) ⎟ = n ⎝ n ⎠ = 20 = 81 - 16 = 65

σy

2

Σy 2 = n

2

⎛ Σy ⎞ 320 −⎜ ⎟ − (2) 2 ⎝ n ⎠ = 20 = 16 - 4 = 12

Cov(xy)

bxy =

σy2

4 16 . = 12 = 3

Cov(xy) 226. (a) byx =

σx 2

16 . = 65

227. (d) In this problem, Y = overheads, and X = units produced X Y XY X2 Y2 40 191 7640 1600 36481 42 170 7140 1764 28900 53 272 14416 2809 73984 35 155 5425 1225 24025 56 280 15680 3136 78400 39 173 6747 1521 29929 48 234 11232 2304 54756 30 116 3480 900 13456 37 153 5661 1369 23409 40 178 7120 1600 31684 Total 420 1922 84541 18228 395024

420 1922 X = 10 = 42, Y = 10 = 192.2

303

∑ XY − nX Y

b

2

∑ X − nX

=

2

84541 − (10) (42) (192.2) 18228 − (10) (42) 2

=

= 6.4915

a = Y − bX = 192.2 - 6.4915 (42) = - 80.4430 The regression equation is ˆ Y

=

- 80.4430 + 6.4915 X. ˆ = -80.4430 + (6.4915) (50) = 244.1320. 228. (a) When X = 50, Y 229. (a) The standard error of the estimate is given by ∑ Y 2 − a ∑ Y − b ∑ XY n−2

395024 − (−80.4430) (1922) − 6.4915 (84541) 8 Se = = = 10.232. 230. (b) When units produced = 165, the overhead expense estimate given by the regression line is ˆ = -80.4430 + 6.4915 (165) = 990.6545 Y

The confidence intervals are given by: Estimate + tSe (where Se is the standard error of the estimate.) ∑ Y 2 − a ∑ Y − b ∑ XY n−2

395024 − (−80.4430) (1922) − (6.4915) (84541) 8 = = 10.232.

Se = At α = 0.05 and 8 degrees of freedom the critical t value = 2.306

The confidence intervals are given by: = 990.6545 ± (2.306) (10.232) = 990.6545 ± 23.595 or 967.0595 to 1014.2495. rσ σ = 0.5 x 6 x 5 = 15. 231. (b) The coefficient of correlation between x and y = x y 232. (d) For any simple liner regression equation, the standard error of estimate = ∑ (Y − Y) 2 125 = = 13.87. n−2 9

233. (b)

Advertising Expenditure (X) 1 2 3 4 5 6 Total 21 21 78 X = 6 = 3.5, Y = 6 = 13 ∑ X − nX

2

X2

XY

Y2

3 15 6 20 9 25 78

1 4 9 16 25 36 91

3 30 18 80 45 150 326

9 225 36 400 81 625 1376

326 − (6 x 3.5 x 130)

∑ XY − nX Y 2

Sales (Y)

=

91 − 6(3.5) 2

b

=

a

= Y - b X = 13 - (3.0286) 3.5 = 2.3999.

53 17 = .5 = 3.0286

The estimated relationship between sales and advertising expenditure is ˆ Y

= 2.3999 + 3.0286 X.

234. (c) The standard error of the estimate is given by

304

Se

=

∑ Y 2 − a ∑ Y − b ∑ XY n−2

=

1376 − (2.3999) (78) − (3.0286) (326) 6−2

= 7.0973.

235. (b) When advertising expenditure = 7 lakh, the estimated sales is ˆ Y

= 2.3999 + 3.0286 (7) = 23.6001.

ˆ ± tS . The confidence intervals are given by Y e

The critical t-value at 0.05 level of significance and with d.f = 4 is 2.776. The 95% confidence interval is given by 23.6001 ± (2.776) (7.0973) = 23.6001 ± 19.7021 or 3.897 to 43.302. 236. (d) We have to test the null hypothesis, H0: B = 0 against the alternative hypothesis H1: B > 0,

where B is the slope of the population regression line. b−B b−0 = Sb Sb Test Statistic: t = Se 7.0973 7.0973 2 2 ∑ X − n.X 91 − 6 (3.5) 2 Where, Sb = = = 4.1833 Sb = 1.6966 b 3.0286 S t = b = 1.6966 = 1.7851 The critical t-value at 0.05 level of significance and with degree of fredom = 4 is 2.132. We observe that t < t0.05, therefore we accept H0, that is, we cannot conclude that the relationship between sales and advertising expenditure is significant positively at 5 percent level of significance. 237. (e) a ∑ Y + b ∑ XY − nY 2

r2 =

∑ Y 2 − nY 2

(2.3999)(78) + (3.0286)(326) − 6(13) 2

=

1376 − 6(13) 2

= 0.4434

Therefore, 44.34% of the variation in sales is explained by variation in the advertising expenditure. 238. (b)

125 Σx x = n = 25 = 5 Σy 100 y = n = 25 = 4 σx2

Σx 2 = n Σy 2 = n

2

⎛ Σx ⎞ 650 −⎜ − (5) 2 ⎟ n ⎝ ⎠ = 25 = 26 - 25 = 1 2

⎛ Σy ⎞ 436 −⎜ ⎟ − (4) 2 σ y2 n ⎝ ⎠ = 25 = 17.44 - 16 = 1.44 Σxy ⎛ Σx ⎞ ⎛ Σy ⎞ −⎜ ⎟⎜ ⎟ Cov (xy) = n ⎝ n ⎠ ⎝ n ⎠ 520 = 25 - (5) (4) = 20.80 - 20 = 0.80

305

Cov(xy) σy2

∴ bxy =

Cov(xy) σx 2

byx = 239. (b) Test Scores (X) 6 9 3 8 7 5 8 10 Total 56

0.80 5 = 1.44 = 9 4 0.80 . = 1 = 0.8 = 5

First Month’s Sales (Y) 30 49 18 42 39 25 41 52 296

XY

X2

Y2

180 441 54 336 273 125 328 520 2257

36 81 9 64 49 25 64 100 428

900 2401 324 1764 1521 625 1681 2704 11920

n( ∑ XY) − ( ∑ X) ( ∑ Y)

b

=

a

=

n∑X

2

− ( ∑ X)

8(2257) − (56) (296)

2

=

8(428) − (56) 2

1480 = 288 = 5.1389

Y - b X = 37 - (5.1389) (7) = 1.0277

Therefore, we have the following linear relationship between the aptitude test score and the first month’s sales ˆ Y

= 1.0277 + 5.1389 X.

240. (b) The standard error of the estimate given by the regression line is given by

Se

=

∑ Y 2 − a ∑ Y − b ∑ XY n−2

11920 − (1.0277) (296) − (5.1389) (2257) 6 = 2.8839 = 1.6982. =

241. (d) For the aptitude test score of 4, the first month’s sales estimate given by the regression line is ˆ Y

= 1.0277 + (5.1389) 4 = 21.5833

The 95% confidence intervals for this estimate are given by: Estimate ± tSe. t0.05 at 6 degrees of freedom

= 2.447

The 95% confidence intervals are: 21.5833 ± (2.447) (1.6982) = 21.5833 ± 4.1555 or = 17.427 to 25.738. 242. (e) The standard error of the regression coefficient is given by

306

Se ∑X

Sb =

2

1.6982

− nX

2

=

428 − 8( 49)

= 0.283

We have to test the null hypothesis H0: B = 0 against the alternative hypothesis H1: B ≠ 0, where B is the slope of the population regression line.

Test Statistic t =

b Sb

5.1389 = 0.283 = 18.1587

t0.05 at 6 degrees of freedom

= 2.447

Since the observed value of t (t = 18.1587) is very much greater than the critical value, we reject H0 and conclude that there is a significant linear relationship between the aptitude test scores and the first month’s sales at 5% level of significance. 243. (a) a ∑ Y + b ∑ XY − nY 2 ∑ Y 2 − nY 2

r2 =

(1.0277) ( 296) + (5.1389) ( 2257) − 8(37) 2 11920 − 8(37) 2

=

950.6965 968 = = 0.9821

Therefore, 98.21% of the variation in the first month’s sales is explained by the variation in the aptitude test results.

244. (c) Number of Years’ Experience X

Sales Commission (in Rs. ’00) Y

XY

X

Y

X−X

Y−Y

(X − X)

6 13 27 15 9 11 21 14 12 Total

50 100 140 110 70 90 140 100 80 880

300 1300 3780 1650 630 990 2940 1400 960 13950

36 169 729 225 81 121 441 196 144 2142

2500 10000 19600 12100 4900 8100 19600 10000 6400 93200

-8.22222 -1.22222 12.77778 0.777778 -5.22222 -3.22222 6.777778 -0.22222 -2.22222

-47.7778 2.222222 42.22222 12.22222 -27.7778 -7.77778 42.22222 2.222222 -17.7778

392.839506 -2.7160494 539.506173 9.50617284 145.061728 25.0617284 286.17284 -0.4938272 39.5061728 1434.4444

2

2

(X − X)2

(Y − Y)2

67.60494 1.493827 163.2716 0.604938 27.2716 10.38272 45.93827 0.049383 4.938272 321.5556

2282.716 4.938272 1782.716 149.3827 771.6049 60.49383 1782.716 4.938272 316.0494 7155.556

(Y − Y)

128

We have to fit a straight line of the form Y = a + bX i.

128 880 X = 9 = 14.222, Y = 9 = 97.7778 ∑ XY − nXY

ii.

b

=

∑ X 2 − nX 2 13950 − 9(97.7778)(14.2222)

=

2142 − 9(14.2222) 2

1434.4612 = 321.5612 = 4.4609

307

a

= Y -bX = 97.7778 - 4.4609 (14.2222) = 34.334.

245. (e) Standard error of the estimate: Se

=

∑ Y 2 − a ∑ Y − b ∑ XY n−2

93200 − (34.334)(880) − (4.4609)(13950) 7 = = 10.3959

Standard error of the regression coefficient, Se 10.3959 Sb

=

∑ X 2 − nX 2

=

2142 − 9(14.2222)

246. (a) Correlation coefficient = Cov XY

∑(X − X ) 2 n −1

2 SX

=

SX

= 6.339 ∑(Y i − Y ) 2

2 SY

=

SY

=

n −1

10.3959 = 17.9321 = 0.5797.

Cov XY SX SY

∑(X − X ) (Y − Y ) n −1

=

2

(1434.4444) ~ 8 = = 179.30555 − 179.3056

(321.5556) 8 = = 40.1944

1 = 8 (7155.556) = 894.4445

29.9073

Correlation coefficient, r =

179.3056 (6.3399) (29.9073)

179.3056 = 189.6092 = 0.9457.

247. (a) Coefficient of determination, r2 = 0.8944

The proportion of variation in commission earnings explained by factors other than years of selling experience = 1 - 0.8944 = 0.1056. ˆ = 34.334 + 4.4609 X 248. (b) The regression line is Y

The commission estimate for a person with 10 years experience would be equal to Rs.7,894.30 ˆ Y

=

34.334 + 4.4609 (10) = 78.943.

249. (a) The return on the market is taken to be the independent variable. X Y XY X2 Y2 11 10 110 121 100 15 12 180 225 144 3 8 24 9 64 18 15 270 324 225 10 9 90 100 81 12 11 132 144 121 6 8 48 36 64 7 10 70 49 100 18 13 234 324 169 13 11 143 169 121 Total 113 107 1301 1501 1189 113 107 ∑Y ∑X X = n = 10 = 11.3, Y = n = 10 = 10.7

308

1301 − (10) (11.3) (10.7)

∑ XY − nXY

b= a=

2

∑ X − nX

Y − bX

2

1501 − 10 (11.3) 2

=

= 0.4101

= 10.7 - (0.4101) (11.3) = 6.0659

Here, we test the hypothesis

H0: B = 1 H1: B < 1

This is a one-tailed test at a significance level of α = 0.05. Standard error of the estimate Se

=

∑ Y 2 − aΣY b ∑ XY n −2

1189 − (6.0659) (107) − (0.4104) (1301) 8 = 0.8674

=

Se

Sb =

2

0.8674

∑ X − nX

2

=

1501 − (10) (11.3) 2

0.8674 = 14.97 = 0.0579.

t − BH0 0.4101 − 1 = = − 10.20. Sb 0.0579

Standardized statistic is t = 250. (e)

The standardized statistic is t =

b − BH 0 Sb

0.4101 − 1 = 0.0579 = -10.20

a ∑ Y + b ∑ XY − nY 2

r2 =

∑ Y 2 − nY 2 (6.0659) (107) + (0.4101) (1301) − (10) (10.7) 2

37.6914 1189 − (10) (10.7) = = 44.1 = 0.8547 2 Systematic risk = r x Variance of security returns Variance of Y = 4.9 Systematic risk = (0.8547) (4.9) = 4.18803 = 4.188. 251. (c) 2

X

40.75 65.65 = 8 = 5.0938; Y = 8 = 8.2062 ∑ XY − nXY

∑ X 2 − nX 2 b = Assessed Selling Value (in Price lakh of Rs.) (in lakh of Rs.) X Y XY

4.03 7.20 3.25 4.48 2.79 5.16 8.04

6.34 11.83 5.52 7.40 4.88 8.11 12.32

25.5502 85.176 17.940 33.152 13.6152 41.8576 99.0528

X2 16.2409 51.84 10.5625 20.0704 7.7841 26.6256 64.6416

Y2 40.1956 139.9489 30.4704 54.76 23.8144 65.7721 151.7824

(X − X)

(Y − Y )

-1.06375 2.10625 -1.84375 -0.61375 -2.30375 0.06625 2.94625

-1.86625 -3.62375 -2.68625 -0.80625 -3.32625 -0.09625 4.11375

(X − X) 2 1.131564 4.436289 3.399414 0.376689 5.307264 0.004389 8.680389

(Y − Y ) 2 3.482889 13.13156 7.215939 0.650039 11.06393 0.009264 16.92293

309

5.80 Total 40.75

9.25 65.65

53.650 33.64 85.5625 0.70625 369.9838 231.4051 592.3063

369.9838 − 8(8.2062) (5.0938) b

231.4051 − 8(5.0938) 2

=

1.04375

0.498789 1.089414 23.83478 53.56598

35.57779 = 23.8307 = 1.4929

a = Y - b X = 8.2062 - (1.4929) (5.0938) = 0.6017 We have the linear relationship ˆ = 0.6017 + 1.4929 X. Y 252. (a) Standard error of the estimate given by the regression line

Se

∑ Y 2 − a ∑ Y − b ∑ XY n −2

= =

592.3063 − (0.6017) (65.65) − (1.4929) (369.9838) 6 = 0.2756 Se 0.2756

231.4051 − 8(5.0938) 2 ∑ X 2 − nX 2 = Sb = = 0.0565 BH 0 + t S = 1.3 + (1.943) (0.0565) = 1.3 + 0.1098, where t b n - 2, d = t6,0.05 = 1.943 Since 1.4929 does not lie within these limits we reject H0 and conclude that B is significantly greater than 1.3. 253. (e) Y

M

(Y − Y)

(M − M)

0.3 0.3 0.4 -0.1 0.3 0.6 Sum1.8

0.2 0.0 0.4 -0.2 0.2 0.6 1.2

0.0 0.0 0.1 -0.4 0.0 0.3 0.0

0.0 -0.2 0.2 -0.4 0.0 0.4 0.0

ˆ = If Y

a + bM, then

b

=

a

=

Cov (Y, M ) Var (M)

(Y − Y) 2 0.00 0.00 0.01 0.16 0.00 0.09 0.26

(M − M) 2

(Y − Y) (M − M)

0.00 0.04 0.04 0.16 0.00 0.16 0.40

0.00 0.00 0.02 0.16 0.00 0.12 0.30

0.30 / 5 0 = .40 / 5 = 0.75

Y - b M = 0.3 - 0.75 x 0.2 = 0.15

ˆ = 0.15 + 0.75M. Therefore, Y

254. (a) S.E =

ˆ )2 ∑(Y − Y n−2

=

0.035 4 = 0.0935.

S.E 2

255. (a) Sb = M 0.2 M2 0.04

∑ M − nM

0 0

2

2

∑ M − n x 0.2 = = 0.4 -0.2 0.2 0.6 0.16 0.04 0.04 0.36

0.0935

Therefore, Sb = 256. (b)

0.0935

0.0935 2

0.64 − 2.24

2

∑ M − 0.24

0.0935

=

0.4 = 0.1479.

310

i.

H0: B = 0; n = 6 H1: B ≠ 0 b−B Sb

ii.

iii.

b

=

Sb

~ tn-2 distribution.

0.05 The critical points ± tn - 2, α/2 = t4, 2 = ± t4, 0.025 = ± 2.776. 0.75 The observed t-value is 0.1479 = 5.07

Since this falls in the critical region, we reject H0. 257. (a) ˆ = 0.15 At M = 0, Y ˆ ± 2.776 x S 95% prediction level is Y e That is, 0.15 ± 2.776 x 0.0935 = (-0.1096, 0.4096). 258. (a) P = 0.75X + 0.25Y rp = 0.75 x 30 + 0.25 x 28 = 22.50 + 7 = 29.5%

σ2

= (0.75)2 (5)2 + (0.25)2 (4)2 + 2 x 0.75 x 0.25 x (-1) x 5 x 4 = 14.0625 + 1 - 7.5625 σ = 2.75%.

259. (e)

= αX + (1 - α) Y, 0 ≤ α ≤ 1

P 2

σ

= α2 25 + (1 - α)2 16 + 2α (1 - α) x 5 x 4 x (-1) = 25α2 + (1 - 2α + α2)

16 - 40 (α - α2) = 81α2 - 72α + 16 Risk less portfolio <=> σ = 0 <==> σ2 = 0 Therefore, 81α2 - 72α + 16 = 0 72 ± (72) 2 − 4 x 81 x 16

⇒α=

2 x 81

72 ± 5184 − 5184 162 ⇒ = 0.4444 Therefore, P = 0.44X + 0.56Y.

260. (c)

Name Clio Fin. Asian Net Smelters Heera Shri Rang Napa Vantage Sagar Kenwood Total Mean Standard Deviation

V 27 49 171 191 193 199 228 422 500 1,980 220

R -15 4 28 32 36 34 43 52 56 270 30

(V - M)2 37,249 29,241 2,401 841 729 441 64 40,804 78,400 1,90,170 21,130 154.1792

(R - M)2 2,025 676 4 4 36 16 169 484 676 4,090 511.25 22.61083

(V - M) x (R - M) 8,685 4,446 98 -58 - 162 -84 104 4,444 7,280 24,753 3,094.125

311

i. Average of returns is 30%. ii. Average of trading volumes is Rs.220 lakh. iii. The standard deviation of the returns is 22.6108. iv. The standard deviation of the trading volumes is 154.1792. 261. (e) The covariance of the returns and the trading volumes is 3,094.125. The correlation coefficient is Cov (V,R)/ (STD (V) x STD (R)) = 0.8876. Since the correlation coefficient is high, there is a strong linear relationship between trading volumes and returns for non-specified scripts. The linear relationship is given by R = A + B x V, where B = Cov(V, R)/Var (V) = 0.1302 A = Mean (R) - B x Mean (V) = 1.3643. Therefore, R = 1.3643 + 0.1302 x V. 262. (e) The trend for the trading volumes of Gujarat cotton is given by 50 + 60 x t.

Therefore, the expected trading volume for the week ending 10.11.95 is 50 + 60 x 5 = 350. This implies that the expected return is 1.3643 + 0.1302 x 350 = 46.9343, or approximately 47%. 263. (c) For calculating the linear regression on equation the following table can be prepared.

X Y X2 65 68 4,225 63 66 3,969 67 68 4,489 64 65 4,096 68 69 4,624 62 66 3,844 70 68 4,900 66 65 4,356 68 71 4,624 67 67 4,489 69 68 4,761 71 70 5,041 Total 800 811 53,418 = 800 ΣX = 811 ΣY 2 = 53418 ΣX 2 = 54849 ΣY ΣXY = 54107

b =

Y2 4,624 4,356 4,624 4,225 4,761 4,356 4,624 4,225 5,041 4,489 4,624 4,900 54,849

54,107 − 12(66.67) (67.58) 53,418 − 12(66.67) (66.67 )

a = 67.58 - (0.5079)(66.67)

XY 4,420 4,158 4,556 4,160 4,692 4,092 4,760 4,290 4,828 4,489 4,692 4,970 54,107

= 0.5079 = 33.72

ˆ = a + b x =33.72 + 0.5079X. The linear regression equation i Y

264. (b) Standard error of estimate

S.E =

∑ Y 2 − a ∑ Y − b ∑ XY n−2

=

54,849 − (33.72) (811) − (0.5079) (54,107) 10 = 1.454.

265. (a) The coefficient of correlation can be calculated as follows: a ∑ Y − b ∑ XY − nY 2

r2 =

∑ Y 2 − nY 2

312

(33.72)(811) + (0.5079) (54,107) − 12(67.58)2 54,849 − (12) (67.58)2

= r

= 0.5232

0.5232 = 0.7233

=

(Since the slope of the regression line is positive, the coefficient of correlation is also positive). 266. (c)Following are the quarterly returns tabulated for BSE Index and Ind Jyoti.

Example: BSE returns for quarter ending Dec. ’96 2713 − 3205 3205 = x 100 = -15.35 Returns on BSE (X) (%) -15.35 35.02 12.83 - 5.03

Returns on Ind Jyoti (Y) (%) 7.89 -5.49 6.45 9.09

Quarter Ending X

Y

(X − X)

(Y − Y)

(X − X)2

(X − X) (Y − Y)

Dec. 96 Mar. 97 Jun. 97 Sep. 97

7.89 -5.49 6.45 9.09

-22.218 28.152 5.962 -11.898

3.405 -9.975 1.965 4.605

493.639 792.535 35.545 141.562 1463.281

-75.652 -280.816 11.697 -54.790 -399.561

-15.35 35.02 12.83 -5.03

X

=

6.868

Y

=

4.485 ∑(X − X ) 2

σx

n

=

Cov (X,Y) =

1463.281 4 = = 19.13

∑(X − X ) (Y − Y ) n − 99.89

Cov (X, Y)

∴φ2 β

=

σ 2x

−399.561 4 = = -99.89

=

(19.13) 2

= -0.273

The investment is not advisable in the fund as the average returns are lower than the market returns. However, negative Beta may help in risk reduction in a portfolio. 267. (a) Portfolio Return (Rp) =

RM =

XA RA + XB RB

0.4 x 18 + 0.6 x 23 = 21%

2 2 2 2 Portfolio risk = [ x A σ A + x B σ B + 2 x A xB x Cov (A, B)]1/2

σM = [0.42 x 3.88 + 0.62 x 8.87 - 2 x 0.4 x 0.6 x 2.59]1/2 = [1.3968 + 1.4192 - 1.2432]1/2 = 1.6034. 268. (a)

313

RN = 0.6 x 18 + 0.4 x 23 = 20% σN

= [0.62 x 3.88 + 0.42 x 8.87 - 2 x 0.6 x 0.4 x 2.59]1/2 = [1.3968 + 1.4192 - 1.2432]1/2 = 1.254.

269. (e ) Preference of a portfolio can be based on co-efficient of variation. Standard deviation Mean

Co-efficient of variation =

1.6034 Co-efficient of variation of M = 21 = 0.0763 1.254 Co-efficient of variation of N = 20 = 0.0627.

270. (b) The risk-free rate of return is the value of ‘A’ in the following regression equation: Y = A + BX The value of A can be found out as follows: Return Risk Y X XY X2 13 1.19 15.47 1.42 15 2.37 35.55 5.62 17 3.49 59.33 12.18 20 5.10 102.00 26.01 23 7.27 167.21 52.85 26 8.95 232.70 80.10 ∑ Y= 114 ∑ X = 28.37 ∑ XY = 612.26 ∑ X2 = 178.18 Y

= 19; X = 4.73

Normal equations are Y

= A + BX ∑ XY = A ∑ X + B ∑ X 2 ∴φ0 19

=

Α + 4.73Β

− (ι)

612.26 =

28.37Α + 178.18Β − (ιι)

539.03 =

28.37Α + 134.19Β − Εθ. (ι) ξ 28.37

73.23

=

0 + 43.99Β

− Εθ. (ιι)

− 28.37 ξ Εθ. (ι) B =

1.664

and A = 19 - 1.664 x 4.73

= 11.12%

The risk-free rate of return = A = 11.12%. 271. (c) The regression equation is found as follows: Y = A + BX Year Sales X Share Price Y (X − X) (Y − Y) (X − X)

1989 1990 1991

2.10 2.70 2.90

11 13 18

-1.11 -0.51 -0.31

-19.6 -17.6 -12.6

2 x (Y − Y) (X − X) 21.756 1.232 8.796 0.260 3.906 0.096

(Y − Y)2

384.16 309.76 158.76

314

1992 1993 1994 1995 1996 1997 1998

2.90 3.10 3.00 3.30 3.70 4.00 4.40

24 28 32 38 42 47 53

-0.31 -0.11 -0.21 0.09 0.49 0.79 1.19 0

X=

ΣX i 32.1 = = 3.21 n 10

Y=

ΣYi 306 = = 30.6 n 10

B=

Σ(X − X) (Y − Y) 82.54 = = 20.491 4.028 Σ(X − X) 2

-6.6 -2.6 1.4 7.4 11.4 16.4 22.4 0

2.046 0.286 -0.294 0.666 5.586 12.956 26.656 82.54

0.096 0.012 0.044 0.008 0.240 0.624 1.416 4.028

43.56 6.76 1.96 54.76 129.96 268.96 501.76 1860.40

A = Y − B X = 30.6 - 20.491 x 3.21 = 30.6 - 65.776 = -35.176 ∴φ0 Τηε ρεγρεσσιον εθυατιον ωιλλ βε Ψ = −35.176 + 20.491 Ξ. 272. (β) Τηε χορρελατιον χοεφφιχιεντ χαν βε χαλχυλατεδ ασ φολλοωσ: A ΣY + BΣXY − nY 2 Σ Y 2 − nY 2

ρ2 =

Σ(X − X) (Y − Y)

= ∴φ0

2

Σ(X − X) x Σ(Y − Y)

82.54 2

=

(4.028) x (1860.40)

= 0.9534

ρ = 0.9764 ορ 97.64%.

273. (α) 2 Γιϖεν Σ(R1 - R 2 ) = 231 2

i.e., Σd = 231 also given R = -0.4 ⎡ 6 Σd 2 ⎤ ⎢ 3 ⎥ n -n ⎦ i.e., 1 - ⎣ = -0.4 ⎡ 6 x 231 ⎤ ⎢ ⎥ i.e., ⎣ 1.4 ⎦ = n3 - n i.e., 990 = n3 - n i.e., 1000 - 10 = n3 - n ∴ Number of pairs, n = 10. 274. (b) x 1 2 3 4 5 15

y 160 180 140 180 200 860

x2 1 4 9 16 25 55

xy 160 360 420 720 1000 2660

315

Σxy ⎛ Σx ⎞ ⎛ Σy ⎞ 2660 ⎛ 15 ⎞ ⎛ 860 ⎞ −⎜ −⎜ ⎟⎜ ⎟⎜ ⎟ ⎟ n ⎝ n ⎠⎝ n ⎠ = 5 ⎝ 5 ⎠⎝ 5 ⎠

Cov(xy) = = 532 - (3 x 172) = 532 - 516 = 16. 275. (a) Regression Coefficient of y on x, σy Cov(xy) σ y Cov(xy) σ . σ σx 2 σ σ y x x x = byx = r x = σx2

Σx 2 = n

2

Cov(xy)

∴ byx =

2

55 ⎛ 15 ⎞ ⎛ Σx ⎞ −⎜ −⎜ ⎟ ⎟ n ⎝ ⎠ = 5 ⎝ 5 ⎠ = 11 - 9 = 2

σx 2

16 = 2 = 8.

276. (a) R.Eqn., of y on x is : y = x + 5 ⇒ byx = 1 R.Eqn of x on y is : 16x - 9y = 94 i.e. 16x = 9y + 94

i.e.

94 9 ⎛9⎞ x =⎜ ⎟ y+ ⇒ bxy = 16 16 ⎝ 16 ⎠

9 3 ⇒r= 16 4 . ∴φ0 ρ = βψξ.βξψ = 1 ξ 2

277. (a) The Mean return and standard deviation of both securities are calculated below.

RA =

0.2 x (-10) + 0.25 x 5 + 0.3 x 25 + 0.25 x 35 = 15.5%

σ 2A =

0.2 (-10 - 15.5)2 + 0.25 (5 - 15.5)2 + 0.3 (25 - 15.5)2 + 0.25 (35 - 15.5)2

σA

=

16.725%.

=

0.2 x (-3.0) + 0.25 x 2 + 0.3 x 15 + 0.25 x 20 = 9.4%

278. (b)

RB

σ 2B =

0.2 (-3.0 - 9.4)2 + 0.25 (2 - 9.4)2 + 0.3 (15 - 9.4)2 + 0.25 (20 - 9.4)2

σB

9.052%.

=

279. (a) The covariance between A and B is Cov (A, B)

=

0.2(-10 - 15.5) (-3.0 - 9.4) + 0.25 (5 - 15.5) (2-9.4) + 0.3 (25 - 15.5) x (15 - 9.4) + 0.25 (35 15.5) (20 - 9.4) = 150.30

Correlation coefficient r =

Cov(A, B) σA σB

=

150.30 16.725 x 0.052

= 0.9927.

280. (a) Let us calculate the NPV of all the three projects

(NPV) A = 0.1 x (-11) + 0.2 x 0.5 + 0.45 x 10 + 0.25 x 21 = 8.75 (NPV) B = 0.1 x (-40) + 0.2 x 2.0 + 0.45 x 10 + 0.25 x 30 = 8.40 (NPV) C = 0.1 x (100) + 0.2 x 5 + 0.45 x 20 + 0.25 x 35 = 8.75. 281. (e) σ NPVA A

= {0.1 (-11 - 8.75)2 + 0.2 (0.5 - 8.75)2 + 0.45 (10 - 8.75)2 + 0.25 (21- 8.75)2}1/2 = {39 + 13.61 + 0.70 + 37.52}1/2 = 9.53%

316

σ NPVA B σ NPVA C

= {0.1 (-40 - 8.4)2 + 0.2 (2 - 8.4)2 + 0.45 (10 - 8.75)2 + 0.25 (30 - 8.4)2}1/2 = {234.26 + 8.19 + 1.15 + 116.64}1/2 = 360.24 = 18.98% = {0.1 (-100 - 8.75)2 + 0.2 (5 - 8.75)2 + 0.45 (20 - 8.75)2 + 0.25 (35 - 8.75)2}1/2 = {1182.66 + 2.81 + 56.95 + 172.26}1/2 = 37.61%.

282. (a) It is noted that NPV of projects A and C is the same, but the variability of project C is more than A indicating the higher risk (from 280 and 281). Therefore, project A should be selected. 283. (b) ˆ 2 ∑(Y − Y) r2 = 1 − ∑(Y − Y) 2 Coefficient of determination, r2 = ˆ denotes the estimated values. Y denotes the observed values and Y Y=

812 = 116 7

Σ (Y − Y) 2 Σ Y 2 − n Y 2 = 95254 − 7 × 1162 = 1062 2

∴φ0 Χοεφφιχιεντ οφ δετερµινατιον, ρ = So r = 0.8744. 284. (c) Year

1−

250 = 0.7645 1062

B Sterlite Price Rs.(Y) XY S E

X2

I n d e x

1999

1998

1997

1996

1995

( X ) 3 245 0 7 0 3 255 2 2 0 3 240 3 7 0 3 390 0 9 7 3 655 5 1

752150

9424900

821100

10368400

808800

11356900

1207830

9591409

2304290

12376324

317

8 3 395 6 3 6 ∑ ∑ Y = 2180 X

1994

n=6

1436220

13220496

∑ XY = 7330390 ∑ XY2 = 66338429

= 1 9 9 1 1 n ∑ XY − (∑ X)(∑ Y) b

n ∑ X 2 − (∑ X) 2

=

6(7330390) − (19911) (2180) 6(66338429) − (19911) 2

= a ⇒ ˆ Y

285. (a) Year

1999 1998 1997 1996 1995 1994

Se =

= a

=

43982340 − 43405980 398030574 − 396447921

576360 = 0.36 = 1582653

Y − bX = =

363.33 - (0.36)(3318.5) 363.33 - 1194.66 = (-) 831.33

=

-831.33 + 0.36X.

Sterlite Price (Rs.) (Y) 245 255 240 390 655 395

BSE Index (X) 3070 3220 3370 3097 3518 3636

ˆ = [(-831.33) + (0.36) (X)] (Y - Y ˆ) Y

ˆ )2 (Y - Y

273.87 327.87 381.87 283.59 435.15 477.63

833.48 5310.04 20127.10 11323.09 48334.02 6827.72 92755.45

-28.87 -72.87 -141.87 106.41 219.85 -82.63 0.02 ≅ 0

ˆ 2 Σ(Y − Y) 92755.45 = =152.28. n−2 4

286. (a) Price (x) Demand (y) 14 84 16 78 17 70 18 75 19 66 20 67 21 62 22 58 23 60

u = (x - 20) v = (y - 70)

u

v

uv

-64 -4 -3 -2 -1 0 1 2 3 -10

36 16 9 4 1 0 1 4 9 80

196 64 0 25 16 9 64 144 100 618

-84 -32 0 -10 4 0 -8 -24 -30 -184

14 8 0 5 -4 -3 -8 -12 -10 -10

318

Σuv ⎛ Σu ⎞ ⎛ Σv ⎞ −184 ⎛ − 10 ⎞ ⎛ −10 ⎞ −⎜ −⎜ ⎟⎜ ⎟ ⎟⎜ ⎟ ⎝ n ⎠ ⎝ n ⎠ = 10 ⎝ 10 ⎠ ⎝ 10 ⎠ = -18.4 - (-1)(-1) = -19.4. Cov(uv) = n

287. (b)

σu

Σu 2 ⎛ Σu ⎞ −⎜ ⎟ n ⎝ n ⎠

=

2

σv

Σu ⎛ Σv ⎞ −⎜ ⎟ n ⎝ n ⎠

=

∴r

Cov(uv) σu . σv

=

2

80 ⎛ −10 ⎞ −⎜ ⎟ 9 ⎝ 9 ⎠

= 2

= −19.4

2

618 ⎛ −10 ⎞ −⎜ ⎟ 9 ⎝ 9 ⎠

7 x 60.8

= 2.77 2

= 8.21

= - 0.95.

288. (b) Year 1998-99 1997-98 1996-97 1995-96 1994-95

Capital receipts (X) (Rs. in crore) 105933 96731 74728 58338 68695 ∑X= 404425

Capital expenditure (Y) (Rs. in crore) 57865 53046 42074 38415 38627 ∑Y = 230027

(X − X) dx = 25048 15846 (6157) (22547) (12190)

(Y − Y) dy = 11859.6 7040.6 (3931.4) (7590.4) (7378.4)

(X − X) (Y − Y)

297059261.0 111565347.6 24205629.8 171140748.8 89942696.0 693913683.2

404425 5 = 80885.0 σx = 19832.86 230027 Y= 5 = 46005.4 σY = 8912.51 1 Σ dx dy (n − 1) 693913683.2 = 0.9814. σx σy 4(19832.86) (8912.51) r = = 289. (a) X=

Debt % Y

Fixed Assets % X

XY

(X − X)

(Y − Y)

(X − X) (Y − Y)

64 69 77 73 75 72 ∑Y = 430

65 67 72 69 71 68 ΣX = 412

4160 4623 5544 5037 5325 4896 29585

-3.67 -1.67 3.33 0.33 2.33 -0.67

-7.67 2.67 5.33 1.33 3.33 0.33

28.15 4.46 17.75 0.44 7.76 -0.22 58.34

Y

= 71.67 X = 68.67 ΣY = 30924 ΣX2 = 28324 σy = 4.633 σx = 2.582 i. The normal equation can be written as follows: 2

ΣY = nA + bΣX or Y = A + b X (we obtain this equation on dividing ΣY = nA + bΣX by n throughout) ΣXY = AΣX + BΣX2 Substituting the respective values, we have 71.67 = A + 68.67B -------------Eq. (1) 29585 = 412A + 28324 B -------------Eq. (2) Dividing equation (2) throughout by 412, we obtain 71.808 = A + 68.747B --- Eq. (3) Subtracting eq. (1) from eq. (3), we get 0.138 = 0.077B

319

B A

= =

1.792 -51.40

ˆ = 1.792 X - 51.40. Therefore the regression equation can be written as follows: Y 1 1 Σ(X − X) (Y − Y) (58.34) n −1 5 σX σY 2.582 x 4.633 290. (a) r = = = 0.975.

291. (b) Since sales depends on advertising expenditure, advertising expenditure has to be taken as independent variable (X) and sales has to be taken as dependent variable (Y). Year 1993 1994 1995 1996 1997 1998 1999

Adv. Exp. (X) (Rs. ’000) 25 28 33 38 45 49 56 ∑X= 274

Sales (Y) (Rs. XY lakh) 7 175 9 252 13 429 16 608 18 810 23 1127 29 1624 Σ XY= 5025 Σ Y= 115

2

X 625 784 1089 1444 2025 2401 3136 Σ X2= 11504

The regression equation is ˆ Y

= a + bX ΣXY − nX Y

;a = Y − bX

b

2 2 = ΣX − n(X)

X

ΣX 274 = = 39.14 n 7 =

Y

ΣY 115 = =16.43 7 = n 5025 − 7(39.14) (16.43)

b a

11504 − 7(39.14) 2

= 0.67

= = 16.43 - (0.67)(39.14) = -9.79 ˆ = -9.79 + 0.67X The regression equation is: Y For sales (Y) = 35 the advertising expenditure (X) can be found out as follows: 35 = -9.79 + 0.67X or 0.67X = 35 + 9.79 = 44.79 44.79 or X = 0.67 = 66.85 The advertising expenditure will be Rs.66.85 thousands i.e. Rs.66,850. 292. (a) Let the following notations be used:

Price (Rs./Unit) : X Quantity that can be sold (’000 units) : Y The relationship between X and Y as per the least squares regression method is Y = a + bX ΣXY − nX Y 2 2 Where, b = ΣX − nX

and a = Y − b X

Total

320

X Y XY X2

28 130 3,640 784

26 145 3,770 676

25 170 4,250 625

24 190 4,560 576

21 215 4,515 441

19 225 4,275 361

143 1,075 25,010 3,463

ΣXY = 25,010; ΣX2 = 3,463; ΣX = 143; ΣX = 1,075 X=

ΣX 143 = = 23.83 n 6

Y=

ΣY 1075 = =179.17 n 6 25,010 − 6(23.83) (179.17) = − 10.89 3,463 − 6(23.83) (23.83)

b=

∴φ2

a = 179.17 - (-10.89)(23.83) = 438.68 ∴φ0 Τηε ρελατιονσηιπ βετωεεν πριχε (Ξ) ανδ θυαντιτψ σολδ (Ψ) ισ: Ψ = 438.68 − 10.89Ξ. 293. (β) Λετ τηε φολλοωινγ νοτατιονσ βε υσεδ. Ξ: Περχενταγε οφ δισχουντ οφφερεδ Ψ: Χηανγε ιν σαλεσ = Αφτερ Προµοτιον − Βεφορε προµοτιον Total X (%) Y (’000 units) XY X- X Y- Y (X − X)

2

(Y − Y)

2

(X − X) (Y − Y)

5 6 30 (19.38)

10 0 0 (14.38)

15 4 60 (9.38)

20 6 120 (4.38)

25 8 200 0.62

30 6 180 5.62

40 8 320 15.62

50 12 600 25.62

(0.25)

(6.25)

(2.25)

(0.25)

1.75

(0.25)

1.75

5.75

375.58

206.78

87.98

19.18

0.38

31.58

243.98

656.38

1621.84

0.06

39.06

5.06

0.06

3.06

0.06

3.06

33.06

83.48

4.85

89.88

21.11

1.10

1.09

(1.41)

27.34

147.32

291.28

X=

ΣX 195 = = 24.38 n 8

Y=

ΣY 50 = = 6.25 n 8

rX, Y =

Cov (X, Y) σX σY

Σ(X − X) (Y − Y)

=

2

Σ(X − X) Σ(Y − Y)

291.28 2

=

(1621.84) (83.48)

195 50 1510

= 0.792.

~ 294. (b) Coefficient of determination, r2 = (0.792)2 = 0.6267 − 0.627 62.7% of the changes in sales is explained by the variations in percentage of discount offered.

295. (e) Let the following notations be used:

X: Price Y: Quantity demanded (’000 units) X Y XY X2

10 30 300 100

9 44 396 81

8 63 504 64

7 72 504 49

6 90 540 36

5 108 540 25

4 125 500 16

3 140 420 9

Total 52 672 3704 380

321

The linear regression equation has the form: ˆ Y

= a + bx ΣXY − nX Y

Where, b =

ΣX 2 − nX 2

X =

ΣX 52 = = 6.50 n 8

Y =

ΣY 672 = = 84 n 8

and a = Y − b X

3704 − (8) (6.50) (84) − 664 = 380 − (8) (6.50) (6.50) 42

= - 15.81

b

=

a

~ = Y − bX = 84 - (-15.81)(6.50) = 186.765 − 186.77

The demand function is ˆ = 186.77 - 15.81X Y

296. (e) Demand for the product at a price of Rs.7.50 per unit = 186.77 - 15.81 (7.50) = 68.195 i.e. 68.195 x 1000 = 68195 units. 297. (a) Let the following notations be used:

X = Urban population index Y = Car registrations A B C D E F G X

X Y X -X Y -Y (C)(D) C2 D2

158.00 490.00 (16.33) (188.83) 3083.59 266.67 35656.77

1046 = 6 = 174.33

163.00 539.00 (11.33) (139.83) 1584.27 128.37 19552.43

175.00 628.00 0.67 (50.83) -34.06 0.45 2583.69

177.00 708.00 2.67 29.17 77.88 7.13 850.89

178.00 810.00 3.67 131.17 481.39 13.47 17205.57

195.00 898.00 20.67 219.17 4530.24 427.25 48035.49

Total 1046.00 4073.00

9723.31 843.34 123884.84

4073 Y = 6 = 678.83 Σ (X − X) (Y − Y)

Coefficient of correlation (r) =

2

Σ (X − X) Σ(Y − Y)

9723.31 2

=

(843.34) (123884.83)

= 0.951.

298. (a) Let the following notations be used:

X: Sales (Rs. in lakh) Y: Operating costs (Rs. in lakh) ˆ = Estimated operating costs (Rs. in lakh) Y

According to linear regression technique ˆ = a + bX Y

n ∑ XY − (∑ X) (∑ Y) where, b =

n ∑ X 2 − (∑ X) 2

322

and,

a=

X Y XY X2

Y − bX

54 60 21 22 1134 1320 2916 3600

n=6 X = b =

55 22 1210 3025

64 24 1536 4096

70 26 1820 4900

82 30 2460 6724

∑X ∑Y ∑XY ∑X2

= = = =

385 145 9,480 25,261

ΣY 145 ΣX 385 = Y= = n 6 n 6

(6) (9,480) − (385) (145) 1,055 = = 0.316 (approx.) (6) (25,261) − (385) (385) 3,341

⎛ 385 ⎞ 145 ⎜ ⎟ 6 ⎠ = 3.89 a = 6 - 0.316 ⎝

The relationship between sales and operating costs is ˆ = 3.89 + 0.316 X. Y

299. (c) For sales (X) = Rs.85 lakh the estimated operating cost is ˆ = 3.89 + 0.316 (85) = 30.75 i.e. Rs.30.75 lakh. Y 300. (d) x Marks in Maths 45 70 65 40 80 40 50 70 85 60 605

y Marks in Science 35 80 70 40 90 45 60 80 80 50 630

x2

y2

xy

2025 4900 4225 1600 6400 1600 2500 4900 7225 3600 38975

1225 6400 4900 1600 8100 2025 3600 6400 6400 2500 43150

1575 5600 4550 1600 7200 1800 3000 5600 6800 3000 40725

Σxy ⎛ Σx ⎞ ⎛ Σy ⎞ 40725 ⎛ 605 ⎞ ⎛ 630 ⎞ −⎜ −⎜ ⎟⎜ ⎟ ⎟⎜ ⎟ ⎝ 10 ⎠ ⎝ 10 ⎠ = 4072.5 - (60.5) (63) = 261. Cov (xy) = n ⎝ n ⎠ ⎝ n ⎠ = 10 301. (a)

σx

σy

=

Σx 2 n

⎛ Σx ⎞ −⎜ ⎟ ⎝ n ⎠

Σy 2 ⎛ Σy ⎞ −⎜ ⎟ n ⎝ n ⎠

2

2

=

38975 ⎛ 605 ⎞ −⎜ ⎟ 10 ⎝ 10 ⎠ 43150 ⎛ 630 ⎞ −⎜ ⎟ 10 ⎝ 10 ⎠

2

=

3897.5 − 3660.25

= 15.4

2

= = = Cov(xy) 261 261 σ . σ 15.4 x18.6 ∴ x y = = 286.44 = 0.91. 302. (b) Let the following notations be used: X: Average receivables balance (Rs. in lakh) Y: Bad debts

4315 − 3969

= 18.6

(Rs. in thousands)

323

A.

X

49

54

B.

Y

94

125 118

C.

-10

-5

-3

-30

1

-6

E.

X- X Y- Y C2

100

25

9

1

25 144

F.

D2

900

1

36

49

676 256

G.

CxD

300

-5

18

-7

130 192

D.

56

Σ (X − X) (Y − Y) r =

2

Σ (X − X) (Y − Y)

60

64

117

150 140

1 -7

628 2

=

304 x1918

71

5

12

26

16

∑X=

∑Y=

354 744

304 ∑( X − X)2 = 1918 ∑(Y − Y)2 = 628 ∑ (X − X)(Y − Y)

= 0.8224.

r2 =

a ∑ y + b ∑ xy − ny 2

∑ y 2 − ny 2 303. (a) To find coefficient of determination Percentage of variations in bad debts that is explained by the variations in accounts receivables = r2 x 100 = 0.6764 x 100 = 67.64%.

304. (b)

=

(1 - r2) 100

=

(1 - 0.6764) 100 = 32.36%.

305. (e) According to linear regression technique ˆ = a + bX Y n ∑ XY − (∑ X) (∑ Y)

Where, b = b=

n ∑ X 2 − (∑ X) 2

, or 10 x 67, 699 − 57 x10, 038 10 x 433 − (57) 2

= 96.692

and, a = Y − bX = 1003.8 ˆ = 1003.8 - 96.692 X. So Y ˆ = a + bX Where b is the slope of the line so the slope of the line in the 306. (c) Equation of line is Y above question is 6.01

The estimated height of a child who is 10 years = 50.3+60.1 = 110 cm The estimated intercept is 50.3 cm but it does not mean that children reach this height when they are 50.3/6.01=8.4 years old it mean minimum height of a children which age is 0 years is 50.3 , so alternatives (c) is not correct. 307. (e) X 65 63 67 64 68 62 70 66 68 67

Y 68 66 68 65 69 66 68 65 71 67

X2 4225 3969 4489 4096 4624 3844 4900 4356 4624 4489

Y2 4624 4356 4624 4225 4761 4356 4624 4225 5041 4489

XY 4420 4158 4556 4160 4692 4092 4760 4290 4828 4489

324

69 71 800

68 70 811

4761 5041 53418

4624 4900 54849

4692 4970 54107

According to linear regression technique ˆ = a + bX Y n ∑ XY − (∑ X) (∑ Y) n ∑ X 2 − (∑ X) 2

Where, b =

b =

, or

12 × 54107 − 811 × 800 12 × 54849 − (811) 2

and,

a = Y − bX = 1.479

So

ˆ = 1.479 + 1.036 X. Y

= 1.036

Σ (X − X) (Y − Y)

308. (a) r = X

Σ (X − X) 2 (Y − Y) 2

Y

XY

65 63 67 64 68 62 70 66 68 67 69 71 800

68 66 68 65 69 66 68 65 71 67 68 70 811

X = 66.66667

Y= 67.58333 =

(X − X)

4420 4158 4556 4160 4692 4092 4760 4290 4828 4489 4692 4970 54107

-1.67 -3.67 0.33 -2.67 1.33 -4.67 3.33 -0.67 1.33 0.33 2.33 4.33 -0.04 40.33

84.66X38.91

(Y − Y)

(X − X)

(X − X) 2

(Y − Y) 2

0.417 -1.583 0.417 -2.583 1.417 -1.583 0.417 -2.583 3.417 -0.583 0.417 2.417 0.004

(Y − Y) -0.69639 5.80961 0.13761 6.89661 1.88461 7.39261 1.38861 1.73061 4.54461 -0.19239 0.97161 10.46561 40.33332

2.7889 13.4689 0.1089 7.1289 1.7689 21.8089 11.0889 0.4489 1.7689 0.1089 5.4289 18.7489 84.6668

0.173889 2.505889 0.173889 6.671889 2.007889 2.505889 0.173889 6.671889 11.67589 0.339889 0.173889 5.841889 38.91667

= 0.70268

309. (b) The percentage of variations in Y that is explained by the variations X=r2 x 100

=

0.70268 x 100 = 70.27%.

310. (c)

x x

350 Σx = n = 10 = 35 310 Σy = n = 10 = 31

325

Σ(x − 35) (y − 31) Σ(x − x) (y − y) 10 n Cov (xy) = = = 9.2. 311. (c)

σx σy

=

Σ(x − x) 2 n =

Σ(x − 35) 2 10 =

162 10 =

16.2

=

Σ(y − y) 2 n =

Σ(y − 31) 2 10 =

222 10 =

22.2

∴r =

Cov(xy) σ x .σ y

9.2 =

16.2 x 22.2

= 0.49.

312. (c) We know that the regression coefficient b =

Cov(X, Y) V(X)

−121.5 = −1.5 ∴φ0 β = 81

Given that, ∑X = 55 and ∑Y = 33 and the number of observations n = 10 Therefore

X=

55 33 = 5.5 Y= = 3.3 10 10 and

Now, The Y-intercept, a = Y − bX = 3.3 - (− 1.5 × 5.5) = 11.55 ˆ = 11.55 - 1.5 X The estimating equation is Y ˆ = 11.55 - 1.5 × 5 = 4.05. The estimate of Y for X = 5, Y 313. (a) We know that the variation in Y explained by X1 if X2 is kept constant is given by the partial

R 212.3 =

coefficient of determination denoted by (0.7 − 0.5 × 0.6)

2

(r12 − r13 .r23 ) 2 (1 − r132 )(1 − r232 )

.

(0.7 − 0.3) 2 (0.75)(0.64)

0.42 = 0.3333 (1 − 0.52 )(1 − 0.62 ) = = 0.48 ≈ 0.33 i.e 33%. Therefore, 314. (d) Let X be the number of soccer games played and Y be the number of accidents during weekends. R 212.3 =

∑X = 146, ∑Y=48, n = 7 ∑XY =1101, ∑X2 = 3550, X = 20.86, Y = 6.86 ∑ XY − nXY 2

2

The coefficient of regression b = ∑ X − nX =

1101 − 7 × 20.86 × 6.86 3550 − 7 × (20.86) 2

= 0.197

The Y-intercept a = Y − b X = 6.86 -0.197 × 20.86 = 2.751 ˆ Therefore the estimating equation is Y = 2.751 + 0.197 X ˆ = 2.751 + 0.197 × 33 = 9.251 ≈ 9.25. The estimate of Y for X =33, is Y 315. (c) Let the number of advertisement shown be X and the number of cans of cola purchased be Y.

Then ∑X = 21, ∑Y= 56, n = 7

326

X = 3, Y = 8, ∑(X − X)(Y − Y) = 52, ∑(X − X)2 = 32 and ∑(Y − Y)2 = 132 ∑(X − X)(Y − Y)

52

∑(X − X)2 ∑(Y − Y) 2

The coefficient of correlation r =

=

32 × 132 = 0.8001.

316. (b) We observe that the number of data points is odd. The working is shown below:

Total

Years (X) 1999 2000 2001 2002 2003 10005

Inventory (Y) 4620 4910 5490 5730 5990 26740

x=X−X -2 -1 0 1 2 0

x.Y -9240 -4910 0 5730 11980 3560

x2 4 1 0 1 4 10

10005 The mean of “X” = 5 = 2001 ∑ xY

3560 = 356 2 ∑ x We have b = = 10

a = Y = 5348 ˆ Therefore the estimating equation is Y = 5, 348 + 356 x (in thousands rupees).

r 317. (b) Slope of a regression equation, b =

σy σx

0.80 × 5 × 8 = 1.28 25 =

318. (d) Proportion of variations in the dependent variable explained by the regression relationship = ESS 1530 1− = 1− = 0.903 TSS 15730 i.e. 90.3 percent. ˆ 319. (d) Regression equation : Y = a + bX

a = Y − bX Given: b = 2.5, ∑ X = 120, ∑ Y = 425, n = 10 ⎛ 425 ⎞ ⎛ 120 ⎞ ⎜ ⎟ − 2.5 ⎜ ⎟ = 12.5 ⎝ 10 ⎠ ∴φ0 α = ⎝ 10 ⎠ ˆ ∴φ0 Ρεγρεσσιον εθυατιον ισ Y = 12.5 + 2.5X

If X = 10, then 320. (c)

ˆ = 12.5 + ( 2.5 × 10 ) = 37.5. Y

Let the following notations be used: X : Age (in years) Y : Number of sick days

(r) = Coefficient of correlation

∑ ( X − X )( Y − Y ) ∑ (X − X) ∑ (Y − Y) 2

2

327

A B C . D . E . F G

X

21

31

33

3 6

41

47

53

56

59

63

∑X =

44 0

Y

2

3

1

4

5

7

6

5

9

8

∑Y =

50

X−X

-23

-13

-11

-3

3

9

12

15

19

Y−Y

-3

-2

-4

0

2

1

0

4

3

2

529

169

12 1

8 1 6 4

9

9

81

14 4

22 5

36 1

∑ (X − X)

2

9

4

16

1

0

4

1

0

16

9

∑ (Y − Y) =

69

26

44

8

0

6

9

0

60

57

∑ ( X − X )( Y − Y ) = 9

(X − X) (Y − Y) C×D

x=

∑ x = 440 = 44

y=

∑ y = 50 = 5

n

2

2

17 12 60 27

10

n

10

279 1712 × 60

∴φ0 ρ =

= 0.871

n ∑ XY − ( ∑ X )( ∑ Y )

b=

n∑ X2 − ( ∑ X )

321. (c)

2

Given: 2 n = 10 ∑ X = 140, ∑ X = 2528,

b=

∴φ0

∑ XY = 21040 , ∑ Y = 1300

(10 × 21040 ) − (140 ×1300 ) 28400 = =5 2 5680 (10 × 2528) − (140 )

Y − bX =

a=

1300 ⎛ 140 ⎞ − 5⎜ ⎟ = 60 10 ⎝ 10 ⎠

Standard error of estimate, Se =

2 ∑ Y − a ∑ Y − b ∑ XY n −2

2 ∑ Y = 184730

∴φ0

184730 − ( 60 × 1300 ) − ( 5 × 21040 )

Se =

10 − 2

= 13.83.

322. (b) Proportion of variation in Y, that is not explained by the regression line = 1- Coefficient of determination (r2) r2 =

a ∑ y+b ∑ XY − nY 2 2 ∑ Y − nY

2

2 ∑ Y = 180, ∑ Y = 5642

n ∑ XY − ∑ X.∑ Y

b=

n∑ X − ( ∑ X ) 2

2

=

( 6×1000 ) − ( 30×180 ) =2 ( 6× 200 ) − ( 30×30 )

328

⎛ 180 ⎞ ⎛ 30 ⎞ Y − bX = ⎜ ⎟ − 2 × ⎜ ⎟ = 20 ⎝ 6 ⎠ ⎝ 6 ⎠ a= r2 =

( 20 ×180 ) + ( 2 ×1000 ) − ( 6 × 302 )

∴φ0

5642 − ( 6 × 302 )

200 = 0.826 = 242

∴φ0 Προπορτιον οφ ϖαριατιον ιν Ψ τηατ ισ νοτ εξπλαινεδ βψ τηε ρεγρεσσιον λινε = 1−ρ2 = 1− 0.826 = 0.174 ⇒ Percentage of variations in Y that is not explained by the regression line = 17.4% 323. (e) Let the following notations be used:

X : Test score Y : Monthly sales (Rs. ‘000) Y = a + bX n ∑ XY − ∑ X.∑ Y

b=

n∑ X2 − ( ∑ X )

2

a = Y − bX 2 n = 10, ∑ XY = 21650, ∑ X = 500, ∑ Y = 400, ∑ X = 27400 b=

∴φ0

(10 × 21650 ) − ( 500 × 400 ) 216500 − 200000 = = 0.6875 (10 × 27400 ) − ( 500 × 500 ) 274000 − 250000

⎛ 400 ⎞ ⎛ 500 ⎞ Y − bX = ⎜ ⎟ − 0.6875 ⎜ ⎟ = 5.625 ⎝ 10 ⎠ ⎝ 10 ⎠ a= ) ∴φ2 Y = 5.625 + 0.6875X ) Y = 5.625 + ( 0.6875 × 90 ) = 67.5 For X = 90, (in Rs.’000)

∴φ0 Εστιµατεδ µοντηλψ σαλεσ = Ρσ.67,500. 324. (χ) Α σιµπλε ρεγρεσσιον εθυατιον ισ οφ τηε φορµ: ˆ = a + bX Y

Given: ˆ =31 If X = 5, Y

∴φ0 31 = α + 5β

(Α)

ˆ =47 Ιφ Ξ = 10, Y

∴φ0 47 = α + 10β

..(Β)

Συβτραχτινγ (Α) φροµ (Β) ωε γετ: 47 − 31 = (10 − 5)β 16 = 5β 16 = 3.2 ορ β = 5

Putting the values of b in (B): 47 = a + 10 × 3.2 = a + 32

329

or a = 47 - 32 = 15 ˆ =15 + 3.2X ∴φ2 Y ⎛ ∑X ⎞ Y − bX = Y − b ⎜ ⎟ ⎝ n ⎠ Now, a = ⎛ ∑X ⎞ ⎛ 60 ⎞ Y = a + b⎜ ⎟ = 15 + 3.2 ⎜ ⎟ = 47 n ⎝ ⎠ ⎝ 6 ⎠ ∴φ2

∴φ0 Μεαν οφ τηε οβσερϖεδ ϖαλυεσ οφ Ψ, ι.ε. Y = 47. 325. (b) Let the following notations be used:

X

: Years

x

: Coded value of year

Y : Production of sugar (thousand quintals) X Y x = X−X 1996 -3 75 1997 -2 85 1998 -1 87 1999 0 80 2000 1 89 2001 2 95 2002 3 90 ∑ Y = 601

xY -225 -170 -87 0 89 190 270 ∑ xY = 67

x2 9 4 1 0 1 4 9 2 ∑ x = 28

∑ X = 13993 ∴φ0

X=

13993 = 1999 7

ˆ = a + bx Y b=

∑ xY 67 = = 2.393 2 28 ∑x

a=Y=

601 = 85.857 7

ˆ ∴φ2 Y = 85.857 + 2.393x

For the year 2003 : x = 2003 - 1999 = 4 ∴φ0 Εστιµατεδ συγαρ προδυχτιον = 85.857 + (2.393 × 4 ) = 95.429 (thousand quintals) i.e. 95429 quintals.

Multiple Regressions 326. (c)

Y 7.88 7.43 8.38 7.42 7.97 7.49 8.84

X1

X2

3 2 4 2 3 2 5

2 1 3 1 2 2 3

X12 9 4 16 4 9 4 25

X 22

X1X2

X1Y

X2Y

4 1 9 1 4 4 9

6 2 12 2 6 4 15

23.64 14.86 33.52 14.84 23.91 14.98 44.20

15.76 7.43 25.14 7.42 15.94 14.98 26.52

330

8.29 Total 63.7

4 25

2 16

16 87

4 36

8 55

33.16 203.11

16.58 129.77

To fit an equation of the form Y = a + b1X1 + b2X2 We use the following normal equations. ΣY

= na + b1ΣX1 + b2ΣX2

ΣX1Y

= aΣX1 + b1

∑ X12

+ b2ΣX1X2

2 ΣX2Y = aΣX2 + b1ΣX1X2 + bΣ X 2 Substituting the respective values, we have 63.7 = 8a + 25b1 + 16b2 .........(1) 203.11 = 25a + 87b1 + 55b2 .........(2) 129.77 = 16a + 55b1 + 36b2 .........(3) (1) x 2 is 127.4 = 16a + 50b1 + 32b2 (3) x 1 is 129.77 = 16a + 55b1 + 36b2 Subtracting, we get -2.37 = -5b1 - 4b2 That is, 2.37 = 5b1 + 4b2 ............ (4) (1) x 25 is 1592.5 = 200a + 625b1 + 400b2 (2) x 8 is 1624.88 - 200a + 696b1 + 440b2 Subtracting, we get -32.38 = -71b1 - 40b2 That is, 32.38 = 71b1 + 40b2 .........(5) Now (5) x 1 is 32.38 = 71b1 + 40b2 (4) x 10 is 23.70 = 50b1 + 40b2 Subtracting, we have 8.68 = 21b1

8.68 = 0.4133 21 Substituting the value of b1 = 0.4133 in (4) ⇒ b1 =

2.37 = 5 (0.4133) + 4 b2 ⇒

4 b2 = 0.3035

⇒ b2 = 0.0759 Substituting the values of b1 = 0.4133 and b2 = 0.0759 in (1) 63.7 = 8a + 25 (0.4133) + (16) (0.0759) 63.7 = 8a + 10.3325 + 1.2144 ⇒ 8a = 52.1531 ⇒ a = 6.5191 ˆ = 6.5191 + 0.4133 X + 0.0759 X . The multiple regression equation is given by Y 1 2

327. (a) Y 2.4 2.1 3.2 5.6 9.3 14.6 21.9

X 0 1 2 3 4 5 6

X2 0 1 4 9 16 25 36

X3 0 1 8 27 64 125 216

X4 0 1 16 81 256 625 1296

XY 0 2.1 6.4 16.8 37.2 73.0 131.4

X2Y 0 2.1 12.8 50.4 148.8 365.0 788.4

331

Total 59.1 21 91

441

2275 266.9

1367.5

We have to fit a parabola of the form Y = a + bX + cX2 The required normal equations are ΣY

= na + bΣX + cΣX2

ΣXY

= aΣX + bΣX2 + cΣX3

ΣX2Y

= aΣX2 + bΣX3 + cΣX4

Substituting the respective values, we have 59.1

= 7a + 21b + 91c

................(1)

266.9

= 21a + 91b + 441c ................(2)

1367.5 = 91a + 441b + 2275c ................(3) (1) x 3 is 177.3 = 21a + 63b + 273c (2) x 1 is 266.9 = 21a + 91b + 441c Subtracting, we get -89.6

= - 28b - 168c

That is, 89.6 = 28b + 168c

...............(4)

(1) x 13 is 768.3 = 91a + 273b + 1183c (3) x 1 is 1367.5 = 91a + 441b + 2275c Subtracting, we get -599.2

= -168b - 1092c

That is, 599.2

= 168b + 1092c ...........(5)

(4) x 6 is 537.6

= 168b + 1008c

(5) x 1 is 599.2

= 168b + 1092c

Subtracting, we get -61.6

= -84c

That is, c = 0.7333 Substituting the value of c in (5), we have 599.2

= 168b + (1092) (0.7333)

-201.5636 = 168b ⇒ b = - 1.1998 Substitute the values of c = 0.7333 and b

=

-1.1998 in (1) 59.1 = 7a + 21 (-1.1998) + 91 (0.7333)

a

=

2.5094

The required parabola is Y

= 2.5094 - 1.9998X + 0.7333X2.

328. (a)

i.

Degrees of freedom of ESS = Degrees of Freedom of TSS - Degrees of Freedom of RSS = 11 - 2 = 9

ii.

ESS

= TSS - RSS = 453.19 - 309.99 = 143.20

332

iii.

RSS 309.99 = 2 = 154.95 Mean RSS = k

[k = Degrees of freedom of RSS]. 329. (b)

i.

Mean ESS =

ESS 143.20 = n − (k + 1) 9

= 15.91

[n = number of data points in the sample]

ii.

iii.

F ratio =

Mean RSS Mean ESS

154.95 = 15.91 = 9.7392

ESS n − (k + 1)

Root MSE =

= 15.91 = 3.9887.

330. (c)

i.

Coefficient of determination, RSS 309.99 = = 0.684 = TSS 453.19

R2

b i − Bi o

ii.

t

=

Here

iii.

t=

Bi o

Sb i

= 0. Therefore, t =

b i 2.139 = Sb i 1.47

bi Sb i

0.625 = 0.558. = 1.12

= 1.455.

331. (e)

i.

Degrees of freedom of RSS = Degrees of freedom of TSS - Degrees of freedom of ESS. = 24 - 20 = 4.

ii.

TSS = RSS + ESS = 1050.78 + 83.88 = 1134.66

iii.

RSS 1050.78 k 4 Mean RSS = = = 262.695.

332. (a)

i.

ii.

iii.

Mean ESS =

F ratio =

ESS 83.88 = n − (k + 1) 20

Mean RSS Mean ESS

Root MSE =

= 4.194

262.695 = 4.194 = 62.6359

ESS n − (k + 1)

=

4.194 = 2.0479.

333. (d) i. Coefficient of determination,

333

R2 =

RSS 1050.78 TSS = 1134.66 = 0.9261 b i − Bi o

ii.

t

Sb i

=

Here, Bi = 0. Therefore t =

iii.

t

=

bi Sb i

bi Sb i

0.4216 = 0.17192 = 2.4523

0.2714 = 0.2184 = 1.2427.

334. (c) Year 1977 1978 1979 1980 1981 1982 Total

a

b

X -5 -3 -1 1 3 5 0

=

=

Y 4.0 4.5 6.0 8.0 8.5 10.0 41.0

Y=

Y2 16.00 20.25 36.00 64.00 72.25 100.00 308.50

X2 25 9 1 1 9 25 70

X2Y 100.0 40.5 6.0 8.0 76.5 250.0 481.0

XY -20.0 -13.5 -6.0 8.0 25.5 50.0 44.0

X4 625 81 1 1 81 625 1414

ΣY 41 = = 6.8333 n 6

ΣXY 44 = = 0.6286 ΣX 2 70

Estimating equation will be Y = 6.8333 + 0.6286X. 335. (d)

The equations required for second degree estimation are na + bΣX + cΣX2

= ΣY

aΣX + bΣX2 + cΣX3 = ΣXY aΣX2 + bΣX3 + cΣX4 = ΣX2Y Substituting the values of X, X2, etc. from the table we have 6a + b(0) + 70c

= 41

.....(1)

a(0) + 70b + c(0)

=4

.....(2)

70a + b(0) + 1414c = 481

.....(3)

Equation (2) yields 70b = 44 i.e., b = 0.6286 Multiplying (1) by 70 and (2) by 6 we have 420a + 4900c = 2870

.....(4)

420a + 8484c = 2886

.....(5)

334

Subtracting (4) from (5) we have 3584c

= 16

or c

= 0.0045

Substituting c = 0.0045 in (1) we have 6a + 70(0.0045) = 41 or 6a

= 40.685

or a

= 6.781

Therefore, the second degree equation will be Y = 6.781 + 0.6286X + 0.0045X2. 336. (e) Estimated value of inventory in the year 1984 (x = 9) will be

i.

ˆ Y

=

6.8333 + 0.6286 (9) = $12.49 lakh (using the linear equation)

ii.

Y

=

6.781 + 0.6281 (9) + 0.0045(92)

=

$12.803 lakh (using the second-degree equation).

337. (e) ˆ = a + b + b be the equation of multiple regression plane Let Y 1x1 2 x2

then -1 = a + 2b1, + 3b2 3 = a + b1 + 5b2

…. (i) .… (ii)

1 = a + b1 - b 2

.…(iii)

Subtracting (ii) from (i) we get 4 = -b1 + 2b2

..... (1)

1 ∴φ0 −2 = −6β2 ⇒ b2 = 3 2 Substituting b2 in (1) we get 4 = -b1 + 3 −10 Hence b1 = 3

Substituting b1 in the equation -1 = a + 2b1 + 3b2 we get 20 3 3 -1 = a = + 3 14 ∴φ0 α = 3

Hence, the required equation is Y=

14 10 1 − X1 + X 2 . 3 3 3

335

338. (a) Y=

14 − 10X1 + X 2 14 10 1 − X1 + X 2 = 3 3 3 3

Y Value table -4.5 -5 59.9/3 64.9/3 60/3 65/3 60.1/3 65.1

X2/X1 0.9 1.0 1.1 Range of

Y=

-5.5 69.9/3 70/3 70.1/3

70.1 − 59.9 10.2 + = = 3.4 3 3 3

339. (d) A multiple regression equation is obtained by solving the following equations:

∑ Y = na + b ∑ X 1

1

+ b 2 ∑ X2

∑ X Y = a∑ X + b ∑ X + b ∑ X X ∑ X Y = a∑ X + b ∑ X X + b ∑ X 2

1

1

1

1

2

1

2

2

2

2

1

1

2

2

2

Putting in the relevant values in the equation : 272

= 10a + 441 b1 + 147b2………………….(A)

12005

= 441a + 19461 b1 + 6485 b2……………….(B)

4013

= 147a + 6485 b1 + 2173 b2 ……………...(C)

Multiplying (A) by 441 and (B) by 10, and subtracting (B) form (A): 119952 = 4410a + 194481b1 + 64827 b2 64850b2 120050 = 4410a + 194610b1 + - 23b2 - 98 = - 129b1 ∴φ0 98 = 129 β1 + 23β2

(∆)

Μυλτιπλψινγ (Β) βψ 147 ανδ (Χ) βψ 441, ανδ συβστραχτινγ (Χ) φροµ (Β): 953295 β2 1764735 64827α + 2860767β1 + = 958293β2 1769733 64827α + − 2859885β1 + − =− − − 4998 = 882β1 − 4998β2 ∴φ0 4998

=

− 882β1

+

4998β2

.(Ε)

Μυλτιπλψινγ (∆) βψ 882 ανδ (Ε) βψ 129, ανδ αδδινγ τηε τωο εθυατιονσ: 86436 = 113778β1 + 20286 β2 644742β2 644742 = − 113778 β1 + 731178 = 665028β2 731178 = 1.099 ∴φ0 β2 = 665028 98 − 25.277 = 0.564 129 Putting the values of b2 in (D) : = 129 b1 + (23 × 1.099) or b1 =

Putting the values of b1 and b2 (A): = 10a + (441 × 0.564) + (147 × 1.099) 272 − 248.724 − 161.553 = −13.828 10 or a =

336

ˆ =a+b x +b x ∴φ0 Τηε µυλτιπλε ρεγρεσσιον εθυατιον ισ : Y 1 1 2 2 ˆ = −13.828 + 0.564X + 1.099X Y 1 2

For X1 = 40 and X 2 = 10 : ˆ = −13.828 + ( 0.564 × 40 ) + (1.099 × 10 ) = 19.722. Y

340. (e)

In multiple regression relationship:

Standard error of estimate, Given :

)

2

n − k −1

=

ESS n − k −1

ESS = 576 n

∴φ0 4.8 =

se =

(

ˆ ∑ Y−Y

=?

k

=2

se

= 4.8 ESS n − 2 −1

ESS = 23.04 n ∴φ2 − 3 576 = 23.04 or n − 3 576 + 3 = 28. 23.04 or n =

Index Numbers 341. (a)

2001 P0 48,500 41,900 38,750 36,300 33,850 1,99,300

2002 2003 P1 P2 A 55,100 63,800 B 46,200 60,150 C 43,500 46,700 D 45,400 39,900 E 38,300 50,200 Total 2,28,500 2,60,750 ΣP1 ΣP0 Unweighted aggregate index = x 100 1,99,300 x 100 = 100 Index for 2001 = 1,99,300 2,28,500 Index for 2002 = 1,99,300 x 100 = 114.6513 2,60,750 x 100 = 130.8329 Index for 2003 = 1,99,300

342. (b) Calculation of Laspeyres Index Model 2001 P0 2002 2003 P2 P1

2001 Q0

P0Q0

P1Q0

P2Q0

337

Vision 2000 Emperor Prince Crown Pearl Total

17.0 13.1 11.2 8.5 5.5

17.9 13.6 11.4 8.7 5.6

Laspeyres price index for 2002 =

18.6 13.7 11.9 8.8 5.8

11.2 8.0 9.6 11.2 12.0

ΣP1Q0 x100 ΣP0 Q0 ΣP2 Q 0 ΣP0 Q 0

190.4 104.8 107.52 95.2 66.0 563.92 583.36 = 563.92

200.48 108.8 109.44 97.44 67.2 583.36

208.32 109.60 114.24 98.56 69.6 600.32

x 100 = 103.4472

600.32 x 100 = 563.92 x 100 = 106.4548.

Laspeyres price index for 2003 = 343. (e) Calculation of Paasche’s index Model 2001 2002 2002 2003 2003 P0Q1 P1Q1 P0Q2 P1 Q1 P2 Q2 P0 Vision 2000 17 17.9 12 18.6 12 204.00 214.00 204.00 Emperor 13.1 13.6 9.6 13.7 11.2 125.76 130.56 146.72 Prince 11.2 11.4 10.4 11.9 12 116.48 118.56 134.40 Crown 8.5 8.7 10.4 8.8 12 88.40 90.48 102.00 Pearl 5.5 5.6 14.4 5.8 18 79.20 80.64 99.00 Total 613.84 635.04 686.12 ΣP Q 1 1 x100 = 635.04 ΣP Q 613.84 0 1 Paasche’s price index for 2002 = x 100 = 103.4536

P2Q2 223.20 153.44 142.80 105.60 104.40 729.44

ΣP Q 2 2 x100 729.44 x100 ΣP Q 0 2 = 686.12 = 106.3137. Paasche’s price index for 2003 = Laspeyres Index x Paasche's Index 344. (c) Fisher’s price index =

Fisher’s price index for 1996 = Fisher’s price index for 1997 = 345. (d) Model

(103.4472) (103.4536) (106.4548) (106.3137)

= 103.4504 = 106.3842.

2001 P0

2002 P1

2003 P2

P1 x100 P0

P2 x100 P0

Vision 2000 17 Emperor 13.1 Prince 11.2 Crown 8.5 Pearl 5.5

17.9 13.6 11.4 8.7 5.6

18.6 13.7 11.9 8.8 5.8

105.2941 103.8167 101.7857 102.3529 101.8181 515.0677

109.4117 104.5801 106.2500 103.5294 105.4545 529.2258

⎛P ⎞ 515.0677 Σ ⎜ 1 x 100 ⎟ = = 103.0135 P 5 ⎠ Unweighted average of relatives price index for 2002= ⎝ 0 529.2258 =105.8452. Σ ⎛⎜ P2 x100 ⎞⎟ = P0 5 ⎝ ⎠ Unweighted average of relatives price index for 2003 = 346. (d) 2001 as the base year. Weighted average of relatives index. P0 P2 Q0 P0Q0 P1 P2 P1 ⎞ ⎞ ⎛ P1 ⎛ P2 x 100 x 100 ⎜ x 100 ⎟ ⎜ x 100 ⎟ ⎟ ⎟ ⎜P ⎜P P0 P0 ⎠ ⎠ ⎝ 0 ⎝ 0 (P0Q0) (P0Q0) 17 17.9 18.6 12.0 94.9721 103.9106 214.80 20,400 22,320

338

13.1 11.2 8.5 5.5 Total

13.6 11.4 8.7 5.6

13.7 11.9 8.8 5.8

9.6 10.4 10.4 14.4

96.3235 98.2456 97.7011 98.2143

⎡⎛ P

=

100.7352 104.3859 101.1494 103.5714

130.56 118.56 90.48 80.64 635.04

12,576 11,648 8,840 7,920 61,384

13,152 12,376 9,152 8,352 65,352

⎞⎤

∑ ⎢⎜ P1 x100 (P0 Q0 ) ⎟ ⎥ ⎣⎝

⎠⎦

0

ΣP0 Q0 61384

= 96.6616 Price index for 2001 = 635.04 65352 =102.9101. Price index for 2002 = 635.04 347. (a) Laspeyres Quantity indices for 2002 and 2003.

2001 Q0 11.2 8.0 9.6 11.2 12.0 Total

2002 Q1 12.0 9.6 10.4 10.4 14.4

2003 Q2 12.0 11.2 12.0 12.0 18.0

2001 P0 17.0 13.1 11.2 8.5 5.5

Laspeyres quantity index for 2002 =

Q0P0 Q1P0 Q2P0 190.40 204.00 204.00 104.80 125.76 146.72 107.52 116.48 134.40 95.20 88.40 102.00 66.00 79.20 99.00 563.92 613.84 686.12 ΣQ1P0 613.84 x 100 = x 100 =108.8523 ΣQ 0 P0 563.92

Laspeyres quantity index for 2003 =

ΣQ 2 P0 686.12 x 100 = x 100 =121.6697 ΣQ 0 P0 563.92

348. (b) Paasche’s Quantity Indices for 2002 and 2003 Q0 Q1 Q2 P1 P2 Q0P1 Q1P1 Q0P2 Q2P2 11.2 12.0 12.0 17.9 18.6 200.48 214.80 208.32 223.20 8.0 9.6 11.2 13.6 13.7 108.80 130.56 109.60 153.44 9.6 10.4 12.0 11.4 11.9 109.44 118.56 114.24 142.80 11.2 10.4 12.0 8.7 8.8 97.44 90.48 98.56 105.60 12.0 14.4 18.0 5.6 5.8 67.20 80.64 69.60 104.40 Total 583.36 635.04 600.32 729.44 ΣQ1P1 635.04 x 100 = x 100 =108.859 ΣQ 0 P1 583.36 Paasche’s Quantity Index for 2002 =

Paasche’s Quantity Index for 2003 = 349. (a) 2001

2002

2003

Q0 11.2 8.0 9.6 11.2 12.0

Q1 12.0 9.6 10.4 10.4 14.4

Q2 12.0 11.2 12.0 12.0 18.0

ΣQ 2 P2 729.44 x 100 = x 100 =121.5085 ΣQ 0 P2 600.32

Q1 x 100 Q0

Q2 x 100 Q0

107.1428 120.0000 108.3333 92.8571 120.0000

107.1428 140.0000 125.0000 107.1428 150.0000

339

Total

548.3333

629.2857

Unweighted average of relatives quantity index using 2001 as the base. ⎛Q ⎞ Σ ⎜⎜ 1 x 100 ⎟⎟ ⎝ Q0 ⎠ = 548.3333 n 5 = 109.6666 Index for 2002 = 629.2857 5 = 125.8571. Index for 2003 =

350. (a) Q0

11.2 8.0 9.6 11.2 12.0 Total

Q1

12.0 9.6 10.4 10.4 14.4

Q2

12.0 11.2 12.0 12.0 18.0

P0

17.0 13.1 11.2 8.5 5.5

Q0P0

190.40 104.80 107.52 95.20 66.00 563.92

Q1 x100 Q0

Q2

107.1428 120.0000 108.3333 92.8571 120.0000

107.1428 140.0000 125.0000 107.1428 150.0000

Q0

x100

⎛ Q1 ⎞ x100 ⎟ ⎜⎜ ⎟ Q ⎝ 0 ⎠

⎛ Q2 ⎞ x100 ⎟ ⎜⎜ ⎟ Q ⎝ 0 ⎠

(P0Q0) 20400 12576 11648 8840 7920 61384

(P0Q0) 20400 14672 13440 10200 9900 68612

Relative quantity indices with 2001 as the base year and 2001 values as weights. 61384 = 108.8523 Index for 2002 = 563.92 68612 = 121.6697. Index for 2003 = 563.92

351. (b) P0 17 13.1 11.2 8.5 5.5 Total

P1 17.9 13.6 11.4 8.7 5.6

P2 18.6 13.7 11.9 8.8 5.8

Q0 11.2 8.0 9.6 11.2 12.0

Q1 12.0 9.6 10.4 10.4 14.4

Q2 12.0 11.2 12.0 12.0 18.0

P0Q0 190.40 104.80 107.52 95.20 66.00 563.92

214.80 130.56 118.56 90.48 80.64 635.04

P2Q2 223.20 153.44 142.80 105.60 104.40 729.44

635.04 563.92 The value index for 2002 with 2003 as the base = x 100 = 112.6117. 729.44 The value index for 2003 with 2002 as the base = 635.04 x 100 = 114.8652

The growth rate for the period 2001-2002 as given by the value index for 2002 is 12.61%. The growth rate for the period 2002-2003 as given by the value index for 2003 is 114.8652 −112.6117 112.6117 = 2.001% = The growth rate for the period 2001-2003 is given by

(1.1261) (1.1786)

- 1 = 0.1521 or 15.21%.

120 + 30 352. (e) The employment index number is Quantity relative = 120 = 1.25 or 125%.

340

2,40,000 + 36,000 2,40,000 353. (d) The labor expense index number is value index = = 1.15 or 115%.

354. (a) P0 Q0 P1 4 16 6 7 10 7 7 25 10 3 18 6 Total Laspeyres Index

Q1 13 14 32 22

P0Q0 P0Q1 P1Q0 P1Q1 64 52 96 78 70 98 70 98 175 224 250 320 54 66 108 132 363 440 524 628 = ΣP1Q0 524 x100 = x100 =144.3526 363 ΣP0 Q0 Paasche’s Index = ΣP1Q1 628 x100 = x100 =142.7273 440 ΣP0 Q1

355. (c) Marshall-Edge worth’s Index =

=

ΣP1Q 0 + ΣP1 Q1 x 100 ΣP0 Q 0 + ΣP0 Q1

356. (d) Fisher’s Index

=

.

Σ(Q 0 + Q1 ) P1 x 100 Σ (Q 0 + Q1 ) P0

524 + 628 x 100 363 + 440

1152 x100 =143.462. = 803

=

Laspeyres Index×Paasches Index

=

144.3526 x142.7273

= 143.5376.

357. (b) Material Average Annual Price Value of P1 P2 Purchase P x 100 P x 100 (in $) P1 (in $) P0 0 0 P2 P0 Q0

Butadiene Styrene Rayon Card Carbon Black Sodium Pyrophosp hate Total

⎛ P1 ⎞ ⎜ x 100 x P0 Q0 ⎟ ⎜P ⎟ ⎝ 0 ⎠

⎛ P2 ⎞ ⎜ x 100 x P0 Q 0 ⎟ ⎜P ⎟ ⎝ 0 ⎠

17 85 348

15 89 358

11 95 331

50 210 1,640

155 90 105

136 94 108

7,750 18,900 1,72,200

(in 000’s of $) 6,800 19,740 1,77,120

62

58

67

630

93

87

58,590

54,810

49

56

67

90

73

84

6,570

7,560

2,64,010

2,66,030

2,620

2,64,010 = 100.78 Index for 1981 = 2,620 2,66,030 = 101.54 Index for 1982 = 2,620 Index for 1983 = 100.

358. (e)

Model

Number Sold 2001 2002

2003

2001

Q0 P0

Q1 P0

Q2 P0 341

Sport Touring Cross-Country Sprint Total

Q1 48 67 35 16

Q0 45 64 28 21

Q2 56 71 27 28

P0 89 104 138 245

4,005 6,656 3,864 5,145 19,670

4,272 6,968 4,830 3,920 19,990

4,984 7,384 3,726 6,860 22,954

∑ Q1 P0 Q P Weighted Average of Relative Quantity Indices = ∑ 0 0 x 100 for 2002

Index for 2002

∑ Q 2 P0 19,990 for 2003 ∑ Q0 P0 19,670 = x 100 = 101.63 and

Index for 2003

22,954 = 19,670 x 100 = 116.70.

359. (b) Material

Prices (in $) 2000 P0 2001 P1 2002 P2 2003 P3 Aluminum 0.96 0.99 1.03 1.06 Steel 1.48 1.54 1.55 1.59 Brass Tubing 0.21 0.25 0.26 0.31 Copper Wire 0.06 0.08 0.07 0.09 Total 2.71 2.86 2.91 3.05 ∑ P1 2.86 P0 ∑ Index for 2001 = x 100 = 2.71 x 100 = 105.5 ∑ P2 2.91 P0 ∑ x 100 = 2.71 x 100 = 107.4 Index for 2002 = ∑ P3 3.05 P0 ∑ x 100 = 2.71 x 100 = 112.5. Index for 2003 =

360. (b) Q0 2,342 676 604 292 266 380 526 165

P0 6.50 4.50 3.90 3.70 3.75 4.20 4.20 14.30

la P01

= 361. (a) Q0 2,342 676 604 292 266 380

∑ P1 Q0 ∑ P0 Q0

P0 6.50 4.50 3.90 3.70 3.75 4.20

Q1 3,061 1,072 750 360 230 440 1,183 255 Total x

P1 9.3 6.5 6.2 6.4 6.2 5.8 11.2 22.5

P0 Q 0 15,223.0 3,042.0 2,355.6 1,080.4 997.5 1,596.0 2,209.2 2,359.5 28,863.2

P1 Q 0 21,780.6 4,394.0 3,744.8 1,868.8 1,649.2 2,204.0 5,894.2 3,712.5 45,245.1

45, 245.1 28,863.2 x 100 = 156.76. 100 ⇒

Q1 3,061 1,072 750 360 230 440

P1 9.3 6.5 6.2 6.4 6.2 5.8

P1 Q1 28,467.3 6,968.0 4,650.0 2,304.0 1,426.0 2,552.0

P0 Q1 19,896.5 4,824.0 2,925.0 133.2 862.5 1,848.0 342

526 4.20 1,183 11.2 13,249.6 165 14.30 255 22.5 5,737.5 Total 65,354.4 65,354.4 pa ∑ P1 Q1 P01 P0 Q1 40,303.1 x 100 = 162.1572. ∑ = x 100 ⇒ 362. (b) Q0 2,342 676 604 292 266 380 526 165 Total ME P01

P0 6.50 4.50 3.90 3.70 3.75 4.20 4.20 14.30

Q1 3,061 1,072 750 360 230 440 1,183 255

P1 9.3 6.5 6.2 6.4 6.2 5.8 11.2 22.5

Q 0 + Q1 5,403 1,748 1,354 652 496 820 1,709 420

∑ P1 (Q0 + Q1 ) P0 (Q0 + Q1 ) ∑ = x 100 ⇒

4,968.6 3,646.5 40,303.1

Pl(Q0 + Ql) P0 (Q0 +Ql) 50,247.9 35,119.5 11,362.0 7,866.0 8,394.6 5,280.6 4,172.6 2,412.4 3,075.2 1,860.0 4,756.0 3,444.0 19,140.8 7,177.8 9,450.0 6,006.0 1,10,599.1 69,166.3 1,10,599.1 69,166.3 x 100 = 159.9.

363. (a) Let X be the percentage price relative for B in 2002 ⎡120 ⎤ ⎡ X ⎤ ⎢100 x 100⎥ + ⎢100 x 100⎥ ⎣ ⎦ ⎣ ⎦ 2 = 130 then

i.e. 120 + X = 260 i.e.

X = 140

Similarly, if Y is the percentage price relative for B in 2003 ⎡135 ⎤ ⎡ Y ⎤ ⎢100 x 100⎥ + ⎢100 x 100⎥ ⎣ ⎦ ⎣ ⎦ 2 = 150

i.e. 135 + Y = 300 i.e.

Y = 165

Calculation of Weighted Average Price Index

Year 2001 2002

A B 100 100 120 140

2003

135 165

Index 100 120 x 3 + 140 x 1 400 x 100 = 125 135 x 3 + 165 x 1 400 x 100 = 142.5

364. (d) Unweighted aggregates price index for 2003 using 2001 as base = Unweighted aggregates quantity index for 2003 using 2001 as base 35 + 25 + 20 80 30 + 25 + 20 x 100 = 75 x 100 = 106.67. = 365. (a) Weighted aggregates price index for 2003 using Laspeyres method

1 + 3 + 3.50 0.50 + 2 + 3

= 136.36

343

(1 x 30) + (3 x 25) + (3.50 x 20) (0.50 x 30) + (2 x 25) + (3 x 20)

175 = x 100 = 125 x 100 = 140 Weighted aggregates price index for 2003 using Paasche method (1 x 35) + (3 x 25) + (3.50 x 20) 180 (0.50 x 35) + (2 x 25) + (3 x 20) 127.50 = x 100 = x 100 = 141.18. 366. (d) Unweighted average of relative price index for 2003

[(1/0.5) x 100]+ (3/2 x 100) + (3.5/3 x 100) 3

=

200 + 150 + 116.67 3 = = 155.56

Unweighted average of relative quantity index for 2003

[(35/30 x 100) + (25/25 x 100) + (20/20 x 100)] 3

=

116.67 + 100 + 100 3 = = 105.56.

367. (c) We know that Fisher index is Geometric mean of Laspeyres and Paasche’s Index ∴ Fisher Index

=

∴ Paasche’s Index =

368. (a) Year

1999 2000 2001 2002 2003

Laspeyres Index × Paasche' s Index (Fisher's Index) 2 Laspeyres Index

Index movement (%) 4.3 6.7 -4.8 2.1 -4.6

(108.91) 2 = 103.28 =114.85.

Value of the index 104.30 111.29 105.95 108.17 103.19

Value of Index of the year = (Previous year Index + Index movement of the year) The year 1993 is taken as the base year. The value of index during the base year is 100. 369. (b) Calculations for Marshall-Edgeworth’s Index Commodity 2001 2002 poqo Price Quantity Price Quantity po qo p1 q1 A 40 10 50 7 400 B 20 5 30 8 100 C 30 6 40 10 180 D 10 9 20 10 90 Total 770 ∑p q 0 0 =

poq1

p1qo

p1q1

280 160 300 100 840 ∑ p q1 0 =

500 150 240 180 1070 ∑p q 1 0 =

350 240 400 200 1190 ∑p q 1 1 =

∑ p1 (q o + q1 ) ∑ p1q o + ∑ p1q1 p q + p q p (q + q ) ∴φ0 Μαρσηαλλ−Εδγεωορτη σ Ινδεξ = ∑ o o 1 x 100 = ∑ o o ∑ o 1 x 100

=

1070 + 1190 770 + 840

2260 x 100 = 1610 x 100 = 140.37.

∑ p1q o 1070 10700 poq o ∑ x 100 = 770 x 100 = 77 = 138.96. 370. (d) =

344

371. (d) The formula for calculating the price index through Marshall-Edgeworth method is as follows: ∑ (Q 0 + Q1 ) P1 ∑ (Q 0 + Q1 )P0 x 100 Commodity P0 Q0 P1 A 5 13 8 B 7 8 12 C 8 6 11 D 3 16 4

Q1 12 9 10 25

Q 0 + Q1 25 17 16 41

P1 (Q0 + Q1 ) 200 204 176 164 ∑P1 (Q0 + Q1) = 744

P0 (Q0 + Q1) 125 119 128 123 ∑P0 (Q0 + Q1) = 495

744 Price Index = 495 x 100 = 150.30. 372. (b) Item P0 P1 Q0 Q1 P1Q0 P1Q1 P0Q0 P0Q1

Bread 6.00 8.00 360.00 500.00 2,880.00 4,000.00 2,160.00 3,000.00

Milk 10.00 15.00 540.00 600.00 8,100.00 9,000.00 5,400.00 6,000.00

Fisher’s ideal price index =

=

Eggs 12.00 18.00 240.00 300.00 4,320.00 5,400.00 2,880.00 3,600.00

Mutton 80.00 120.00 100.00 80.00 12,000.00 9,600.00 8,000.00 6,400.00

∑ P1 Q 0 ∑ P1 Q1 x ∑ P0 Q 0 ∑ P0 Q1 27,300 28,000 x 18,440 19,000

Total

∑P1Q0 = 27,300 ∑P1Q1 = 28,000 ∑P0Q0 = 18,440 ∑P0Q1 = 19,000

x 100

x 100 = 147.71.

373. (a)

Sales person A B C D E

2000

2001

2002

E0 48500 41900 38750 36300 33850 ∑ E0 = 199300

E1 55100 46200 43500 45400 38300 ∑ E1 = 228500

E2 63800 60150 46700 39900 50200 ∑ E2 = 260750

Unweighted aggregates index: ∑ E1 228500 E 2001: ∑ 0 x 100 = 199300 x 100 = 114.65 ∑ E2 260750 E 2002: ∑ 0 x 100 = 199300 x 100 = 130.83.

374. (a) Let the following notations be used: Base year = 2001 Current year = 2002 Base year prices (Rs./kg) = P0 Current year prices (Rs./kg) = P1 Base year quantities (kgs) = Q0 Current year quantities (kgs) = Q1

345

Fruit

Prices P0 Mangoes 15 Oranges 45 Apples 60 Papaya 8 Grapes

P1 25 65 65 12

Quantity Q1 26 30 25 45

P0Q1 390 1,350 1,500 360 18

P1Q1 650 1,950 1,625 540 2 28 5

346

Part III: Model Question Papers (with Suggested Answers) The model question paper consists of two parts – A and B. Part A is intended to test the conceptual understanding of the students. It contains around 40 multiple-choice questions carrying one point each. Part B contains problems with an aggregate weightage of 60 points. Students are requested to note that this is an indicative format of the question paper in general and that ICFAI University reserves the right to change, at any time, the format and the pattern without any notice. Hence, the students are advised to use the model question papers for practice purposes only and not to develop any exam-related patterns out of these model question papers. The suggested answers given herein do not constitute the basis of evaluation of the students’ answers in the examination. These answers have been prepared by the faculty members of ICFAI University with a view to assist the students in their studies. And, they may not be taken as the only answers for the questions given.

Model Question Paper I Time: 3 Hours

Total Points: 100 Part A: Basic Concepts (40 Points)

Answer all the questions. Each question carries one point. 1. The maximax criterion is used to make decision under

2.

a.

Conditions of certainty irrespective of the nature of the decision maker

b.

Conditions of risk irrespective of the nature of the decision maker

c.

Conditions of uncertainty, the decision maker being perfectly optimistic

d.

Conditions of uncertainty, the decision maker being perfectly pessimistic

e.

Conditions of uncertainty, the decision maker being neither perfectly optimistic nor perfectly pessimistic.

Which of the following distributions can best describe a process or an experiment which has only two possible outcomes and the probability of any outcome remains constant over time, and the outcome of one trial does not influence the outcome of another? a.

3.

b.

Standard normal distribution.

c.

Asymmetric distribution.

d.

t-distribution.

e.


Which of the following is an example of Type II error? a.

4.


The null hypothesis is false and accepted.

b.

The null hypothesis is false and rejected.

c.

The alternative hypothesis is true and accepted.

d.

The alternative hypothesis is not framed correctly.

e.

The null hypothesis is not framed correctly.

An estimator which has a lower standard error than another is said to be a. A more efficient estimator b. A more consistent estimator c. A more sufficient estimator d. A more unbiased estimator e.

An accurate estimator.

5.

Repetitive and predictable movement around the trend line in one year or less is known as a.

6.

7.

Secular trend

b.

Cyclical fluctuation

c.

Irregular variation

d.

Seasonal variation

e.

Temporary variation.

Which of the following can be identified using the percent of trend measure? a.

Secular trend.

b.

Cyclical variation.

c.

Seasonal variation.

d.


e.

All of the above.

A multiple regression equation will a.

Have only one dependent variable

b.

Have many dependent variables

c.

Have only one independent variable

d.

Have the same coefficient for all the independent variables

e. Not have any constants. 8. Which of the following is true when the slope of a regression line is negative? a. The correlation coefficient between the dependent and independent variables is zero. b. There is a positive correlation between the dependent and independent variables. c. There is a negative correlation between the dependent and independent variables. d. The regression line is perpendicular to the horizontal axis. e. The regression line is parallel to the horizontal axis. 9. Which of the following is true with regard to a given coefficient of correlation and its corresponding coefficient of determination? a. The coefficient of determination is always greater than or equal to zero. b. The coefficient of determination is always negative. c. The coefficient of determination is always zero. d. The coefficient of determination always has the same sign as the coefficient of correlation. e. The coefficient of determination is always higher in magnitude than the coefficient of correlation. 10. Which of the following true when the slope of a regression line is zero? a. Variance of the independent variable is zero. b. Variance of the dependent variable is zero. c. Covariance between the dependent and independent variables is zero. d. Covariance between the dependent and independent variables is positive. e. Covariance between the dependent and independent variables is negative. 11. In regression analysis, the graphical plot of the numerical values of the dependent and independent variables is called a a. Histogram b. Frequency polygon c. Scatter diagram x chart d. e. 360

Ogive.

Part III

12. Testing a hypothesis at a significance level of 10% means that a. The null hypothesis will be true 10% of the times b. The null hypothesis will be false 10% of the times c. The alternative hypothesis will be true 10% of the times d. The alternative hypothesis will be false 10% of the times e. The null hypothesis will be rejected 10% of the times though it may be true. 13. Which of the following is described by the coefficient of multiple correlation? a.

It is the degree of correlation between some specified independent variables.

b.

It is the degree of correlation between all the independent variables.

c.

It is the degree of correlation between the dependent variable and one independent variable when other independent variables are held constant.

d.

It is the degree of correlation between the dependent variable and all the independent variables.

e.

It indicates existence of cause and effect relationship between the dependent variable and the independent variables.

14. Which of the following is true with regard to index numbers? a.

Unweighted aggregates index numbers do not link the price changes to the usage or consumption levels.

b.

Laspeyres price index tends to underestimate the rise in prices.

c.

Paasche’s price index tends to overestimate the rise in prices.

d.


e.

All of the above.

15. Which of the following price indices uses only base year quantities as weights? a.

Fixed weight aggregates price index.

b.


c.


d.


e.

Fisher’s ideal price index.

16. Which of the following types of variations observed in time series analysis is described by using the least squares method? a.

Seasonal variations.

b.

Cyclical variations.

c.

Secular trend.

d.


e. All of the above. 17. Which of the following is true in the context of quality control? a. Variations in quality of output occur only due to the existence of random or inherent variations in the process. b. Variations in quality of output occur only due to the existence of non-random or special cause variations in the process. c. All non-random variations must be identified and brought under control before redesigning the process to reduce the random or inherent variations. d. The process must be redesigned to eliminate all random variations before identifying the non-random variations and eliminating the same. e. All of the above. 361

18. In quality control, which of the following is used to monitor variations in attributes? a.

x -Charts.

b.

p Charts.

c.

R Charts.

d.

Histograms.

e.

Ogives.

19. For which of the following purposes chi-square test cannot be used? a.

To test whether more than two population proportions can be considered equal.

b.

To test whether more than two population means can be considered equal.

c.

To test whether two attributes with respect to which a population is categorized are independent of each other.

d.

To determine the goodness of fit of a theoritical distribution.

e.

To test whether there is a significant difference between an observed frequency distribution and a theoritical distribution.

20. Which of the following statements is not true with respect to ANOVA? a.

It asumes that populations from which the samples are drawn are normal.

b.

It assumes that the populations from which the samples are drawn, have the same variance.

c.

It is based on a comparison between the variance among the sample means and the variance that exists within the samples.

d.

The nearer the F-ratio comes to 1, the greater are the chances that the null hypothesis will be accepted.

e.

The larger the F-ratio, the greater are the chances that the null hypothesis will be accepted.

21. Which of the following is/are true? i.

When we accept a null hypothesis, it is because, that there is no statistical evidence to reject it.

ii.

The significance level will indicate the percentage of sample means, that is outside certain limits, when we assume that the hypothesis is correct.

iii.

The higher the significance level, we use for testing a hypothesis, the lower the probability of committing Type I error.

a.

Only (i) above.

b.

Only (ii) above.

c.

Only (iii) above.

d.


e.


22. We are using t-distribution. Which of the following is/are true?

362

i.

The sample size is equal to or less than 30.

ii.

The standard deviation of the population is known.

iii.

The underlying population is normal or approximately so.

a.

Only (i) above.

b.

Only (ii) above.

c.

Only (iii) above.

d.


e.

All of (i), (ii) and (iii) above.

Part III

23. Which of the following is/are false? i. Sometimes, a test does not reject a null hypothesis when it is false. ii. When the null hypothesis is false, H0: μ ≠ μ0. iii. Usually we prefer a high value of β. (β = probability that null Hypothesis is false but sample static falls in the acceptance region) a. Only (i) above. b. Only (ii) above. c. Only (iii) above. d. Both (i) and (iii) above. e. Both (ii) and (iii) above. 24. Confidence level, refers to a. The probability that we associate with a point estimate b. The probability that we associate with an interval estimate c. The probability that we associate with the test statistic d. Both (a) and (c) above e. All of the above. 25. Owing to the fact that σ x varies inversely with the square-root of sample size ‘n’, what will be the impact on the return on investment that goes into collecting additional information, if sampling size is going to be increased indiscriminately? a. Diminishes. b. Increases. c. Slightly increases. d. Remains same. e. All of the above. 26. Why in random sampling, it is possible to describe mathematically how objective our estimates are? a. The chance that any population element will be included in the sample is always known. b. Every sample always has an equal chance of being selected. c. All the samples are of exactly the same size and can be counted. d. (a) and (b) but not (c). e. All of the above. 27. In testing of hypothesis the decision to use a single-tailed or two-tailed test depends upon a. The size of the sample b. The alternative hypothesis c. The values of the sample statistic d. The null hypothesis e. All of the above. 28. The regression line of two variables X and Y (Y being dependent) is as shown in the given diagram. Then, ↑

Y

a a. b. c. d. e.

{

X→ The intercept of the regression line is zero and the slope is zero There is no relation between X and Y The slope of the line is zero All of (a), (b) and (c) above Both (b) and (c) above. 363

29. If the regression equation is not a perfect estimator of the dependent variable then which of the following statement is true? a. The standard error of estimate is not zero. b. The data points do not lie on the regression line. c. The coefficient of determination is not one. d. Error sum of squares is not zero. e. All of the above. 30. The standard error remains the same at all points on a regression line because i. Observed values are normally distributed around each estimated value of y ii. The variance of the distributions around each possible value of y is the same iii. All available data points were taken into account when the regression line was calculated. a. Only (i) above b. Only (ii) above c. Only (iii) above d. Both (i) and (ii) above e. All of (i), (ii) and (iii) above. 31. A person was drawing Rs.4,500 per month, during 1995-96. The CPI in 1995-96 was 337. His real salary in 1995-96 approximately was Rs.________ a. 1,275 b. 1,298 c. 1,335 d. 1,385 e. 1,415. 32. Which of the following is/are false? i. The link relatives, calculated by using chain base method, enable comparisons over long range. ii. Seasonal variations, have maximum impact on chain index numbers. iii. Chain base method enables, the introduction of new items and the deletion of old items without altering the original series. a. Only (ii) above. b. Only (iii) above. c. Both (i) and (ii) above. d. Both (ii) and (iii) above. e. All of (i), (ii) and (iii) above. 33. Given Cov (x,y) = 14, σX = 5 and σy = 4, what is the value of correlation coefficient? a. b. c. d. e.

0.29. 0.36. 0.70. 0.80. 1.0.

2 ˆ ˆ = a + bX , the quantity ∑ (Y − Y) represents 34. Given an estimating equation Y 2 ∑ (Y − Y)

a. b. c. d. e. 364

Fraction of the total variation in Y that is unexplained Fraction of the total variation in Y that is explained The strength of a linear relationship between two variables The variation of the Y values around their mean Both (b) and (c) above.

Deleted:

ˆ 2 /(Y-Y 1- (Σ- (Y-Y)

Part III

35. The normal distribution has wide applications to the practical situations, because a.

It is a perfectly symmetrical distribution

b.

It serves as a good approximation, for the distribution of various physical, financial and human characteristics

c.

For large samples, the normal distribution is considered to be the limiting distribution of the average of the sample values

d.

Both (b) and (c) above

e.


36. There are three variables X, Y and Z with the following relationships Y = a+z, X = b+z the correlation between X and Y is a.

Positive

b.

Negative

c.

Zero

d.

Depend on ‘a’ and ‘b’

e.

Insufficient information.

37. A two tailed test means, a test a.

Where a second sample is taken based on the first

b.

Where null hypothesis is accepted if test statistic exceeds a given value

c.

Where both upper and lower limits of confidence interval are used to take a decision of accepting null hypothesis

d.

Used only when the sample size is small

e.

Used only to measure the difference between two proportions.

38. Each Bernoulli process has a characteristic probability, which is a.

Unique

b.

Equal to that of others

c.

Greater than others

d.

Less than others

e.

All of the above.

39. When the probability of failure is 0.9, then the binomial distribution is a.

Negatively skewed

b.

Positively skewed

c.

Symmetrical

d.

Skewers to the right is less noticeable

e.

All of the above.

40. Which of the following is/are true? i.

There is a different t distribution for every possible sample size.

ii.

A t-distribution is higher at the mean and lower at the tails like a normal distribution.

iii.

In a t-distribution, the interval widths are wider than those used for normal distribution.

a.

Only (i) above.

b.

Only (iii) above.

c.


d.


e.


365

Part B: Problems (60 Points) Solve all the problems. Points are indicated against each problem. 1.

2.

For generating an x chart, 20 samples are taken. The sum of all the observations in the samples is 360 and the value to be used for the centre line is 1.5. The size of each sample is a. 5 b. 10 c. 12 d. 20 e. 36. (1 point) Given below are the probabilities associated with the possible prices of a share in future, in the secondary market: Possible Prices

(Rs.)

Probability (%) The expected price of the share is a. Rs.27.5 b. Rs.28.5 c. Rs.31.5 d. Rs.32.5 e. Rs.37.5.

3.

4.

5.

366

25

30

35

40

20

40

30

10

(1 point) If the sample size is 100 and the proportion of defective pieces in a batch of output is 0.10 then, the upper control limit of p chart will be a. 0.01 b. 0.03 c. 0.09 d. 0.10 e. 0.19. (1 point) In a test of ANOVA, the following are found: Estimated population variance based on the variance among sample means = 21.60 F ratio = 1.50 What is the estimated population variance based on the variance within the samples? a. 12.25. b. 14.40. c. 24.25. d. 26.75. e. 31.88. (1 point) A multiple regression relationship contains two independent variables. The standard error of ˆ 2 = 129.96. The number of observations is estimate is 2.85. ∑ (Y − Y) a. 11 b. 14 c. 16 d. 19 e. 20. (1 point)

Part III

6.

7.

8.

Two variables, X and Y, are related by a simple regression equation such that X is the independent variable and Y is the dependent variable. The following details are available: Coefficient of correlation between X and Y = –0.80 Standard deviation of X = 16.4 Standard deviation of Y = 24.6 The slope of the regression equation is a. –1.20 b. 1.20 c. 10.37 d. 16.20 e. 25.31. (1 point) Sixty percent of the medium sized software firms based in the northern region of the country planned to send their representatives to Europe in order to explore business opportunities there. An export-import group in Manchester, UK., has invited 16 medium sized software firms based in the northern region of the country, to participate in a conference to explore the possibilities of developing trading relationships. What is the probability that at least 10 of the invited software firms will send their representatives to the conference? a. 19.83%. b. 31.05%. c. 32.88%. d. 52.71%. e. 47.29%. (2 points) A wine producer is contemplating an expansion in its sales territory up to the southern part of the state. In order to do so, it must build a new brewery in that part of the state so that it can avoid the problems related with transportation. The producer has to make a decision on the capacity of the brewery that should be constructed. Three capacities of the brewery are under consideration viz, Small, Medium and Large. The management of the firm has advised that the capacity should be based on the projected net profit for the third year of operation of each of the three capacities under consideration. The marketing department of the firm has made an assessment of the future demand level during the third year of operation of the brewery, and according to it the sales during the third year of operation may grow by 5 percent, 10 percent or 15 percent. The projected net profits from the three proposed brewery capacities in the third year of operation under the three types of growth in demand, are given below: Net profit (Rs. millions)

Growth in Demand 5% 10% 15% Small 75 110 125 Medium 55 100 140 Large 45 75 155 If the regret criterion for decision-making is used then what should be the brewery capacity? a. Small. b. Medium. c. Large. d. Medium or large. e. Cannot be determined. (2 points) Capacity

367

9.

A sample of paired observations of two random variables, X and Y, is given below: X

3

6

2

4

5

4

7

1

Y

25

32

16

28

30

13

30

10

What is the covariance between X and Y? a. 10.12. b. 11.25. c. 13.86. d. 14.35. e. 16.48. (2 points) 10. The Western Delight Restaurant prepares a Chicken Dish, Chicken Fun, every Sunday. The manager of the restaurant, wants to ensure that the restaurant earns the maximum profit on this dish. From each whole chicken that is cooked, two portions of Chicken Fun are prepared. The selling price per portion of the dish is Rs.50, which is considered to be a bargain price, and this has made the dish very popular. Each portion of the dish costs Rs.32 including labor and other costs of preparation. It is observed over a long period of time that the demand for the dish is normally distributed with a mean of 210 portions and a standard deviation of 50 portions. The dish is perishable and the unsold portions of the dish cannot be preserved and sold later. How many whole chickens should be ordered by the manager of the restaurant? a. 88. b. 90. c. 94. d. 95. e. 96. (2 points) 11. Given below are the probabilities attached to the possible prices of a stock in the secondary market: Possible Prices 25 30 35 40 Probability (%) 20 40 30 10 The expected price of the stock is a. 27.5 b. 28.5 c. 31.5 d. 32.5 e. 37.5. (2 points) 12. Mr. Vijayan is a young graduate engineer working as shift-in-charge in a switchgear factory. The new production manager has asked him to conduct a study of how different speeds of the conveyor belt affect the number of defective units produced in an 8-hour shift. To examine this, he ran the conveyor belt at three different speeds and for each speed he took the observations of the number of defective units produced in four different shifts. The observations taken by him are given below: Speed 1 5 3 1 8 368

Speed 2 6 6 2 9

Speed 3 4 4 3 6

Part III

Which of the following choices best estimate the population variance using the variance within the samples? a.

8.917.

b.

6.25.

c.

7.806.

d.

4.75.

e.

24.75. (3 points)

13. New Era Ventures Ltd. (NEVL) is a venture capital company. The company has identified three ventures involving new technologies in three different fields viz. biochemistry, telecommunications and solar power. The business environment may assume any one of three states in future viz. high growth, moderate growth and low growth, and the company has no information about what the state of business environment will be. For an investment of Rs.10 crores made in each of the three types of venture the expected gains are given below: (Rs. in crore) Business environment

Type of venture Biochemistry

Telecommunications

Solar power

High growth

6.0

5.0

4.0

Moderate growth

3.5

4.0

4.0

Low growth

1.5

2.0

2.5

Using the regret criterion of decision making, find out the type of venture in which NEVL should invest. a.

Biochemistry

b.

Telecommunication

c.

Solar Power

d.


e.

Both (b) and (c) above. (4 points)

14. If the variances of X and Y are 2.3 and 4.1, and the coefficient of determination between X and Y is 0.8 what is the approximate value of covariance between X and Y? a.

1.25.

b.

2.75.

c.

7.54.

d.

8.43.

e.

9.43.

(2 points) 2 ˆ 15. For a simple linear regression equation Σ(Y – Y ) = 178.56 and there are 11 pairs of observations. The standard error of estimate is a.

4.03

b.

4.45

c.

12.60

d.

13.36

e.

19.84. (2 points) 369

16. For a basket of commodities, Laspeyres price index is 110.10 and Paasche’s price index is 122.33. The value of Fisher’s ideal price index for the same basket of commodities would be a.

111.11

b.

112.47

c.

114.80

d.

116.05

e.

118.85. (2 points)

17. For generating a x chart 20 samples are taken. The sum of all the observations is 360 and the value to be used for the centre line is 1.8. The size of each sample is a.

5

b.

10

c.

18

d.

20

e.

36. (3 points)

18. In a p chart, if n = 60 and p = 0.33, the value of Upper Control Limit (UCL) will be given by a.

0.347

b.

0.368

c.

0.391

d.

0.451

e.

0.512. (2 points)

19. Two samples A and B are drawn seemingly from the same population. Sample

Size

Standard Deviation

A

16

3.2

B

10

4.8

σ and σ 2B represent the variances of the respective populations to which the samples belong. 2 A

For testing the hypothesis, H0 : σ 2A = σ B2 , the value of F statistic will be a.

0.44

b.

0.67

c.

0.82

d.

1.50

e.

2.25. (3 points)

20. The slope and the intercept on X-axis of a regression line are 3 and 4 respectively. The equation of the line is

a.

4y = x + 3

b.

3y = 4x + 1

c.

y = 3x + 4

d.

y = 3x – 12

e.

3y = 4x. (3 points)

370

Part III

21. Western Garments Company is in the business of trading men’s garments. It purchases ready made garments from different manufacturers and sells them in the retail market. The company sells a particular brand of T-shirts named “Macho” every year during the summer season. It has to decide on the number of the XL size of these T-shirts to stock for this season. Each T-shirt costs Rs.150 and sells for Rs.200. The salesman at the counter is given a commission of Rs.10 on every T-shirt sold. The stock of this type of T-shirts is maintained for a period of 4 months. In the past the sales of this brand and size of T-shirts during the same period of time have averaged 400 with a standard deviation of 60. The T-shirts which remain unsold at the end of 4 months are sold at 45% discount from the selling price. Further, on each T-shirt sold at discount the salesman at the counter is given a commission of Rs.20. The sales of the T-shirts have been found to be normally distributed. How many of “Macho” brand T-shirts of XL size should be stocked by the company? a. 412. b. 387. c. 385. d. 415. e. 210. (4 points) 22. City Finance Company Ltd. is a non-banking finance company. In the recent years there has been variations in the demand for loans as well as in the rate of interest. The general manager of the company wants to know whether the variations in the amount of advances explain the variations in the profit after tax of the company. The following data pertain to the operations of the company in the recent years: Year 2002 2001 2000 1999 1998 1997 Amount of advances (Rs. in crores) 7.0 5.5 5.0 4.5 4.0 4.0 Profit after tax (Rs. In lakhs) 54 48 42 27 19 14 What is the coefficient of correlation between the amount of advances and profit after tax? a. 1.315. b. 2.340. c. 0.9182. d. 1.425. e. 0.1283. (4 points) 23. A wholesaler requires an assessment of the change in prices of the commodities he trades in. The following data is provided by him: Year Commodity 1995 2001 1997 Price (Rs./kg) Price (Rs./kg) Price (Rs./kg) Quantity (kg.) Rice 9.50 11.25 10.00 1000 Wheat 6.50 8.50 6.75 600 Salt 2.50 4.00 3.25 400 Sugar 10.00 14.50 12.00 800 Pulses 22.50 31.00 26.50 500

What is the weighted average of relatives price index for the year 2001 using the year 1997 for weightage and the year 1995 as the base year? a.

134.54.

b.

124.54.

c.

114.54.

d.

84.54.

e.

74.54. (4 points) 371

24. The Bombay Transport Company runs a fleet of buses for transportation of passengers within an Indian city. The following data shows the number of passengers for various ticket prices:

Ticket price (Rs.)

5

5.50

6.00

6.50

7.00

Passengers per 100 kms

700

680

620

560

500

What is the slope of the estimating equation to predict the number of passengers produced per 100 kms from the ticket price? a.

30.

b.

104.

c.

–104.

d.

1236.

e.

–1236. (4 points)

25. A survey was conducted on the retail prices of a specific type of pulses in a city. At a confidence level of 95 percent it was estimated that the mean retail price of the commodity fell between Rs.25.50 per kg and Rs.27.46 per kg. It is assumed that the confidence interval is spread equally on either side of the mean. The sample size was 36 and less than 5 percent of the population of the grain retailers in the city were sampled. Which of the following gives the mean and standard deviation of the sample respectively?

a. b. c. d. e.

24.68, 2.0. 24.68, 3.0. 26.48, 2.0. 26.48, 3.0. 22.38, 4.0. (4 points)

372

Part III

Model Question Paper I Suggested Answers Part A: Basic Concepts 1.

(c) Maximax criteria is used to make decision under conditions of uncertainty, when the decision maker is optimistic.

2.

(e) The experiment or process stated in the question is a Bernoulli process and the distribution which can describe such a process is the binominal distribution.

3.

(a) Type II error is the error of accepting the null hypothesis when it is false.

4.

(a) An estimator which has lower standard error than another is said to be a more efficient estimator.

5.

(d) Repetitive and predictable movement around the trend line in one year or less is known as seasonal variation.

6.

(b) Cyclical variation can be identified using the percent of trend measure.

7.

(a) A multiple regression equation has only one dependent variable and multiple independent variables – the value of the dependent variable is derived from the values of the independent variables. Further all the independent variables need not have the same coefficients.

8.

(c) When the slope of a regression line is negative, the dependent variable decreases as the independent variable increases. Hence increase in the independent variable can be associated with decrease in the dependent variable and vice versa. Hence, the correlation between the dependent and the independent variables is negative.

9.

(a) Coefficient of determination (c.d.) = r2 (where ‘r’ is the coefficient of correlation and –1 ≤ r ≤ 1) ∴ c.d. = r2 ≥ 0 Because square of a positive or, negative number is always positive. Further if r = 0 then r2 = 0, and if r ≠ 0 then r2 ≠ 0 Also if –1 ≤ r ≤ 1 then 0 ≤ r2 ≤ 1 Hence all other alternatives except (a) are false.

10. (c) b =

Cov (X,Y) σ 2X

. If b = 0 then,

Cov (X,Y) σ 2X

= 0 i.e. Cov(X, Y) = 0

All other alternatives except (c) are false. 11. (c) In regression analysis the graphical plot of the numerical values of the dependent and independent variables is called a scatter diagram. 12. (e) A significance level of 10% implies a rejection area of 10% in the normal distribution curve. This area consists of 10% of all possible values of the statistic when the null hypothesis is true. Hence, the answer is alternative. 13. (d) Coefficient of multiple correlation describes the correlation between the dependent variable and all the independent variables. 14. (a) All other alternatives are false. Unweighted aggregates index numbers do not link price changes to the usage or consumption levels. Hence alternative (a) is true. 15. (b) Laspeyeres price index =

∑ P1Qo x 100. Qo represents base year quantities which are ∑ Po Qo

used as weights. 16. (c) The method of regression analysis used for finding out the best fitting equation for the secular trend, incorporates the method of least squares.


17. (c) As long as non-random variations are present the process remains out-of-control. Once the non-random variations are eliminated the process is brought in-control, and the entire process can be redesigned to reduce the incidence of random variation. 18. (b) In quality control p-charts are used to monitor variations in attributes. 19. (b) All other alternatives are the purposes for which chi-square test can be used. However, the purpose stated in alternative (b) can be accomplished by ANOVA (and not chi-square test). 20. (e) All other alternatives are true with respect to ANOVA except alternative (e). 21. (d) Higher the significance level, higher is the probability of committing type I error. 22. (d) t-distribution or student’s t-distribution is used when the sample size is 30 or less and the population standard deviation is not known. 23. (c) β refers to type-II error. We usually prefers to have a high value of (1 - β). 24. (b) Confidence level refers to the probability associated with an interval estimate. 25. (a) With the increased efforts beyond a certain limit, benefit start reducing. 26. (d) All the samples may not be of exactly same size. 27. (b) In testing of hypothesis the decision to use a single tailed or two-tailed test depends upon the alternative hypothesis. 28. (e) Since Y = a, no relationship exists between the two variables. The slope of the horizontal line is equal to 0. 29. (e) If the regression equation is not a perfect estimator of the dependent variable all the statements become valid. 30. (b) The standard error is the same at all points on a regression line because we assume that the variance of the distributions around each possible value of y is same. 31. (c) (4500/337) x 100 = 1,335 32. (c) As percentages are chained together in chain index numbers, long-range comparisons are not very accurate. Chain base method enables the introduction of new items and the deletion of old items without altering the original series. Seasonal variations have minimal impact on chain index numbers. 33. (c) Coefficient of correlation, r =

=

34. (e)

Cov(x,y) σx σ y

14 14 = = 0.7 5x4 20

ˆ − Y)2 ∑ (Y represent total variation i.e is explained. 2 ∑ (Y − Y)

35. (e) Statements (a), (b) and (c) about normal distribution are correct. Hence (e) is the answer. 36. (a) Since Z is related with X and Y in same fashion X and Y are having positive correlation. 37. (c) Two tailed test primarily indicates inclusion of upper and lower limits of confidence interval to take a decision of accepting null hypothesis. 38. (a) Each Bernalli process has a unique characteristic probability. 39. (b) Probability of failure is given as 0.9 (i.e., q = 0.9), which implies that probability of success, p = 1– 0.9 = 0.1. For a binomial distribution, when p is small then the binomial distribution is skewed to the right, i.e., it is positively skewed. 40. (d) Depending upon degree of freedom. There is different t–distribution for every possible sample size. Area under the curves depends on DOF and it is not like the standard normal probability distribution. 374

Part B: Problems 1.

(c) Let the size of each sample be x.

()

The value to be used for the centre line is the grand mean. In this case the grand mean x = 1.8.

Sum of all the observations = Σx = 360 (given) Σx n×k

Now, x =

Where, n = Sample size = ? k = number of samples = 20 (given)

2.

∴n =

Σx x.k

or n =

360 = 12. 20 × 1.5

(c) The expected price = Σ (Possible Price × Probability)

= 25 × 0.20 + 30 × 0.40 + 35 × 0.30 + 40 × 0.10 = 31.5. 3.

4.

5.

(e) The lower control limit

(d) The number of data points can be obtained as

ˆ Σ (y − y) 2

se

2

+ (k + 1)

129.96 + 2 + 1 = 19. 2.85 × 2.85

(a) The slope (b) of a regression equation is given by

b= 7.

0.1× 0.9 100

= 0.1 + 3 × 0.03 = 0.10 + 0.09 = 0.19. (b) The required estimate of population variance based on the variance within the sample will be 21.6 = 14.40. 1.5

= 6.

pq = 0.1 + 3 × n

=p+3

Cov(x, y)

σ

2 x

σ

= r

σ

y x

= –0.80 ×

24.6 16.4

= –1.20.

(d) P (at least 10) = P(10) + P(11) + P(12) + P(13) + P(14) + P(15) + P(16)

P(10)

= 16C10 (0.60)10 (0.40)16–10 =

0.19833

P(11)

= 16C11 (0.60)11 (0.40)16–11 =

0.16223

P(12)

= 16C12 (0.60)12 (0.40)16–12 =

0.10142

P(13)

= 16C13 (0.60)13 (0.40)16–13 =

0.04681

P(14)

= 16C14 (0.60)14 (0.40)16–14 =

0.01505

P(15)

= 16C15 (0.60)15 (0.40)16–15 =

0.00301

P(16)

= 16C16 (0.60)16 (0.40)16–16 =

0.00028

P(10) + P(11) + P(12) + P(13) + P(14) + P(15) + P(16) = 0.52713 i.e. 52.71% (approx). 375

8.

Net profit (Rs millions)

(b)

10%

15%

Maximum regret

Small

75–75 = 0

110 –110 = 0

155 –125 = 30

30

Medium

75–55 = 20

110 –100 = 10

155 –140 = 15

20

Large

75–45 = 30 110 –75 = 35 155 –155 = 0 35 The decision should minimize the maximum regret. This happens

(c) Cov (X,Y) =

9.

Growth in demand 5%

Capacity

∑(X − X) (Y − Y) n −1

A

X

3

6

2

4

5

4

7

1

∑X = 32

B

Y

25

32

16

28

30

13

30

10

∑Y = 184

C

X– X

–1

2

–2

0

1

0

3

–3

D

Y– Y

2

9

–7

5

7

–10

7

–13

E

C×D

–2

18

14

0

7

0

21

39

X

=

∑X n

=

32 =4 8

Y

=

∑Y n

=

184 = 23 8

∴ Cov (X,Y) =

∑(X– X ) (Y– Y )=97

97 = 13.86 (approx). 8 −1

10. (e) Minimum required probability of selling each portion of the dish,

p*

=

ML MP + ML

Marginal Profit, MP = 50 – 32 = Rs.18 Marginal Loss, ML = Rs.32 ∴ p* =

32 = 0.64 32 + 18

The Z-value for the area 0.14 to the left of the mean is – 0.36. ∴ – 0.36 =

x − 210 50

x = 210 – (0.36 × 50) = 192 ∴ The restaurant should prepare 192 portions at the optimal level. ∴ The number of chickens that should be ordered = 376

192 = 96. 2

11. (c) The expected price = Σ (Possible Price x Probability) = 25 x 0.20 + 30 x 0.40 + 35 x 0.30 + 40 x 0.10 = 31.5 2

12. (b) Estimated population variance using the variance within the samples ( σˆ ) ⎛ n j −1 ⎞ 2 = ∑⎜ s ⎜ n − k ⎟⎟ j ⎝ T ⎠ s12

s22

s32

2

=

(5 − 4.25)2 + (3 − 4.25)2 + (1 − 4.25)2 + (8 − 4.25)2 4 −1

=

26.75 = 8.917 3

=

(6 − 4.25)2 + (3 − 4.25)2 + (1 − 4.25)2 + (8 − 4.25)2 4 −1

=

24.75 = 8.25 3

=

(4 − 4.25)2 + (4 − 4.25)2 + (3 − 4.25)2 + (6 − 4.25)2 4 −1

=

4.75 = 1.583 3

∴ σˆ =

⎛ 4 −1 ⎞ ⎛ 4 −1 ⎞ ⎛ 4 −1 ⎞ ⎜ ⎟ (8.917) + ⎜ ⎟ (8.25) + ⎜ ⎟ (1.583) ⎜ 12 − 3 ⎟ ⎜ 12 − 3 ⎟ ⎜ 12 − 3 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

= 6.25 13. (b) Regret criteria: In each row = Maximum value–Existing value

(Rs. in crore) Maximum Regret

Business Environment

Venture type High growth

Moderate growth

Low growth

6.0 – 6.0 = 0

4.0 – 3.5 = 1.5

2.5 – 1.5 = 1.0

1.5

Tele -communications

6.0 – 5.0 = 1.0

4.0 – 4.0 = 0

2.5 – 2.0 = 0.5

1.0

Solar Power

6.0 – 4.0 = 2.0

4.0 – 3.0 = 1.0

2.5 – 2.5 = 0

2.0

Biochemistry

According to the regret criteria, the alternative which minimizes the largest regret, should be selected. In the above case, the telecommunications venture minimizes the largest regret. Hence according to the regret criteria, NEVL should invest in the telecommunications ventures. 14. (b) Coefficient of correlation, rXY =

Coefficient of determination = 0.8

= 0.894 (taking positive value) also Cov(x,y) = r x σ x σ xy x y = 0.894x 2.3 x 4.1 = 2.75.

377

15. (b) The standard error of estimate for the simple regression relationship ∑(Y − Y) 2 178.56 = 4.45. = n−2 11 − 2

=

16. (d) Fisher’s ideal price index = Laspeyres ' price index x Paasche ' price index

= 110.10 x 122.33 = 116.05. 17. (b) Let the size of each sample be x.

The value to be used for the centre line is the grand mean. In this case the grand mean x = 1.8.

()

Sum of all the observations = Σx = 360 (given) Now, x =

∑x nxk

Where, n

= Sample size = ?

k

= number of samples = 20 (given)

∴n

=

or n =

360 = 10 20 x 1.8

∑x x.k

18. (e) Upper Control Limit (UCL) = p + 3

= 0.33 + 3

p x (1 − p) n

0.33 x 0.67 = 0.512. 60

19. (a)

F=

s 12 s 22

2

⎛ 3. 2 ⎞ =⎜ ⎟ = 0.44. ⎝ 4.8 ⎠

20. (d) y = a + bx

Or x =

(1)

y−a y a = − b b b

⎛−a ⎞ y Or x = ⎜ ⎟+ ⎝ b ⎠ b

(2)

From equation (2) the intercept on the X-axis Slope, in equation (1), b = 3 (given) ∴

−a = 4 or a = – (4 × 3) = –12 3

∴ We get : a = –12 and b = 3. ∴ y = –12 + 3x = 3x – 12.

378

=

−a = 4 (given) b

21. (c) Marginal profit (MP)

= Selling price – Cost – Commission = 200 – 150 – 10 = Rs.40 Marginal Loss (ML):

Discounted price of the T-shirt = 200 (1 – 0.45)

= Rs.110

Less: Commission payable

= Rs. 20

Amount receivable from discount sale = Rs. 90 Marginal loss (ML) =

Cost of the T-shirt – Amount receivable from the discount sale

=

150 – 90 = Rs.60

Minimum required probability of selling an additional T-shirt. (p*) =

ML 60 = 0.60 = 60%. = ML + MP 60 + 40

Since 0.50 of the area under the normal distribution is located between the mean and the right tail 0.10 of the shaded area in the diagram must be to the left of the mean, (0.60 – 0.50 = 0.10). From the standard normal distribution table we find that the nearest value to 0.10 is 0.0987 which corresponds to a Z-value of 0.25. Let the value of the variable be x for which we get a Z-value of 0.25. ∴

x −μ = –0.25 (–ve because it is on the left side of the mean). σ

Given, μ = 400, σ = 60 ∴

x − 400 = –0.25 60

or x = 400 + (–0.25) (60) = 385 ∴ The company should stock 385 T-shirts of XL size of the brand. 22. (c)

Let the following notations be used: X : Amount of advances (Rs. in crore) Y : Profit after tax (Rs. in lakh) Coefficient of correlation is given by: r

=

Cov(X, Y) = σx .σ y

∑ (X − X) (Y − Y) ∑ (X − X) 2 (Y − Y) 2

379

A.

X

7.0

5.5

5.0

4.5

4.0

4.0

∑ X = 30

B.

Y

54

48

42

27

19

14

∑ Y = 204

C.

(X – X )

2

0.5

0

–0.5

–1

–1

D.

(Y – Y )

20

14

8

–7

–15

–20

4

0.25

0

0.25

1

1

400

196

64

49

225

400 ∑(Y − Y) 2 = 1334

40

7

0

3.5

15

20

2

E.

(X – X )

F.

(Y – Y )2 (X − X) (Y − Y)

G.

∑(X - X) 2 = 6.5

∑(X − X) (Y − Y) = 85.5

X =

∑ X 30 = n 6

Y =

∑ Y 204 = = 34 n 6

∴

r==

= 5.0

85.5

= 0.9182.

6.5 x 1334

23. (a) Weighted average of relatives price index ⎡⎛ P ⎤ ⎞ ∑ ⎢⎜⎜ 1 x 100 ⎟⎟ Pn Q n ⎥ ⎢⎣⎝ P0 ⎥⎦ ⎠ = ∑ Pn Q n

Where, P1 = Prices in the current year (2001) P0 = Prices in the base year (1995) Pn = Prices in the year (1997) to be used for weightage Qn = Quantities in the year (1997) to be used for weighting. Commodity

P0

P1

Pn

Qn

P1 x100 P0

Pn Q n = (4) × (5)

⎛ P1 ⎞ ⎜ P x 100 ⎟ Pn Q n ⎝ 0 ⎠

(1)

(2)

(3)

(4)

(5)

(6)

(7)

9.50 6.50 2.50 10.00 22.50

11.25 8.50 4.00 14.50 31.00

10.00 6.75 3.25 12.00 26.50

1000 600 400 800 500

118.42 130.77 160 145 137.78

10,000 4,050 1,300 9,600 13250

(8) = (6 ) × (7) 11,84,200 529618.50 208,000 13,92,000 18,25,585

Σ (7) = 38,200

Σ(8) = 51,39,403.50

Rice Wheat Salt Sugar Pulses Total

⎡⎛ P ⎤ ⎞ Σ ⎢⎜ 1 x100 ⎟ Pn Q n ⎥ = Σ (8) ⎠ ⎣⎢⎝ P0 ⎦⎥

= 51,39,381.89

Σ PnQn = Σ (7) = 38,200. ∴ Weighted average of price relative index for year 2001. =

380

51,39,403.50 = 134.54. 38,200

24. (c) Let the following notations be used:

Ticket price (Rs.)

:

X

Passengers per 100 kms

:

Y

The regression relationship may be represented as:

ˆ = a + bX Y Where, b=

n ∑ XY − ∑ X ∑ Y n ∑ X 2 − (∑ X) 2

a = Y − bX

ΣX = 30 ΣY = 3060 ΣX2 = 182.50 ΣXY = 18,100 n=5

∴b =

5 (18,100) − 30 (3,060)

= – 104.

5 (182.50) − (30) 2

a = Y − bX Y =

∑ Y 3,060 = = 612 n 5

X

=

∴a =

ΣX 30 = =6 n 5

612 – (–104) (6) = 1,236

∴ The regression relationship is

ˆ = 1,236 – 104X Y 25. (d) The sample size is greater than 30 and the standard deviation of population is not known. Hence the appropriate sampling distribution applicable is the normal distribution. A confidence interval of 95 percent implies that 0.475 of the area of the normal curve is covered on either side of the sample mean. From the normal distribution table it is found that the appropriate z-value is 1.96.

x = ? s = ? σˆ x =? For the given confidence interval: Lower limit = Rs.25.50 per kg Upper limit = Rs.27.46 per kg.

∴ x – 1.96 σˆ x = 25.50

…….(1)

x + 1.96 σˆ x = 27.46

……. (2)

Adding equations (1) and (2) we get:

2x = 52.96 or x = 26.48 (Rs. per kg.) 381

Subtracting equation (1) from equation (2) we get: 3.92 σˆ x = 1.96 1.96 ˆ σ ∴ x = 3.92 = 0.50 (Rs. per kg)

s Now, σˆ x = n

∴ 0.50

=

or s = 0.50

s 36 36 = 3 (Rs. per kg)

Mean price of the sample = Rs.26.48 per kg Standard deviation of the sample = Rs.3.00 per kg.

382

Model Question Paper II Time: 3 Hours


Answer all the questions. Each question carries one point. 1. Which of the following elements is/are commonly found in problems of decision making under uncertain conditions? a. The decision maker has an objective to be achieved. b. There are several courses of action. c. There is a calculable measure of benefit or worth of the various courses of action. d. There are events which are not controllable by the decision maker. e. All of the above. 2. Which of the following is not true with regard to the expected value of a random variable? a. It is a weighted average of the values that the random variable can assume. b. The probabilities assigned to the various values that the random variable can assume are used as weights. c. It is not a prediction of the value of the random variable. d. If the expected value of a random variable ‘A’ is more than the expected value of another random variable ‘B’, then it means that the random variable ‘A’ will always assume values higher than the random variable ‘B’. e. All of the above. 3. Which of the following is not true with regard to normal distribution? a. It is a continuous distribution. b. The normal distribution closely fits with the observed frequency distributions of many phenomena. c. The normal distribution curve remains the same for different normally distributed populations. d. It is a symmetrical distribution. e. It is unimodel distribution. 4. Which of the following is true with regard to the expected value of perfect information? a. It is the expected value which can be achieved when all the uncertainties are removed from consideration. b. It is the maximum expected value which can be achieved under the presence of uncertainties. c. It is the minimum expected value which can be achieved under the presence of uncertainties. d. It is the difference between the expected profit with perfect information and the highest expected profit without perfect information. e. 5.

All of the above.

Which of the following is used to make decisions under uncertain conditions when the decision maker is optimistic? a.

Maximax criterion.

b.

Maximin criterion.

c.

Expected value.

d.

Expected value with perfect information.

e.

Expected value of perfect information.


6.

7.

8.

9.

Which of the following distributions can be identified by a pair of degrees of freedom? a.

t-distribution.

b.


c.


d.

F distribution.

e.

Chi square distribution.

Which of the following types of sampling is suitable when the population consists of well defined groups such that the elements of each group are homogeneous and the groups vary from each other? a. Judgmental sampling. b. Stratified sampling. c. Systematic sampling. d. Cluster sampling. e. Simple random sampling. The means of two populations are represented by μ1 and μ2. It has to be tested whether population mean μ2 is greater than population mean μ1. Which of the following represents the alternative hypothesis for such a test correctly? a. H0 : μ1 – μ2 = 0. b. H0 : μ1 – μ2 ≠ 0. c. H1 : μ1 – μ2 ≠ 0. d. H1 : μ1 – μ2 > 0. e. H1 : μ1 – μ2 < 0 . A type I error occurs in hypothesis testing when the a.

Null hypothesis is false

b.

Alternative hypothesis is false

c.

Null hypothesis is false and accepted

d.

Null hypothesis is true and rejected

e.

Null hypothesis is false and rejected.

10. Which of the following is not true? a.

The purpose of hypothesis testing is to make a judgment about the difference between the sample statistic and the hypothesized population parameter.

b.

When the null hypothesis is rejected the alternative hypothesis is accepted.

c.

The purpose of hypothesis testing is to test the correctness of the computed value of the sample statistic.

d.

The acceptance of the null hypothesis does not indicate that the null hypothesis is true.

e.

Hypothesis testing begins by making an assumption about the population parameter.

11. Which of the following is true with regard to a two tailed test of hypothesis at a significance level of 10 percent? a. b. c. d. e.

384

The rejection region lies entirely in the right tail of the sampling distribution. The rejection region lies entirely in the left tail of the sampling distribution. The rejection region lies entirely in the middle of the sampling distribution. Five percent of the area in each of the left and right tails of the sampling distribution serves as the rejection regions. Ten percent of the area in the middle of the sampling distribution is considered to be the acceptance region.

12. Which of the following is true with regard to the standard error of a sample statistic? a.

It is the standard deviation of the population.

b.

It is the standard deviation of a sample taken from the population.

c.

It indicates the extent of non-sampling error.

d.

It indicates the variability arising out of sampling error due to chance.

e.

All of the above.

13. Which of the following is false? a.

Interval estimates are used for small populations only.

b.

An interval estimate is a range of values within which the actual value of the population parameter may fall.

c.

An interval estimate indicates the probability of the true population parameter falling within the range of values given.

d.

A point estimate is a single number that is used to estimate an unknown population parameter.

e.

A point estimate may be either right or wrong.

14. Which of the following is not true with regard to correlation analysis? a.

It deals with the relationship between two or more series.

b.

It compares the variation in two or more series.

c.

It indicates the presence of a cause and effect relationship between two or more series.

d.

It helps in determining the degree of association between two or more series.

e.

All of the above.

15. In which of the following conditions will the problem of multicollinearity arise in a multiple ˆ = a + b1X1 + b2X2? regression relationship given as: Y a.

There is a high degree of correlation between the variables Y and X1.

b.

There is a high degree of correlation between the variables Y and X2.

c.

There is a high degree of correlation between the variables X1 and X2.

d.

Both the regression coefficients ‘b1’ and ‘b2’ are positive.

e.

The regression coefficient ‘a’ is positive.

16. Which of the following is not true with regard to the coefficient of multiple correlation R1.23 between the dependent variable, Y, and the independent variables, X1 and X2? a.

R1.23 is a non-negative number.

b.

R1.23 can take values between 0 and 1.

c.


d.


e.

It is independent of the coefficient of correlation between X1 and X2.

17. If the coefficient of correlation between two variables lies between –1 and 0, then the covariance between them is a.

Positive

b.

Negative

c.

Zero

d.

Greater in magnitude than the variance of each of the variables

e.

Lesser in magnitude than the variance of each of the variables.

385


18. Which of the following is not true with regard to a value index? a.

It measures changes in net monetary worth.

b.

It combines price changes and quantity changes.

c.

It is not as useful as price indices or quantity indices.

d.

It cannot distinguish between the price changes and quantity changes.

e.

An increase in the value index indicates with certainty that the prices have increased.

19. Which of the following is a quantity index? a.

Index number of industrial production.

b.

Index number of wholesale prices.

c.

Index number of industrial security prices.

d.

Consumer price index for industrial workers.

e.

All of the above.

20. Which of the following is not true? a.

An unweighted aggregate price index does not attach any importance to the consumption or usage levels of the items covered.

b.

Laspeyres price index is an unweighted aggregates price index.

c.

Paasches price index is a weighted aggregates price index.

d.

A price index reflects the overall change in price for the basket of goods considered.

e.

Fisher’s ideal index is the geometric mean of the Laspeyres and Paasche indices.

21. Which of the following is a technique for studying seasonal variations? a.


b.

Residual method.

c.

Hypothesis testing.

d.

Ratio to moving average method.

e.

Percent of trend measure.

22. The movement of the dependent variable above and below the secular trend line over periods longer than one year is known as a.

Cyclical variation

b.

Secular trend

c.

Seasonal variation

d.

Irregular variation

e.

Medium term variation.

23. Which of the following patterns in the x charts indicates the presence of non-random variations in the process?

386

a.

Increasing trend.

b.

Cycles.

c.


d.


e.

Jumps.

24. Which of the following techniques enables us to test whether proportions of more than two populations can be considered equal? a.


b.


c.

Chi-square test.

d.


e.

Marginal analysis.

25. For a portfolio with investments in two securities with equal weightages, the risk is zero when, the a.

Mean returns of both securities is same

b.

Standard deviation of both securities is same

b.

Covariance between these two securities is -1

d.

Standard deviation of either of them is zero

e.

All of the above.

26. If σ is the standard deviation of the normal distribution and μ is the mean. What percentage of the observations will in the interval be μ + 1.28? a.

65%.

b.

76%.

c.

80%.

d.

95%.

e.

98%.

27. The possible returns on a bond and an equity share under three scenarios are given below. Return Scenario

Bond

Share

Bullish

12%

24%

Stable

12%

15%

Bearish

12%

8%

What conclusion can be drawn about the correlation between bond and share returns? a.

Positive.

b.

Negative.

c.

No correlation.

d.

Will depend upon probabilities of given scenarios.

e.


28. The Director of Quality Control for the ABC Company wishes to check the defects in their machines. Out of 200 machines tested 50 were found to be faultless and 150 defectives. Assuming ‘no defect’ as success which of the following is true? a.

p = 0.33.

b.

p = 0.25.

c.

p = 0.20.

d.

p = 0.95.

e.

p = 0.15.

387


29. Regression analysis is used to know the relationship between return on a particular stock Y and overall return on the market X. It is given that the slope coefficient of the regression line is 2 and the intercept of line on the Y axis is 4. If the independent variable has a value of 2, what is the expected value of the dependent variable? a.

10.

b.

8.

c.

0.

d.

– 1.

e.

– 6.

30. Which of the following is/are true? a. b. c.

F ratio is defined as the ratio of the between-column variance and the within-column variance. F ratio is employed to compare the estimates of two population variances. S22 also has F distribution with n2 – 1 degrees of freedom in the numerator S12 and n1 – 1 degrees of freedom in the denominator. F ratio is not governed by the degrees of freedom of the numerator and the denominator.

The ratio

d. e. (a), (b) and (c) above. 31. In P-chart if the value of LCL is below zero then it is taken as a.

–1.0

b.

–0.5

c.

0

d.

0.5

e.

1.0.

32. The standard error of estimate in regression analysis is measured a.

Along the Y-axis

b.

Perpendicular to the regression line

c.

Perpendicular to the X-axis

d.

Parallel to the X-axis

e.

Both (a) and (c) above.

33. The relative frequency of occurrence defines probability as the a.

Observed relative frequency of an event in a very large number of trials

b.

Observed relative frequency of an event in a small number of trials

c.

Proportion of times that an event occurs in the long run, when conditions are stable

d.

Proportion of times that an event occurs in the short run when conditions are stable

e.

Both (a) and (c) above.

34. We will be wasting resources by taking larger samples, if the estimator is not

388

a.

Unbiased

b.

Efficient

c.

Consistent

d.

Sufficient

e.

Inconsistent.

35. The sample median in symmetrically distributed population is not the efficient estimator of the mean, because in large samples, the standard error of the median is _________ than that of the sample mean. a.

Larger

b.

Smaller

c.

Equal to

d.

Larger than or equal to

e.

All of the above.

36. Theoretically, which distribution is the correct one to use in constructing confidence intervals in estimating population proportions? a.


b.

Hypergeometric distribution.

c.


d.

Continuous distribution.

e.

t-distribution.

37. Which of the following class of variations can be identified with the help of time series analysis? a.

Secular trend.

b.

Cyclical fluctuation.

c.

Seasonal variation.

d.


e.

All of the above.

38. The long-term moving average of a variable indicates its a.

Secular trend

b.

Cyclical trend

c.

Seasonal trend

d.

Irregular trend

e.


39. The variation due to events like floods, strikes and war etc., in a time series data can be classified as a. Secular trend b. Cyclical variation c. Seasonal variation d. Irregular variation e.

Both (b) and (d) above.

40. Patterns of change in a variable occurring within a year and which tend to be repeated from year to year are known as a. Secular trend b. Cyclical fluctuation c. Seasonal variation d. Irregular variation e.

All of the above.

389



The following data pertain to three commodities: Commodity

Price in 2002 (Rs./kg)

Price in 1997 (Rs./kg)

Rice

12.50

10.50

Wheat

13.50

8.50

Pulses

25

20

The base year is 1997. The unweighted aggregates price index for the year 2002 is approximately a.

115.62

b.

125.45

c.

130.77

d.

250.55

e.

410.00. (1 point)

2.

The following index values pertain to a year: Fisher’s ideal price index =124.50 Laspeyres price index =122.00 Paasche’s price index for the year is approximately a.

102.90

b.

115.50

c.

120.50

d.

127.05

e.

129.10. (1 point)

3.

The probability of failure in a Bernoulli experiment is 0.60. The experiment is repeated 25 times. The mean of the binomial distribution of the number of successes is a.

10

b.

20

c.

30

d.

40

e.

50. (1 point)

4.

The slope and the intercept on y-axis for a simple regression line are 3 and 5 respectively. If the independent variable has a value of 10, then the estimated value of the dependent variable is a.

3

b.

5

c.

20

d.

35

e.

50. (1 point)

390

5.

A variable X, which has a normal distribution, has a mean of 24 and a standard deviation of 5. The probability that the value of X lies in the interval 20 to 28 is approximately a.

24.7%

b.

31.5%

c.

38.5%

d.

49.5%

e.

57.6%. (1 point)

6.

Southern Garments, a local manufacturer of women’s clothings, markets its products in Hyderabad and adjoining places. From its past records it knows that its products are carried in 25 percent of the women’s clothing stores in Hyderabad. The marketing manager of the firm wants to know if the percentage of women’s clothing stores in Secunderabad, which carry its products is less than the percentage in Hyderabad. He conducted a sample survey of the women’s clothing stores in Secunderabad. It was found out that the ratio of the number of clothing stores in Secunderabad which did not carry its products, to the number of clothing stores in Secunderabad which carried its products, was 4:1. Eight out of the women’s clothing stores in Secunderabad which were sampled, carried its products. Which of the following statements are correct? i.

A right tailed test of hypothesis is appropriate.

ii.

A two tailed test of hypothesis is appropriate.

iii.

A left tailed test of hypothesis is appropriate.

iv.

The alternative hypothesis is p = 0.25.

v.

The alternative hypothesis is p > 0.25.

vi.

The alternative hypothesis is p < 0.25.

vii. The standard error of proportion is 0.06325. viii. The standard error of proportion is 0.25. ix.


x.


xi.

The rejection region falls under the right tail.

xii. The rejection region falls in the middle of the sampling distribution. xiii. The rejection region falls under the left tail. xiv. The critical value for the test is 1.28 at a significance level of 10%. xv. The critical value for the test is –1.28 at a significance level of 10%. xvi. The standardized value of the sample proportion is – 0.791. xvii. The standardized value of the sample proportion is – 0.73. xviii. At a significance level of 10%, it can be concluded that the percentage of women’s clothing stores in Secunderabad which carry the firm’s products is significantly less than the percentage in Hyderabad. xix. At a significance level of 10% it can be concluded that the percentage of women’s clothing stores in Secunderabad, which carry the firm’s products is not significantly less than the percentage in Hyderabad. a.

Only (i), (iv), (vii), (xiv), (xvi) and (xviii)

b.

Only (ii), (iv), (viii), (xii), (xiv), (xvi) and (xviii)

c.

Only (ii), (v), (ix), (xi), (xiv) and (xvi)

d.

Only (iii), (vi), (x), (xiii), (xv), (xvii) and (xix)

e.

Only (i), (vi), (viii), (xi), (xvi) and (xviii). (3 points) 391


7.

There is a population with known variance. If a sample of size m is taken from this population then the standard error of mean will be 2.25. If a sample of size n is taken from this population then the standard error of mean will be 1.5. What will be the standard error of mean if a sample of size m+n is taken from this population? a.

1.50.

b.

2.167.

c.

1.444.

d.

0.692.

e.

1.248. (2 points)

8.

Pyramid Networks Ltd. manufactures and sells network access software products. Its sales have been increasing in the recent years. The company is planning to expand its work force in response to the increasing trend shown by the sales of its products. The future manpower requirement is expected to be dependent on the level of future sales. The company requires an estimate for the sales in the year 2003. The following data are collected: Year

Sales (Rs. in lakh)

2002

160

2001

135

2000

105

1999

85

1998

60

1997

35

The sales for the year 2003, estimated on the basis of a linear trend equation, is a.

Rs.150.16 lakh

b.

Rs.168.84 lakh

c.

Rs.175.34 lakh

d.

Rs.183.68 lakh

e.

Rs.195.25 lakh. (2 points)

9.

The following data shows the sales and operating costs of a firm over a period of eight years: Year

1992

1993

1994

1995

1996

1997

1998

1999

Sales (Rs. in lakh)

106

140

144

155

136

150

146

143

Operating costs (Rs. in lakh)

100

114

120

125

105

124

122

118

What is the coefficient of correlation between sales and operating costs of the firm? a.

0.676.

b.

0.374.

c.

0.553.

d.

0.854.

e.

0.904. (3 points)

392

10. A simple regression relationship was developed between the variables X and Y, with X as the independent variable. The estimated value of Y when X = 2, is 7.0; and the estimated value of Y when X = 8, is 11.8. Σ Y2 = 791.31; ΣXY = 452.7; ΣY = 78.1; n = 8. The standard error of estimate for the regression line is a.

1.1113

b.

8.4572

c.

7.8482

d.

7.5123

e.

6.5893. (3 points)

11. If n = 36 and p = 0.18, what are the upper and lower control limit of the p chart? a. 0.305 and – 0.012. b. 0.305 and 0.012. c. 0.372 and – 0.012. d. 0.372 and 0.012. e. 0.418 and 0.012. (3 points) 12. Consider the following data: fo 9 14 fe 5 20 The χ2 statistic in the above case is a. 1/4 b. 3/4 c. 5/4 d. 7/4 e. 5. (3 points) 13. Following are price and quantities of 3 commodities used as a basket of goods to measure index. Base Year

Current Year

Commodity

Price (P0) Rs

Quantity (Q0) Kg

Price (P1) Rs

Quantity (Q1) Kg

P

4

10

6

11

Q

6

5

7

7

R

9

6

12

5

What is the price index for current year using Marshall-Edgeworth method? a.

109.52.

b.

131.58.

c.

134.12.

d.

149.12.

e.

158.23. (3 points) 393


14. An institution is offering short-term management program. The time taken by the participants to complete this program is normally distributed with a mean of 1000 hours and a standard deviation of 200 hours. What is the probability that a participant selected at random will require more than 1100 hours to complete the program? a.

0.1915.

b.

0.3085.

c.

0.4085.

d.

0.50.

e.

0.6915. (3 points)

15. The weights of carrier box used by a transport agency is normally distributed with a mean of 95kg and standard deviation of 10kg. The maximum permissible load is 400kg. If four boxes are loaded what is the probability that the maximum load is exceeded? a. 0.1587. b. 0.3413. c. 0.500. d. 0.6587. e. 0.8413. (2 points) 16. The beta coefficient of a stock is 0.8 and the standard deviation of market returns is 15%. The unsystematic risk of the stock is a. 81 b. 144 c. 225 d. 324 e. 360. (2 points) 17. If the standard error of mean for a sample of size 16 is 3, then the population standard deviation will be equal to a. 2 b. 3 c. 6 d. 12 e. 18. (2 points) 18. The possible returns from a security are given below: Return: 5 10 15 Probability: 10% 60% 30% The variance of return from the security is a. 3% b. 6% c. 9% d. 12% e. 16%. (3 points) 394

19. In a Bernoulli experiment the probability of success is 0.2 and the number of trials is 7. The approximate probability of getting one success is a. 0.294 b. 0.367 c. 0.425 d. 0.550 e. 0.625. (3 points) 20. A multiple regression equation has two independent variables. The number of observations is 84 and the error sum of squares is 116.64. The standard error of estimate is a. 0.75 b. 1.20 c. 1.50 d. 1.85 e. 2.15. (2 points) 21. An economist believes that the rent charged by the owners of apartments is determined by the number of rooms and the distance from the city center. The following data pertains to a randomly collected sample of tenants living in apartments situated at various parts of the city: Rent (Rs.) (Y)

Number of rooms (X1)

Distance from city center (kms) (X2)

3300 7500 3750 4200 3300 2750

2 5 3 4 4 3

2 4 2 3 10 6

What is the co-efficient of X1 in the multiple regression equation that best relates the three variables? a. 96.45. b. –290.48. c. 290.48. d. –1526.87. e. 1526.87. (4 points) 22. A financial analyst has collected the following data on sales and net profits of five companies manufacturing electronic components: Sales (Rs. in crores) Net Profit (Rs. in crores)

38

18

45

24

55

1.50

1.50

3.00

2.00

4.00

What is the coefficient of determination between sales and net profit? a. 0.212. b. 0.643. c. 0.704. d. 0.839. e. 0.914. (3 points) 395


23. The manager of Modern Electricals Co. wants to formulate a relationship between total cost and sales revenues which may be used to estimate the total cost for a given level of sales. The following information is collected by him from the past records. Sales (Rs. in lakh) 20 23 38 28 Total cost (Rs. in lakh) 17 22 35 22 What is the standard error of estimate? (Rs. in lakh) a. 0.105. b. 1.517. c. 1.662. d. 1.858. e. 2.056.

45 39

40 36

(3 points) 24. Southern Suppliers Ltd., supplies processed agricultural materials to agro-based industries. At present, the company has submitted quotations in response to tenders invited by 6 parties for a specific material. The probability that it will be able to secure supply contract with exactly one party is 18.66% and the probability that it will be able to secure supply contracts with exactly two parties is 31.10%. The invitations for tenders are independent of each other and the probability of securing supply contract with the party inviting the tender is assumed to remain fixed over time. What is the probability that the company will secure supply contracts with at least 3 parties? a. 54.43. b.

45.34.

c.

45.57.

d.

54.75.

e.

45.43. (3 points)

25. Holy Cross Public School has been experiencing a growth in admissions to its kindergarten classes. The data on admissions to the kindergarten classes of the school during the past are given below: Year Number of admissions

1997

1998

1999

2000

2001

2002

60

70

100

145

165

210

According to the data, what would be the number of admissions to the kindergarten classes of the school for the year 2003? a.

132.

b.

140.

c.

167.

d.

233.

e.

358. (3 points)

396

Model Question Paper II Suggested Answers Part A: Basic Concepts 1.

(e) All the elements stated in the alternatives are commonly found in problems of decision making under uncertain conditions.

2.

(d) If the expected value of a random variable ‘A’ is more than the expected value of another random variable ‘B’, then it means that the random variable ‘A’ is more likely (and not always) to assume higher values than the random variable ‘B’. Alternatives (a), (b) and (c) are true.

3.

(c) The normal distribution curve changes for different values of mean and standard deviation. Thus normally distributed populations having different means and standard deviations will be represented by different normal curves. Hence alternative (c) is not true. Alternatives (a), (b) and (d) are true.

4.

(d) Alternative (a) is not true because when all the alternatives are removed we get the expected profit with perfect information. Alternatives (b) and (c) are not true because they are incorrect interpretations of the expected value of perfect information. Alternative (d) is the true meaning of expected value of perfect information.

5.

(a) An optimistic decision maker uses the maximax criterion under uncertain conditions. Maximin criterion is used by a pessimistic decision maker under uncertain conditions. Expected value criterion is used under conditions of risk. Alternatives (d) and (e) are not correct answers.

6.

(d) F distribution is identified by a pair of degrees of freedom.

7.

(b) When the population consists of well defined groups such that the elements of each group are homogeneous and the groups vary from each other, stratified sampling technique is used.

8.

(e) H1 : μ1 – μ2 < 0. H1 represents alternative hypothesis. If μ2 > μ1 then μ1 – μ2 < 0.

9.

(d) Type I error occurs when the null hypothesis is true and rejected.

10. (c) The purpose of hypothesis testing is to make a judgement about the difference between the sample statistic and the hypothesized population parameter. The purpose of hypothesis testing is not to test the correctness of the sample statistics. Alternative (c) is not true. Alternatives (a), (b) and (d) are true. 11. (d) For a two tailed test of hypothesis at a significance level of 10%, the rejection region lies in each of the left and right tails to the extent of 5% of the area of the normal distribution curve.

12. (d) Standard error of a sample statistic indicates the sampling error arising out of sampling error due to chance. Standard error is not the standard deviation of the population; nor is it the standard deviation of the sample. Standard error does not indicate the extent of non-sampling error.


13. (a) Interval estimates can be used for all populations small or large. Alternatives (b), (c), (d) and (e) are true. 14. (c) Correlation analysis does not indicate the presence of cause and effect relationship between two or more series. Alternatives (a), (b) and (d) are true. 15. (c) In a multiple regression relationship, the problem of multicollearity arises when there is a high degree of correlation between the independent variables. In this case the problem of multicollinearity will arise if there is a high degree of correlation between the variations of X1 and X2. 16. (e) The coefficient of multiple correlation is dependent on the coefficients of correlation between Y and X1, between Y and X2, and between X1 and X2. Alternatives (a), (b), (c) and (d) are true. 17. (b) r = Cov (x, y) . By convention σx and σy (standard deviation of variables ‘x’ and ‘y’) are xy σx σ y

positive. Thus if ‘rxy’ is negative then it indicates that the covariance between x and y is negative. Alternatives (d) and (e) cannot be inferred. 18. (e) A value index cannot distinguish between the price changes and quantity changes. Hence an increase in the value index cannot indicate with certainty that the prices have increased. 19. (a) Index number of industrial production is a quantity index. The other indices are price indices. 20. (b) Laspeyres index is a weighted aggregates price index. Alternatives (a), (c) and (d) are true. 21. (d) The technique used for studying seasonal variations is known as ratio to moving average method. 22. (a) The movement of the dependent variable above and below the secular trend line over periods longer than one year is known as cyclical variation. 23. (a) The ‘increasing trend’ pattern in x charts indicates the presence of non-random variations in the process because it indicates that the process mean is increasing over time. 24. (c) The chi-square test enables us to test whether the proportions of more than two populations can be considered equal. 25. (e) For a portfolio with investments in two securities with equal weightages, the risk is zero, when the standard deviation of both securities is same and the two securities are perfectly negatively correlated. 26. (c) From the normal distribution table 80% of the observation will be in the interval μ – 1.28σ to μ + 1.28σ. 27. (c) Since there is no dispersion in the returns on the bond. σbond = 0 this implies ρ = 0. 28. (b) Probability of success = 50/200 = 1/4= 0.25. 29. (b) In a regression line, Y = a + bX, ‘a’ and ‘b’ represent y-intercept and slope of the regression line respectively. Y = 4 + 2(2) = 8. 30. (e) F ratio is governed by the degrees of freedom of the numerator and the denominator. 31. (c) If the value of LCL is below zero, it is considered as zero. 32. (e) The standard error of estimate in regression analysis is measured perpendicular to the X-axis i.e., line drawn along the Y-axis from the observed point to regression. 33. (e) According to relative frequency concept probability is defined as proportion of times that an event occurs in the long run, when conditions are stable. It is also defined as observed relative frequency of an event in a very large number of trials. 34. (c) If the estimator is not consistent (that is, as the sample size increases, the statistic must get closer to the parameter) we will be wasting resources by taking larger samples. 398

Part III

35. (a) The sample median in symmetrically distributed population is not efficient estimator of the mean, because in large samples, the standard error of the median is larger than that of the sample mean. (Hint: Efficiency of an estimator refers to the size of the standard error of the estimator. Greater the standard error, lesser is the efficiency of the estimator or vice versa.) 36. (c) Theoretically, binomial distribution is the correct one to use in constructing confidence intervals in estimating population proportions since the distribution has two possible outcomes. 37. (e) Time series analysis is helpful in identifying the pattern of changes in the data collected over a period of time. It is possible to determine secular trends, cyclical fluctuations, seasonal variation and irregular variation. 38. (a) Under secular trend of variation, we look the long-term behavior of the variable. This is important in the sense that one should be able to conclude whether the value of the variable has been increasing or decreasing over the years, keeping aside the variation observed in the individual years, which constitute the series. 39. (d) The variations caused by the events like floods, strikes and war etc., can be termed as irregular variation. These variations are inconsistent and highly unpredictable. 40. (c) Under seasonal variation the variable under consideration shows a similar pattern during certain months of the successive years.

Part B: Problems (60 Points) 1.

(c) Unweighted aggregates price index = =

2.

3.

Σp n × 100 Σp 0 (12.50 + 13.50 + 25) × 100 = 130.77. (10.50 + 8.50 + 20)

(d) Fisher’s ideal price index =

Laspeyres price index × Paasches price index

Paasches price index

(124.5)2 (Fisher 's ideal price index) 2 = = 127.05 122 Laspeyres price index

=

(a) Mean of a binomial distribution = n . p ∴The mean of the binomial distribution of the number of successes is = 25(1–0.60) = 10.

4.

5.

6.

(d) If y is dependent on variable x then we can write that the estimated value of y, yˆ = 3x + 5. When x = 5 the estimated value of y would be 3 × 10 + 5 = 35. 20 − 24 ⎛ 28 − 24 ⎞ (e) X lies within ± 0.8 times standard deviation ⎜ = 0.8 and = −0.8 ⎟ of mean. 5 ⎝ 5 ⎠ So the probability is 2 × 0.2881 = 0.5762 i.e. 57.6%.

(d) Proportion of women’s clothing stores in Secunderabad which were sampled and found to 1 carry the products of the firm, p = = 0.20 4 +1 Sample size =

8 = 40 0.20

H0 : p = 0.25 H1 : p < 0.25 Standard error of proportion, σ p =

p H0 q H0 n

=

0.25(1 − 0.25) = 0.06847 40

399


Since both np and nq are more than 5 we can use the normal distribution as an approximation to he sampling distribution. At a significance level of 10%, the critical value for the test is –1.28. The standardized value of the sample proportion, Z=

p − p H0 σp

=

0.20 − 0.25 = – 0.73 0.06847

From above we can see that the standardized proportion falls in the acceptance region. Hence the null hypothesis is accepted. It can be concluded at a significance level of 10%, that the percentage of women’s clothing stores in Secunderabad which carry its products is not significantly less than the percentage in Hyderabad. 7.

σ

(e) Standard error of mean for sample size ‘m’ = Standard error of mean for sample size ‘n’ =

m

σ n

= 2.25 = A

= 1.5 = B

A 2.25 = 1.5 B

or

or

or

σ/ m σ/ n

σ m

×

= 1.5 n = 1.5 σ

n = 1.5 m

or n = 2.25m Sample size = m + n = m + 2.25m = 3.25m. ∴ Standard error of mean for sample size ‘m + n’ = 8.

σ m+n

=

σ 3.25m

1 3.25

×

σ

(d) Let the following notations be used: X : Year x : Coded value for year Y : Sales

400

=

m

=

1 3.25

× 2.25 = 1.2481

Part III

X =

Year

x = (X– X )× 2

Y

xY

x2

1997

–2.5 × 2 = –5

35

–175

25

1998

–1.5 × 2 = –3

60

–180

9

1999

–0.5 × 2 = –1

85

–85

1

2000

0.5 × 2 = 1

105

105

1

2001

1.5 × 2 = 3

135

405

9

2002

2.5 × 2 = 5

160

800

25

580

870

Σx2 = 70

∑X = 1999.5 n

ˆ = a + bx Y ∑ xY

b =

∑x

2

a = Y =

=

870 = 12.43 70

∑Y 580 = = 96.67 6 n

∴ The linear trend equation is

ˆ = 96.67 + 12.43x Y The estimated sales for the year 2003: X

= 2003 ∴ x = (2003–1999.5) × 2 = 7

Estimated sales for the year 2003 = 96.67 + 12.43 × 7 = Rs.183.68 lakh. 9.

(e) Let the following notations be used: X: Sales (Rs. in lakh) Y: Operating costs (Rs. in lakh)

A

X

106

140

144

155

136

150

146

143

ΣX = 1120

B

Y

100

114

120

125

105

124

122

118

ΣY = 928

C

X– X

– 34

0

4

15

–4

10

6

3

D

Y– Y

– 16

–2

4

9

–11

8

6

2

E

(X– X )2

1156

0

16

225

16

100

36

9

Σ(X – X )2 = 1558

F

(Y– Y )2

256

4

16

81

121

64

36

4

Σ(Y – Y )2 = 582

G

C×D

544

0

16

135

44

80

36

6

Σ(X – X ) (Y – Y ) =861

X =

∑ X 1120 = = 140 8 n

Y =

∑Y 928 = = 116 8 n

401


∑(X − X) (Y − Y)

r=

2

∑(X − X) Σ(Y − Y)

861

=

2

1558 × 582

= 0.9042.

ˆ = a + bX 10. (a) Y Given: 7 = a + b (2) (For X = 2) Or 7 = a + 2 b………………….(A) 11.8 = a + b (8) (For X = 8) Or 11.8 = a + 8b……………….(B) ∴b =

(11.8 − 7 ) (8 − 2 )

= 0.80

Putting b = 0.80 in (A) 7 = a + 2 (0.80) or a = 7 – 1.6 = 5.4

ˆ = 5.4 + 0.80X ∴ The regression equation is Y Standard error of estimate; Se =

∑ Y 2 − a ∑ Y − b ∑ XY n−2

ΣY2 = 791.31; ΣXY = 452.7 ΣY = 78.1; n = 8 ∴ Se =

791.31 − (5.4 × 78.1) − (0.8 × 452.7) = 1.1113. 8−2

Upper Control Limit (UCL) of a p chart = p+3

11. (c)

pq n

P = p = 0.18 (0.18) (1 − 0.18) = 0.372. UCL = 0.18 + 3 36

Lower Control Limit (LCL) = p − 3

pq

n = 0.18 – 0.192 = – 0.012 = 0 (Taken as zero of LCL is negative).

12. (e)

⎡ (f − f )2 ⎤ ⎡ (9 − 5) 2 (14 − 20)2 ⎤ + χ2 = Σ ⎢ 0 e ⎥ = ⎢ ⎥ fe 20 ⎣ 5 ⎦ ⎣⎢ ⎦⎥ = 5.0 13.

(c)

Commodity

P0

Qo

P1

Q1

Q0+Q1

P0 (Q0+Q1)

P1(Q0+Q1)

P

4

10

6

11

21

84

126

Q

6

5

7

7

12

72

84

R

9

6

12

5

11

99

132

ΣP0(Q0+Q1)= 255

ΣP1 (Q0+Q1) = 342

Price index =

402

∑(Q0 + Q1 )P1 ∑(Q0 + Q1 )P0

x100

Part III

342 ´ 100 255 = 134.12.

= 14. (b)

Given X =

1100

μ

=

1000

σ

=

200

required p (Y>1100) We have Z =

1100 − 1000 = 0.5 0.55

For Z = 0.5 area under the curve = 0.1915. Probability that participant will require more than 1100 hours = 0.5 – 0.1915 = 0.3085. 15. (a) We have σ x =

σ n

=

10 4

=5

If the mean weight of the boxes exceed 100 kg, the maximum load of 400 kg will be exceeded, therefore required probability is P ( x >100) ⇒Z=

X − μ 100 − 95 = =1 5 σX

P( X >100) = P(Z > 1)

= 0.5 – 0.3413 = 0.1587.

16. (a)

2

Systematic risk = β x Variance of market returns = (0.8 x 0.8)(15 x 15) = 144

Unsystematic risk

= Market variance – Systematic risk

⇒ 225 – 144 = 81 17. (d) Given: Standard error of mean, = 3

n

= 16 (given)

σx =

σ n

⇒ σ x x n = 3 x 16

or σ = 3 × 4 = 12. 18. (c) Expected rate of return

=

Σxi pi

=

(5)(0.10) + (10)(0.60) + (15)(0.30)

=

11%

Variance = Σpi(Ri – R i )2 =

0.10(5 – 11)2 + 0.6(10 – 11)2 + 0.3(15 – 11)2 403


=

3.6 + 0.6 + 4.8

=

9.

19. (b) In a Bernoulli experiment the probability of ‘r’ successes in ‘n’ trials is given by

P (r) = n Cr .p r .q n − r Given:

Probability of success, p = 0.20 No. of successes, r = 1 Number of trials, n = 7

∴Probability of failure, q = 1 – p = 0.80 7

∴P(r = 1) =

20.

C1 7C1.(0.20)1 (0.80) (7–1)

=

7 (0.20) (0.80)6

=

0.367.

(b) Standard error of estimate of the multiple regression relationship

ˆ Σ(Y − Y) = n − k −1

=

116.64 = 1.20. 84 − 2 −1

Let the following notations be used: Y

:

Rent (Rs.)

X1

:

Number of rooms

X2

:

Distance (kms) from the city center

21. (e)

X1

2

5

3

4

4

3

ΣX1

=

21

X2

2

4

2

3

10

6

ΣX2

=

27

Y

3300

7500

3750

4200

3300

2750

ΣY

=

24800

X1Y

6600

37500

11250

16800

13200

8250

ΣX1Y

=

93600

X2Y

6600

30000

7500

12600

33000

16500 ΣX2Y

=

106200

2 X1

4

25

9

16

16

9

2 Σ X1

=

79

X2

2

4

16

4

9

100

36

Σ X2

=

169

X1 X2

4

20

6

12

40

18

ΣX1 X2

=

100

The multiple regression relationship can be written as:

ˆ Y

= a + b1X1 + b2X2

The coefficients ‘a’, ‘b1’ and ‘b2’ can be found out as: ΣY = na + b1 ΣX1 + b2ΣX2 2

ΣX1Y = aΣX1 + b1Σ X1 + b2Σ X1 X2 2

ΣX2Y = aΣX2 + b1ΣX1 X2 + b2Σ X 2 Putting the values we get– 24800 = 6a + 21b1 + 27b2 404

….(1)

2

Part III

93600 = 21a + 79b1 + 100b2

….(2)

106200 = 27a + 100b1 + 169b2

….(3)

Multiplying equation (1) with 7 and equation (2) with 2 and subtracting equation (2) from equation (1): 173600 = 42a + 147b1 + 189b2 187200 = 42a + 158b1 + 200b2 – – – – –13600 = – 11b1 – 11b2 13600 = 11b1 + 11b2 ….(4) Multiplying equation (1) with 9 and equation (3) with 2 and subtracting equation (3) from equation (1): 223200 = 54a + 189b1 + 243b2 212400 = 54a + 200b1 + 338b2 – – – – ….(5) 10800 = – 11b1 – 95b2 Adding equations (4) and (5) we get – 13600 = 11b1 + 11b2 10800 = –11b1 – 95b2 24400 = – 84b2 or b2 = –

24400 = –290.48 84

Putting the value of b2 equation (E) we get– 10800 = –11b1 –95 (–290.476) or 10800 = –11b1 + 27595.6 27595.6 −10800 = 1526.87 11 Substituting the values of b1 and b2 in equation (1) we get – 24800 = 6a + 21 (1526.87) + 27 (–290.48) or 24800 = 6a + 32064.27 – 7842.96

or b1 =

24800 − 32064.27 + 7842.96 = 96.45 6

or a =

∴ yˆ = 96.45 + 1526.87x1 – 290.48x2 22. (c) Let the following notations be used: X: Sales (Rs. in crores) Y: Net profit (Rs. in crores)

Coefficient of correlation, r = Σ(X − X)(Y − Y) / Σ(X − X) 2 Σ(Y − Y) 2 A. B. C.

X Y X−X

38.00 1.50 2.00

18.00 1.50 –18.00

45.00 3.00 9.00

24.00 2.00 –12.00

55.00 4.00 19.00

D.

Y−Y

–0.90

–0.90

0.60

–0.40

1.60

E.

CxD

ΣX ΣY

= =

180 12

Σ(X − X)(Y − Y)

=

55

–1.80

16.20

5.40

4.80

30.40

F.

2

C

4.00

324.00

81.00

144.00

361.00

Σ(X − X)2

=

914

G.

D2

0.81

0.81

0.36

0.16

2.56

Σ(Y − Y)2

=

4.70

405


r

55

=

914 x 4.70

∴r = ⇒ r2 =

0.839 0.704 (approx.)

23. (d) Finding the values of a and b: = a + bX Where, X = Sales (Rs. in lakh) Y = Total costs (Rs. in lakh) b = 194 a = 171 n = 6 = 28.5 = 5987 = 6782 b = 0.899 a = –0.568 = –0.568 + 0.899x Calculation of Standard error

Standard error of estimate (se) =

ΣY 2 − aΣY − bΣXY n−2

ΣY 2 = 5299

=

Se

5299 − (−0.568)(171) − (0.899)(5987) 6−2

= 1.858 i.e., Rs.1.858 lakh. 24. (c) This is a case of Bernoulli process. For a Bernoulli process the probability of ‘r’ successes in ‘n’ trials = n Cr .pr .q n − r Given: P(r = 1) = 18.66% = 0.1866 P(r = 2) = 31.10% = 0.3110 n=6 Let the probability of securing the supply contract with the party inviting the tender be ‘p’. ∴ Probability of not securing the supply contract, q = 1–p P(r = 1) = 6 C1 p1 (1 − p)6 −1 = 0.1866 (given) 6p (1 – p)5 =

or

6

0.1866 2

And, P(r = 2) = C2 p (1 − p)

…. (1) 6−2

= 0.3110 (given) or 15p (1 – p)4 = 0.3110 …. (2) Dividing equation (2) by equation (1) we get: 2

15p 2 (1 − p) 4 6p(1 − p)

or or

406

5 p x 2 1− p

p 1− p

or p

5

0.3110 = 0.1866

=

0.3110 0.1866

=

2 ⎛ 0.3110 ⎞ ⎜ ⎟ 5 ⎝ 0.1866 ⎠

= 0.66667 (1 – p)

= 0.66667

Part III

or 1.66667p

= 0.66667

or p

=

0.66667 1.66667

= 0.4000012

≅ 0.40 ∴ Probability of securing supply contract = 0.40 ∴ Probability of not securing supply contract, q = 1 – 0.40 = 0.60 Probability of securing supply contracts with a maximum of 2 parties i.e., P (r ≤ 2) = P (r =0) + P (r=1) + P (r =2) P (r = 2) = 0.3110 (given) P (r = 1) = 0.1866 (given) P(r = 0)

= 6 C0 p0 (1 − p)6 − 0 = 0.0467

∴P (r ≤ 2) = 0.0467 + 0.1866 + 0.3110 = 0.5443 P (r ≥ 3) = P (r = 3) + P (r = 4) + P(r = 5) + P (r = 6) = 1 – P (r ≤ 2) = 1 – 0.5443 = 0.4557 i.e., 45.57%. 25. (d) Let the following notations be used: X : Year Y : Number of admissions x : Coded value of the time variable (i.e.,year) x-x x = (x - x) x 2 X Y 1997 1998 1999 2000 2001 2002 ΣX = 11997 The linear trend equation is ˆ = Y a + bx Where, x

=

b

=

X

=

ΣxY =

60 70 100 145 165 210 ΣY = 750

–2.5 –1.5 –0.5 0.5 1.5 2.5

–5 –3 –1 1 3 5 ΣX = 0

(X − X) × 2

ΣxY

a = Y Σx 2 ΣX 11997 = = 1999.5 6 n 1080

Σx 2 = 70 Σx = 0 ΣY = 750 1080 = 15.43 b = 70 ΣY 750 = = 125 a = Y = 6 n ∴ The linear trend equation is:

ˆ = 125 + 15.43x Y Year (X) = 2003 ∴

x

= (2003 –1999.5) × 2 = 7 407


ˆ Y

= 125 + (15.43 × 7) = 233.01 i.e., 233 (approx.)

∴The estimated number of admission to the kindergarten classes in year 2003 is 233.

408

Model Question Paper III Time: 3 Hours


Answer all the questions. Each question carries one point. 1.

2.

3.

4.

5.

The control chart which is plotted using the variability of individual characteristics of the output is known as a.

x chart

b.

R chart

c.

P chart

d.

K chart

e.

C chart.

If the pattern of x chart of a process is ‘Hugging the center line’, then it indicates that the a.

Process mean may be drifting

b.

Process mean may have shifted

c.

Two distinct populations are being observed

d.

Variation has been reduced

e.

Few individual outliers are present.

The value of chi-square statistic can never be negative, because a.

The difference between the observed and expected frequencies is always positive

b.

The difference between the observed and expected frequencies is always squared

c.

The observed frequency can never be negative

d.

The expected frequency can never be negative

e.

The sum of observed and expected frequencies is always positive.

Which of the following is true with regard to Type II error in hypothesis testing? a.

It is the sampling error due to chance.

b.

It is also known as procedural bias.

c.

It is the error of accepting a null hypothesis when it is false.

d.

It is the error of rejecting a null hypothesis when it is true.

e.

All of the above.

The method of least squares indicates the smallest possible a.

Sum of the squared values of the vertical distances from the regression line to the X-axis at the corresponding ‘x’ values of the plotted points

b.

Sum of the squared values of the horizontal distances from the regression line to the Y-axis at the corresponding ‘y’ values of the plotted points

c.

Sum of the squared values of the vertical distance from each plotted point to the regression line

d.

Sum of the squared values of the horizontal distance from each plotted point to the regression line

e.

All of the above.

6.

7.

Which of the following is the correct representation of the alternative hypothesis for left-tailed tests? (μ0 is the hypothesized value of the population mean) a.

H1 : μ = μ0.

b.

H1 : μ ≠ μ0.

c.

H1 : μ > μ0.

d. H1 : μ < μ0. e. All of the above. Which of the following distributions should be used in hypothesis testing when the population standard deviation is known and the sample size in less than 30? a. Normal distribution. b. t-distribution. c. F-distribution. d. Binomial distribution. e. All of the above.

ˆ = a + b1X1 + b2X2, where X1 and X2 are two independent variables, then the problem of If Y multicollinearity will arise when the a. Degree of correlation between Y and X1 is high b. Degree of correlation between Y and X2 is high c. Degree of correlation between X1 and X2 is high d. Value of a and b1 are higher compared to b2 e. Value of a and b2 are higher compared to b1. 9. Which of the following is/are the use(s) of index numbers? a. Establishing trends. b. Policy making regarding wage payment in an organization. c. Determining the purchasing power of the currency. d. Deflating the time series data. e. All of the above. 10. Which of the following is false? a. Laspeyres’s price index has a downward bias. b. Paasche’s price index generally tends to underestimate the prices. c. In fixed weight aggregate method of index, the weights are from a representative period. d. Fisher’s ideal index is the geometric mean of the Laspeyres and Paasches indices. e. Value index number combines price and quantity changes. 11. In which of the following types of time series, the long-term behavior of a variable is noticed? a. Secular trend. b. Cyclical variation. c. Seasonal variation. d. Irregular variation. e. Estimated variation. 12. The business cycle is an example of a. Secular variation b. Cyclical variation c. Seasonal variation d. Irregular variation e. Relative variation.

8.

409


13. Which component of a time series can be isolated by the ‘Residual Method’? a. Secular. b. Cyclical. c. Seasonal. d. Irregular. e. All of the above. 14. Which of the following is not a characteristic of Bernoulli process? a. Each trial has only two outcomes. b. The probability of outcome of any trial remains fixed over time. c. The trials are statistically independent. d. The probability of occurrence of one outcome is equal to the probability of occurrence of the other. e. All of the above. 15. Which of the following is not a characteristic of normal probability distribution? a. The mean of the normally distributed population lies at the center of its normal curve. b. It is multi-modal. c. The mean, median and mode are equal. d. Both the tails extend indefinitely and never meet the horizontal axis. e. All of the above. 16. The total area under the normal curve is a. Zero b. μxσ c. μ/σ d. σ/μ e. One. 17. Under conditions of uncertainty, an optimistic decision maker normally adopts the ___________ criterion in decision-making. a.

Maximax

b.

Maximin

c.

Hurwicz

d.

Regret

e.

Minimax.

18. In which of the following, the decision is made based upon weighted profits? a.

Maximax criterion.

b.

Maximin criterion.

c.

Hurwicz criterion.

d.

Regret criterion.

e.

Minimax criterion.

19. Which sampling method selects samples that allow each possible sample to have an equal probability of being chosen and each item in the entire population to have an equal probability of being included in the sample?

410

a.

Simple Random.

b.

Systematic.

c.

Stratified.

d.

Cluster.

e.

Judgmental.

20. The standard error of mean indicates the a.

Variance of the distribution of all possible sample means for a specific sample size

b.

Standard deviation of the distribution of all possible sample means for a specific sample size

c.

Mean Absolute Deviation of the distribution of all possible sample means

d.

Range of the distribution of all possible sample means

e.

Coefficient of variation of the distribution of all possible Sample Means.


The mean of the sampling distribution of mean will equal the population mean.

b.

The sampling distribution of mean approaches normality as the sample size increases.

c.

The central limit theorem enables us to make use of the properties of the normal distribution in many cases where the underlying data may not be normally distributed.

d.

As the standard error of mean increases, the value of any sample mean will be closer to the value of the population mean.

e.

As the standard error decreases, the precision with which the sample mean can be used to estimate the population mean increases.


A point estimate is a single number that is used to estimate an unknown population parameter.

b.

A point estimate is often insufficient, because it is either right or wrong.

c.

An interval estimate is a range of values used to estimate a population parameter.

d.

An interval estimate indicates the probability of the true population parameter lying within the range of values given.

e.

Point estimates are used for large populations.

23. For which of the following it is possible to change the base year without changing the weights? a.

The weighted aggregates method.

b.

The Laspeyres method.

c.

The Paasche method.

d.

Fixed weight aggregater method.

e.

Value index.

24. The advantage(s) of using the Laspeyres method is/are a.

One index can be easily compared with another

b.

Changes in consumption patterns are taken into account

c.

The satisfaction of the circular test

d.

Quantity measures need not be tabulated for every period

e.


25. Coefficient of determination is employed to calculate a.

Beta

b.

Systematic risk

c.

Unsystematic risk

d.

Financial risk

e.

Business risk. 411


26. Which of the following determines the sampling precision? a. Absolute size of the sample. b. Fraction of the population sampled. c. Finite population multiplier. d. Both (a) and (b) above. e. All of (a), (b) and (c) above. 27. The method(s), which satisfy circular test of consistency is/are: a. Fisher’s ideal index b. Fixed weight aggregates method c. Laspeyres and Paasche’s method d. Simple aggregates method e. Both (b) and (d) above. 28. D4 in R control charts represents a. 1 + 3d3 / d2 b. 1 – 3d2 / d3 c. 1 + d2 / 3d3 d. 1 – d3 /3d2 e. 1 – 3d3/d2. 29. Which of the following charts are used to study the characteristics of attributes? a. p Charts.

x Charts. b. c. R Charts. d. Both (a) and (b) above. e. All of (a), (b) and (c) above. 30. Which of the following is/are false? a. Chi-square value is negative if the expected value is greater than the actual value. b. The mean calculated when all the individual observations are taken into account is called as grand mean. c. Chi-square distribution consists of a family of distributions. d. F distribution is unimodel in nature. e. All of the above. 31. Which of the following is used to reduce the sampling error in the study of a heterogeneous population? a. Cluster sampling. b. Systematic sampling. c. Survey sampling. d. Stratified sampling. e. Simple random sampling. 32. The estimating equation regarding the sale of refrigerators for three years 1996, 1997 and 1998 taking 1997 as a base year was found to be y = 2.65 + 1.10X. Then the estimate of sales for 1999 is _________. a. 4.85 b. 10.75 c. 11.00 d. 11.25 e. 11.60. 412

33. The weighted average of relative price index, when calculated with the base values is same as a.

Laspeyre’s method

b.

Paasche’s method

c.


d.

Marshall edgeworth index

e.

All of the above.

34. Which of the following series indicates the moving average of two data points of the observations 4, 5, 7, 8, 5, 2? a.

4.5, 6, 7.5, 6.5, 3.5.

b.

9, 12, 15, 13, 7.

c.

1, 2, 1, 3, 3.

d.

2, 2.5, 3.5, 4, 2.5.

e.

3, 4, 5, 4, 2.

35. Percent of trend measure is used to a.

Study secular trend

b.

De-seasonalizing a time series

c.

Relative cyclical residual

d.

Observe the variations from a trend line

e.

Study irregular variation.

36. We want to determine whether the population mean is significantly larger or smaller than 5. The alternative hypothesis is a.

μ=5

b.

μ<5

c.

μ>5

d.

μ≠5

e.

Cannot be determined from the given information.

37. If the dependent variable increases as the independent variable increases in an estimating equation, the co-efficient of determination will be in the range a.

–1 to 0

b.

0 to 1

c.

1

d.

0 to 0.5

e.

0.5 to 1

38. Which of the following statements is/are true? a. Secular trend allows us to describe a historical pattern in the data. b. Secular trend represents the short-to medium-term variation in time series data. c. The component of a time series that oscillate above and below the trend line for a period smaller than one year is known as irregular variation. d. Seasonal variations are periodic and repetitive in nature. e. Both (a) and (d) above. 39. Systematic risk of a security is proportional to a. β b. β2 c. 1/β d. 1/β2 e. (1–β). 413


40. The variables whose value is to be determined through the process of simulation is called as a. System variables b. Environmental variables c. Decision variables d. Endogenous variables e. Random variables.


For a binomial experiment, the standard deviation of the distribution of the number of successes is 3.6 and the probability of success is 0.4. What is the mean of the distribution of the number of successes? a.

1.44.

b.

3.60.

c.

6.00.

d.

12.96.

e.

21.60. (1 point)

2.

The possible returns from a security are given below: Return

(%)

–5

8

12

16

Probability

(%)

10

20

40

30

What is the expected return from the security? a.

10.20%.

b.

10.70%.

c.

11.40%.

d.

12.30%.

e.

13.40%. (1 point)

3.

The variances of variables, X and Y, are 49 and 16 respectively, and the covariance between them is 14. What is the coefficient of correlation between X and Y? a.

0.2857.

b.

0.50.

c.

0.875.

d.

0.05.

e.

Cannot be determined. (1 point)

4.

A fund manager’s index of optimism is 0.6 and, the maximum and minimum profits from a portfolio are Rs.20,000 and Rs.3,000 respectively. According to Hurwicz criterion, how much profit from this portfolio will be considered? a.

Rs.13,200.

b.

Rs.13,800.

c.

Rs.15,000.

d.

Rs.18,000.

e.

Cannot be determined. (1 point)

414

5.

The population variance is 576 and the size of a sample taken from the population is 36. What is the standard error of mean? a.

0.67.

b.

1.50.

c.

4.00.

d.

16.00.

e.

96.00. (1 point)

Based on the following information answer the Questions from 6 to 9. Softech Solutions Ltd. has recorded the following levels of Return On Equity (ROE) over the last six years: Year

1998

1999

2000

2001

2002

ROE (%)

15

19

21

27

35

2

A second degree estimating equation of the form Y = a + bx + cx , is required to be formulated for

(

)

estimating the ROE for a given year. The time variable (X) has to be coded as x = X − X . 6.

The ‘b’ coefficient of the estimating equation is approximately a.

1.2

b.

2.4

c.

3.6

d.

4.8

e.

5.4. (2 points)

7.

The ‘c’ coefficient of the estimating equation is approximately a. 0.857 b. 1.125 c. 2.250 d. 3.126 e. 4.196.

8.

The ‘a’ coefficient of the estimating equation is approximately

(2 points) a.

4.8

b.

8.6

c.

12.25

d.

16.56

e.

21.69. (2 points)

9.

The estimated return on equity for the year 2003 is approximately a. 37.5% b. 40.36% c. 43.80% d. 45.25% e. 50.65%. (1 points) 415


Based on the following information answer the Questions from 10 to 13. The advertisement expenditure and the sales of Magna Products Ltd. for the last six years, are given below: (Rs. in lakh) Advertisement expenditure

37

63

60

88

80

73

Sales

480

540

590

870

630

790

An estimating equation to predict sales from advertisement expenditure has to be formulated using the simple regression technique. 10. The slope of the estimating equation will be a. 3.252 b. 4.186 c. 6.120 d. 6.928 e. 7.236. (2 points) 11. The intercept of the estimating equation will be a. 6.928 b. 69.285 c. 120.560 d. 186.979 e. 228.462. (1 point) 12. If the advertisement expenditure is Rs.78 lakh then the estimated sales will be a. Rs.601.354 lakh b. Rs.650.246 lakh c. Rs.727.363 lakh d. Rs.752.624 lakh e. Rs.787.265 lakh. (1 point) 13. The standard error of estimate for the estimating equation will be a.

Rs.26.356 lakh

b.

Rs.45.219 lakh

c.

Rs.55.126 lakh

d.

Rs.75.113 lakh

e.

Rs.94.449 lakh. (2 points)

14. If the probability of success in a Bernoulli experiment is 0.60 and the test is repeated 67 times, then the standard deviation of the results of this experiment is a.

3.37

b.

3.61

c.

3.76

d.

3.92

e.

4.01. (2 points)

416

15. If a regression line is represented by y = 0.5 – 0.02x and the co-efficient of determination is 0.4624 then, the co-efficient of correlation is a. – 0.68 b. – 0.68 and 0.68 c. – 0.02 d. 0.50 e. 0.68. (2 points) 16. New Era Ventures Ltd. (NEVL) is a venture capital company. The company has identified three ventures. In three different fields viz., bio-technology, media and web services. The business environment may assume any one of three states in future viz., high growth, moderate growth and low growth, and the company has no information about what the state of business environment will be. For an investment of Rs.10 crore made in each of the three types of venture the expected gains are given below: (Rs. in crore) Business environment

Type of venture Bio-technology

Media

Web services

High growth

6.0

5.0

4.0

Moderate growth

3.5

4.0

3.0

Low growth

1.5

2.0

2.5

Using the maximin criterion of decision-making, in which venture NEVL should invest? a. Bio-technology. b. Media. c. Web services. d. Both (a) and (b) above. e. Both (b) and (c) above. (3 points) 17. The manager of Popular Fast Foods is deliberating on the number of pieces of a popular snack, “Magic Puff,” to prepare for the evening. Each piece of “Magic Puff” costs Rs.10 to prepare and is sold at Rs.16. Any unsold pieces cannot be stored because the item is highly perishable. The manager of the shop has arrived at the following probability distribution of sales on the basis of past experience. Probability Number of pieces that can be sold 150 0.05 151 0.10 152 0.15 153 0.30 154 0.20 155 0.12 156 0.08 What is the minimum required probability of selling one or more additional pieces of the item in order to justify preparing them for the evening? a. 0.4. b. 0.6. c. 0.625. d. 0.375. e. 10. (3 points) 417


18. A portfolio consists of two stocks A and B in which funds have been invested in the ratio of 2:3. If the expected returns on stocks A and B are 15% and 25% respectively then, the expected return on the portfolio is a.

14%

b.

17%

c.

18%

d.

21%

e.

23%. (1 point)

19. Consider the following three samples. Sample 1 Sample 2 Sample 3 6

6

4

2

6

6

4

2

5

9

1

Which of the following choices best estimate the population variance using the variance within the samples? a.

3.25.

b.

4.00.

c.

4.63.

d.

5.84.

e.

6.25. (3 points)

20. A variable X, which has a normal distribution, has a mean of 24 and a standard deviation of 5. The probability that the value of X lies in the interval 20 to 28 is approximately a.

24.7%

b.

31.5%

c.

38.5%

d.

49.5%

e.

57.6%. (2 points)

21. An investor, who is neither completely optimistic nor completely pessimistic, follows Hurwicz criterion in selecting the portfolio. If his index of optimism is 0.7 and, the maximum and minimum profit from the portfolio under consideration are Rs.15,000 and Rs.2,000 respectively, then the profit from this portfolio that will be considered for final selection is a.

Rs.5,900

b.

Rs.8,500

c.

Rs.11,100

d.

Rs.12,300

e.

Rs.15,000. (2 points)

418

22. The population standard deviation is 20 and the sample size is 16. What is the standard error of mean if sample is drawn with replacement? a.

1.12.

b.

1.25.

c.

5.00.

d.

6.75.

e.

9.23. (2 points)

23. On a 30 question, 5 answer multiple choice examination taken by a candidate who knows absolutely nothing about the subject and must guess, independently each answer. What is the approximate probability of answering exactly 5 questions correctly? a.

0.0320.

b.

0.1667.

c.

0.1723.

d.

0.2000.

e.

0.2334. (2 points)

24. Two independent samples were collected on the consultation fees charged by the general physicians in two cities in Andhra Pradesh. The observations on consultation fees have been summarized and presented below: City

Number of doctors sampled

Sum of the observations (Rs.)

Sum of the squares of the observations (Rs.2)

Hyderabad

169

10140

640000

Vishakapatnam

144

7200

380000

Which of the following statements are correct? a.

b.

c.

The mean consultation fees charged by the general physicians in Hyderabad is higher than the mean consultation fees charged by the general physicians in Vishakapatnam. The mean consultation fees charged by the general physicians in Vishakapatnam is higher than the mean consultation fees charged by the general physicians in Hyderabad. The mean consultation fees charged by the general physicians in Hyderabad is equal to the mean consultation fees charged by the general physicians in Vishakapatnam.

d.

Number of samples are insufficient to give the conclusion.

e.

No conclusion can be made.

(4 points) 25. The following data pertain to the operations of Metro Finance Ltd. in the recent years: Year

2002 2001 2000 1999 1998 1997

Amount of advances (Rs. in crore)

7.0

5.5

5.0

4.5

4.0

4.0

Profit after tax (Rs. in lakh)

54

48

42

27

19

14

419


If a simple regression equation is developed on the basis of the given data, with the amount of advances as the independent variable, then what will be the standard error of estimate? a.

7.23 lakh.

b.

5.906 lakh.

c.

10.22 lakh.

d.

20.9 lakh.

e.

10.1 lakh. (4 points)

26. A dealer of grocery-items requires an overall comparison of the prices of the commodities he deals in, between the years 1996 and 2000. The following information is provided by him: Year Commodity Rice Wheat Pulses Sugar Salt

1 Price (Rs./Kg) 10.00 8.00 20.00 12.00 4.00

2 Quantity (Kg) 900 700 500 100 50

Price (Rs./Kg) 16.00 9.00 30.00 15.00 5.00

Quantity (Kg) 800 700 400 80 40

What is the Fisher’s ideal price index, using year 1 as the base year? a.

102.87.

b.

105.83.

c.

143.54.

d.

206.03.

e.

217.82. (4 points)

27. A newly formed company is marketing their product in four districts. The marketing manager of the company is concerned that their market share is unevenly distributed in the four districts. In a survey, a random sample of 100 consumers in each district was taken. The following results were obtained: District A

District B

District C

District D

Number of consumers who purchase the product

30

40

45

35

Number of consumers who don’t purchase the product

70

60

55

65

Which of the following best represents the χ2 value? a.

6.486.

b.

5.486.

c.

6.224.

d.

5.334.

e.

3.224. (4 points)

420

28. Northern Transport Company is planning to start an overnight luxury bus service between Delhi and Jaipur from March, 2002. A survey was conducted to assess the number of passengers available per day at different ticket prices. The results are given below: Price per ticket (Rs.)

250

275

300

325

350

Number of passengers per day

110

90

80

70

50

If equation of straight line is formed to predict number of passangers per day what will be the slope of that line? a.

0.43.

b.

–0.56.

c.

0.68.

d.

–0.79.

e.

–0.96. (4 points)

421

Part IV

Model Question Paper III Suggested Answers Part A: Basic Concepts 1. (b) The control chart which is plotted using the variability of individual characteristics of the output is known as R chart. 2. (d) When the pattern of x chart of a process is ‘Hugging the center line’, then it indicates that the variation has been reduced to a great extent. 3. (b) χ 2 = ∑ positive.

(f0 − fe ) fe

2

2

, so, the main reason of χ being positive is that numerator is always

4. (c) In case of committing a Type II error, the null hypothesis is false but sample statistic falls in the acceptance region. 5. (c) The method of least squares indicates the smallest possible sum of the squared values of the vertical distance from each plotted point to the regression line. Other options are incorrect. 6. (d) In the left tailed test, the critical region lies entirely in the left tail of the sampling distribution, which indicates that H1: μ < μ0. 7. (a) In hypothesis testing when the population standard deviation is known and sample size is less than 30, then normal distribution to be used. 8. (c) As the degree of correlation between the independent variable increases, the problem of multicollinearity arises. 9. (e) All the alternatives are stating the different uses of an index number. 10. (a) Laspeyres tends to over estimate the rise in prices or has an upward bias. Rest all the options are correct. 11. (a) Secular trend helps us to conclude whether the value of a variable has been increasing or decreasing over a long time. 12. (b) Reaching a high, its gradual slow down, a depression and then recovery – it is a perfect example of cyclical variation. 13. (b) Residual method facilitates in isolating cyclical variation from a time series. 14. (d) Alternative (a), (b) and (c) are the characteristics of a Bernoulli process. The probability of two outcomes not necessarily should be same. 15. (b) There is only one mode in case of a normal probability distribution. 16. (e) The total area under a normal curve indicates the sum total of the probability, which is 1 or 100%. 17. (a) An optimistic decision maker identifies the maximum profit associated with the selection of each alternative and the maximum of those chosen as the decision. This criteria is known Maximax Criterion. 18. (c) The decision maker, lies in between two extremes, neither complete optimistic nor complete pessimistic, chooses the maximum of the weighted profits. This approach is known as Hurwicz criterion. 19. (a) In simple random sampling each possible sample has an equal chance of being selected. 20. (b) Standard error of mean indicates the standard deviation of the distribution of all possible sample means for a specific sample size. 21. (d) As the standard error increases, the value of sample mean will be away from the value of the population mean. 22. (e) Point estimates are used for all sizes of population.

Part III

23. (d) In fixed weight aggregates method, the weights used are neither from base period nor from current period but from a representative period. 24. (e) Laspeyres method uses the quantity consumed during the base period in computing the index number therefore quantity measures need not be tabulated for every period, besides this comparability is an added advantage. 25. (b) Systematic risk of the security = R2 (coefficient of determination) x Variance of security returns. 26. (a) Absolute size of the sample determines the sampling precision. 27. (e) Only the fixed weight aggregates method and simple aggregates method satisfy circular test of consistency. 28. (a) In R Chart Lower and Upper control limits are given by : ⎛ 3d ⎞ LCL = R ⎜ 1 − 3 ⎟ and d2 ⎠ ⎝ ⎛ 3d ⎞ VCL = R ⎜ 1 + 3 ⎟ d2 ⎠ ⎝ 1−

3d 3d 3 is denoted by D3 and 1 + 3 by D4. d2 d2

29. (a) p charts are used to study the characteristics of attributes. 30. (a) The value of chi-square is always positive. 31. (d) Stratified sampling is generally used when the population is heterogeneous. In this case, the population is first subdivided into several parts (or small groups) called strata according to some relevant characteristics so that each stratum is more or less homogeneous. Each stratum is called a sub-population. Then a small sample (called sub-sample) is selected from each stratum at random. All the sub-samples combined together form the stratified sample. The sample selection is made by drawing equal numbers of elements from each stratum and weight the results or by drawing number of elements from each stratum proportional to their weights in the population. 32. (a) The value of x for 1996, 1997 and 1998 are –1, 0 and 1 respectively. Thus, the value of x for the year 1999 is 2. Y = 2.65 + 1.1(2) = 4.85. 33. (a) The weighted average of relative’s price index, when calculated with the base values is same as Laspeyre’s method. 34. (a) The moving average of two data points of the observations 4, 5, 7, 8, 5 and 2 are (4 + 5)/2, (5 + 7)/2, (7 + 8)/2, (8 + 5)/2 and (5 + 2)/2, that is 4.5, 6, 7.5, 6.5 and 3.5. 35. (d) When percent of trends are calculated and plotted on a graph, variations from the trend line can be observed. 36. (d) Since we are testing whether the mean is significantly larger or smaller than 5, the alternative hypothesis is equal to μ ≠ 5 and H0 = 5. 37. (b) Coefficient of Determination R2 lies between 0 and 1. 38. (e) Under secular trend, long-term behavior of the variable. It allows us to describe a historical pattern in the data. In seasonal variation the variable under consideration shows a similar pattern during certain months of the successive years. 39. (b) Systematic risk of a security = β2 x variance of the market returns ⇒ Systematic risk α β2. 40. (c) Decision variables are those variables whose value is to be determined through the process of simulation. These may assume differing values under differing sets of circumstances.

423


Part B: Problems 1.

(e) Here, p = 0.4 ∴

q = 0.6

Standard deviation (s) = 3.6 ∴Variance = s2 = npq = 3.62 ∴Mean = np =

npq q

=

3.62 = 21.6. 0.6

2.

(b) The expected return = (–5) x 0.10 + 8 x 0.20 + 12 x 0.4 + 16 x 0.3 = 10.70 %.

3.

(b) The coefficient of correlation between x and y =

4.

(a) According to Hurwicz criterion, profit from this portfolio

Cov(x,y) v(x). v(y)

=

14 49 ×16

= 0.5 .

= 20,000 x 0.6 + 3,000 x 0.4 = Rs.13,200. 5.

(c) Variance = σ 2 = 576 ∴ σ = 576 = 24 The standard error of mean is σ x =

6.

σ n

=

24 36

= 4.00 .

(d) Let the estimated equation be Y= a + bx + cx2 The values of a, b, and c can be obtained from the following equations: ∑Y = na + c ∑x2

-------------------

1

∑x2 Y = a.∑x2 + c∑x4

-------------------

2

------------------

3

b=

∑ xY 2 ∑x

Y

X

X−X = x

x2

x4

xY

x2 Y

15

1998

–2

4

16

–30

60

19

1999

–1

1

1

–19

19

21

2000

0

0

0

0

0

27

2001

1

1

1

27

27

35

2002

2

4

16

70

140

0

10

34

48

246

117 X=

10, 000 = 2, 000 5

From the equation (3), we get b = 48 / 10 = 4.8.

424

Part III

7.

(a) Refer Solution 6. From the equations 1 and 2 we get, 117 = 5a + 10c

………(1)

246 = 10a + 34c

………(2)

Multiplying equation 1 by 2 and subtracting equation 2 from it: 10a + 20c = 234 10a + 34c = 246 –

–

–

– 14c = – 12 12 = 0.857 14 ∴ 10a + 20 (0.857) = 234 or, 10a = 234 – 17.14 or 10a = 216.86 216.86 ∴a= = 21.69 approx 10

∴ c=

8. 9.

The estimating equation will be Y = 21.69 + 4.8x + 0.857x2. (e) Refer Solution 7. (c) The estimated return on equity for the year 2003 will be: X = 2003 ∴ x = 2003 – 2000 = 3 Y = 21.69 + 4.8 × 3 + 0.857 × 9 or, Y = 43.803 ≈ 43.80 ∴ The return on equity for the year 2003 = 43.80%.

10. (d) Let the following notations be used: X = Advertisement expenditure Y = Sales ˆ = a + bX Y

b =

n ∑ XY - ∑ X ∑ Y n ∑ X 2 - (∑ X) 2

∑ XY = 2, 71,810; ∑ X = 401; ∑ Y = 3,900 2 ∑ X = 28, 411 ; n = 6

b=

6(2, 71,810) − (401)(3,900) 66,960 = = 6.928 . 9, 665 6(28, 411) − (401) 2

⎛ 3,900 ⎞ ⎛ 401 ⎞ 11. (d) a = Y − bX = ⎜ ⎟ − 6.928 ⎜ 6 ⎟ = 186.979 . 6 ⎝ ⎠ ⎝ ⎠

ˆ = 186.979 + (6.928 × 78) = Rs.727.363 lakh. 12. (c) For X=78, Y 13. (e) Standard error of estimate, s e =

2 ∑ Y - a ∑ Y - b∑ XY n-2

2 ∑ Y = 26, 48, 000; ∑ Y = 3,900; ∑ XY = 2,71,810

n = 6; a = 186.979; b = 6.928

425


∴ se =

26, 48, 000 − (186.979× 3, 900) − (6.928× 2, 71,810) = Rs.94.449 lakh. 6−2

14. (e) Standard deviation of binomial distribution npq = 67 x 0.6 x (1 − 0.6) = 4.01.

15. (a) Coefficient of correlation = Square root of coefficient of determination. Further the sign taken by the coefficient of correlation is the same as the sign of the ‘b’ coefficient in the regression relationship. So in this case the coefficient of correlation is the negative square root of the coefficient of determination. Coefficient of correlation = –0.68 16. (c) Maximin Criterion: (Rs. in crore) Venture type Biotechnology Media Web service

Business Environment High growth Moderate growth Low growth 6.0 3.5 1.5 5.0 4.0 2.0 4.0 3.0 2.5

Minimum Gain 1.5 2.0 2.5

According to the maximin criterion, the alternative which provides the maximum of the minimum gains, should be selected. In the above case, the web services provides the maximum of minimum gains. Hence according to the maximin criterion, NEVL should invest in the web services. 17. (c) Marginal Profit (MP) = 16 – 10 = Rs.6 Marginal Loss (ML) = Rs.10 Minimum required probability of selling one or more additional pieces to justify their preparation, p* =

ML 10 = = 0.625. ML + MP 10 + 6

18. (d) Expected return on the portfolio = wAkA + wBkB B

B

= (2/5)(15) + (3/5)(25) = 21%. 19. (d) Estimated population variance using the variance within the samples (σˆ 2 )

⎛ n j −1 ⎞ 2 ⎟s j ⎝ nT − k ⎠

= ∑⎜

(6 − 4)2 + (2 − 4)2 + (4 − 4)2 3 −1 =4 2 2 2 2 s 22 = (6 − 5.75) + (6 − 5.75) + (2 − 5.75) + (9 − 5.75) 4 −1

s12

s 32

426

=

=

24.75 = 8.25 3

=

(4 − 4)2 + (6 − 4) 2 + (5 − 4)2 + (1 − 4)2 = 4.67 4 −1

Part III 2

∴ σˆ =

⎛ 3 −1 ⎞ ⎜ ⎟ (4) + ⎝ 11 − 3 ⎠

⎛ 4 −1 ⎞ ⎜ ⎟ (8.25) + ⎝ 11 − 3 ⎠

⎛ 4 −1 ⎞ ⎜ ⎟ (4.67) = 5.84. ⎝ 11 − 3 ⎠

20 − 24 ⎛ 28 − 24 ⎞ 20. (e) X lies within ± 0.8 times standard deviation ⎜ = 0.8 and = − 0.8 ⎟ of mean. 5 ⎝ 5 ⎠ So the probability is 2 x 0.2881 = 0.5762 57.6%.

21. (c) 15,000 x 0.7 + 2,000 x (1 – 0.7) = 11,100. 22. (c) Standard error of mean =

σ n

=

20 16

= 5.

23. (c) We have p = 0.20 (success) and 9 = 0.80 (failure) the probability of answering exactly r = 5 questions out of n = 30 questions correctly is given by

⎛n⎞ P (r) = ⎜ ⎟ ( p r )( q n − r ) ⎝r⎠ 5 25 ⎛ 30 ⎞ = ⎜ ⎟ ( 0.20 ) ( 0.80 ) ⎝ 5 ⎠

= 0.1723. 24. (a) Let the following quotations be used: X1 =

Sample mean of general physicians’ fees at Hyderabad

μ1

=

Population mean of general physicians’ fees at Hyderabad

n1

=

Size of sample taken at Hyderabad

X2 =

Sample mean of general physicians’ fees at Vishakapatnam

μ2 =

Population mean of general physicians’ fees at Vishakapatnam

=

n2

Size of sample taken at Vishakapatnam.

Given: n1 = 169 ΣX1 = 10140 ΣX12 = 640000 n2 = 144 ΣX 2 = 7200 ∴ X1 = S12 =

ΣX 22 = 380000

ΣX 1 10140 = = 60 i.e., Rs.60 n1 169

ΣX12 n 1 x 12 640000 169 (60) 2 − = − n 1 − 1 n 1 − 1 169 − 1 169 − 1

= 188.095 (Rs.2) (approx.) X2 =

S 22 =

ΣX 2 7200 = = 50 i.e., Rs.50 n2 144

ΣX 22 n 2 x 22 380000 144 (50) 2 − = − n 2 − 1 n 2 − 1 144 − 1 144 − 1

= 139.860 (Rs.2) (approx.) H0 : μ 1 = μ 2 H1

: μ1 > μ 2

Since the standard deviations of the two populations are not known, we use the standard deviations of the samples as the estimates for the standard deviations of the two populations. ∴ σˆ 1 = s1 Hence σˆ 12 = s12 = 188.095

427


σˆ 2 = s2 Hence σˆ 22 = s 22 = 139.860

Estimated standard error of the difference between the two means, σˆ 12 σˆ 22 188.095 139.860 + = + = 1.44 (approx) n1 n 2 169 144

σˆ x1 − x 2 =

z=

(x 1 − x 2 ) − (μ 1 − μ 2 ) H0 σˆ x − x 1

2

(μ1 − μ 2 )H = 0 0

(60 − 50) − 0 z= = 6.94 1.44 Since we are using large sample sizes (greater than 30) we will use the normal distribution.

Critical value of z at 0.05 level of significance in right tail = +1.64

The z-value for the difference between the two samples is greater than the critical value of z. Hence the given difference falls in the rejection region. We infer that the mean consultation fees charged by the general physicians in Hyderabad is higher than the mean consultation fees charged by the general physicians in Vishakapatnam. 25. (a) Let the following notations be used: X : Amount of advances (Rs. in crore) Y : Profit after tax (Rs. in lakh) Coefficient of correlation is given by: r

=

∑ (X − X)2 (Y − Y) 2

A.

X

7.0

5.5

5.0

4.5

4.0

4.0

ΣX = 30

B.

Y

54

48

42

27

19

14

ΣY = 204

C.

(X – X )

2

0.5

0

–0.5

–1

–1

D.

(Y – Y )

20

14

8

–7

–15

–20

E.

(X – X )2 4

0.25

0

0.25

1

1

Σ (X − X) 2 = 6.5

F.

(Y – Y )2 400

196

64

49

225

400

Σ(Y − Y)2 = 1334

7

0

3.5

15

20

Σ(X − X) (Y − Y) = 85.5

G. (X − X) (Y − Y)

40

X =

∑ X 30 = 5.0 = n 6

Y =

∑ Y 204 = 34 = n 6

∴ 428

∑ (X − X) (Y − Y)

Cov(X, Y) = σ x .σ y

r =

85.5 6.5 x 1334

= 0.9182

Part III

Proportion of variations in profit after tax that is explained by the variations in the amount of advances = r2 = (0.9182)2 = 0.8431 ∴ Percentage of variations in profit after tax that is explained by the variations in the amount of advances = 0.8431 x 100 = 84.31%. Proportion of variations in profit after tax that is explained by variations in the amount of advances = r2 = 1 –

ˆ 2 Σ(Y − Y) Σ(Y − Y) 2

From above, we have the following: r2

= 0.8431

Σ(Y − Y) 2 = 1334

∴ 0.8431 = 1 – or

ˆ )2 Σ(Y − Y 1334

ˆ )2 Σ(Y − Y = 1 – 0.8431 = 0.1569 1334

ˆ ) 2 = 209.305 ∴ Σ(Y − Y

Standard error of estimate =

ˆ )2 Σ(Y − Y 209.305 = n−2 6−2

= 7.23 i.e. Rs.7.23 lakh. 26. (c)

1 Commodity

Rice Wheat

2

P0Q1

P1Q0

P1Q1

Price Quantity Price Quantity (Q0) (P1) (Q1) (P0) 10.00 900

16.00 800

9000 8000

14400

12800

8.00 700

9.00 700

5600 5600

6300

6300

1 Commodity

P0Q0

2

P0Q0

P0Q1

P1Q0

P1Q1

Price Quantity Price Quantity (Q0) (P1) (Q1) (P0)

Sugar

12.00

100

15.00

80

1200

960

1500

1200

Salt

4.00

50

5.00

40

200

160

250

200

ΣP0Q0

ΣP1Q0 ΣP0Q1 = = 22720 = 37450 26000

ΣP1Q1 = 32500

Fisher’s ideal price index =

ΣP1Q 0 ΣP1Q1 x x 100 ΣP0 Q 0 ΣP0 Q1

=

37450 32500 x x 100 26000 22720

429


= 1.4354 x 100 = 143.54. 27. (d)

f0

fe

f0 – fe

(f 0 − f e )2

(f 0 − f e ) 2 fe

30

37.5

–7.5

56.25

1.500

40

37.5

2.5

6.25

0.167

45

37.5

7.5

56.25

1.500

35

37.5

–2.5

6.25

0.167

70

62.5

7.5

56.25

0.900

60

62.5

–2.5

6.25

0.10

55

62.5

–7.5

56.25

0.90

65

62.5

2.5

6.25

0.10

χ2 = ∑

(f 0 - f e ) 2 = 5.334. fe

28. (b) Let the following notations be used: Price per ticket: x Number of passengers per day: y

Sl. No.

Ticket Prices (x)

Number of passengers per day (y)

x2

xy

1

250

110

62,500

27,500

2

275

90

75,625

24,750

3

300

80

90,000

24,000

4

325

70

105,625

22,750

5

350

50

122,500

17,500

Σx = 1,500

Σy = 400

Σx 1,500 = 300 = n 5 Σy 400 = 80 = y = n 5 n ∑ xy − ∑ x ∑ y b= n ∑ x 2 − (∑ x) 2

x =

=

5 x 1,16,500 − 1,500x 400 5 x 4,56,250 − (1,500) 2

−17,500 = – 0.56 31,250 a = y − bx =

= 80 – (–0.56) x 300 = 248 ∴ The regression equation is yˆ = 248 – 0.56.

430

2

Σx = 4,56,250

Σxy = 1,16,500

Model Question Paper IV Time: 3 Hours


Answer all the questions. Each question carries one point. 1. Which of the following statements is true? a. The t distribution approximates the binomial distribution as the sample size increases. b. The same t distribution is applicable to all sample sizes. c. The standard normal distribution has a variance of 1. d. The standard normal distribution has a mean of 1. e. All of the above. 2. In which of the following methods of sampling the population is divided into groups so that the elements in each group are homogeneous and the groups vary from each other? a. Judgmental sampling. b. Stratified sampling. c. Systematic sampling. d. Cluster sampling. e. Random sampling. 3. Which of the following is an instance of Type II error in hypothesis testing? a. The sample statistic is incorrectly calculated. b. The population parameter is incorrectly calculated. c. The sample size is small. d. The null hypothesis is false and accepted. e. The alternative hypothesis is false. 4. When decisions are made under uncertain conditions, which of the following criteria for decision making is based on the opportunity costs of making particular decisions? a. Maximin criterion. b. Maximax criterion. c. Hurwicz criterion. d. Regret criterion. e. Expected value. 5. Which of the following distributions can represent the experiments or processes in which each trial has only two possible outcomes, the probability of the outcome of any trial remains fixed over time and the trials are statistically independent? a. t distribution. b. Normal distribution. c. F distribution. d. Chi-square distribution. e. Binomial distribution. 6. Which of the following is not a characteristic of the normal distribution? a. The normal distribution is unimodel. b. The mean of the normal distribution lies at the center of the normal distribution curve. c. The normal distribution is a symmetrical distribution. d. The two tails of the normal distribution curve intersect the horizontal axis. e. All of the above.


7.

8.

9.

Which of the following is false? a. Only point estimates are used for estimating the parameters of large populations. b. A point estimate provides a single number as the estimated value of the population parameter. c. An interval estimate provides a range of values within which the population parameter may fall. d. An interval estimate also indicates the probability of the true population parameter falling within the range of values specified. e. All of the above. Which of the following components of a time series is expressed as a percentage of the estimated values of the dependent variable as per the secular trend equation? a. Irregular variation. b. Seasonal variation. c. Cyclical variation. d. Short term variation. e. All of the above. Which of the following is true with regard to the concept of expected value? a. It is the value that the random variable will always assume. b. It is used for making decisions under conditions of certainty. c. It is based on the degree of optimism of the decision maker. d. It is a weighted average of the values that the random variable can assume. e.

It is the arithmetic average of the values that the random variable can assume.

10. Which of the following represents the alternative hypothesis for a right tailed test of mean for one sample? (‘ μ 0 ’ is the hypothesized value of the population mean) a.

H1 : μ ≠ μ 0

b.

H1 : μ < μ 0

c.

H1 : μ > μ 0

d.

H1 : μ = μ 0

e.

H1 : μ ≤ μ 0 .

11. Which of the following distributions have to be used for testing the equality of means of more than two populations? a.

Chi-square distribution.

b.

F distribution.

c.


d.

t distribution.

e.

All of the above.

12. Which of the following patterns of x charts indicates that two distinct populations are being observed?

432

a.

Increasing trend.

b.

Jumps in process levels.

c.

Cycles in process level.

d.


e.


Part III

13. Which of the following is true with regard to multicollinearity in multiple regression analysis? a. It increases the influence of the individual variables in the model. b. It reduces the predictive power of the model. c. It increases the reliability of the regression coefficients. d. It reduces the effectiveness of sensitivity analysis of the model. e. All of the above. 14. Which of the following is a control chart plotted on the basis of qualitative factors possessed by the output? a. R chart. b. p chart. c. x chart. d. Scatter diagram. e. Histogram. 15. Which of the following is true if the coefficient of correlation between two variables ‘X’ and ‘Y’ is zero? a. Covariance of the variables X and Y is zero. b. Covariance of the variables X and Y is positive. c. Covariance of the variables X and Y is negative. d. Variance of the variable X is zero. e. Variance of the variable Y is zero. 16. Which of the following price indices has an upward bias? a. Laspeyres prices index. b. Paasche’s price index. c. Fisher’s ideal price index. d. Marshall-Edgeworth price index. e. Value index. 17. Which of the following is true with regard to the method of least squares used in simple regression analysis? a. It minimizes the sum of the squared values of the independent variable. b. It minimizes the sum of the squared differences between the values of the independent variable and their mean. c. It minimizes the sum of the squared differences between the actual values of the dependent variable and the estimated values of the dependent variable as per the regression equation. d. It minimizes the sum of the squared differences between the values of the dependent variable and their mean. e. It minimizes the sum of the squared values of the dependent variable. 18. Which of the following statements is true? a. The mean of the sampling distribution of mean is greater than the population mean. b. The sampling distribution of mean approaches normality as the sample size decreases. c. As the standard error of mean increases, the value of any sample mean will be closer to the value of the population mean. d. As the standard error of mean decreases, the precision of the sample mean as an estimator of the population mean increases. e. Larger the standard error of estimate, the smaller the dispersion of points around the regression line. 433


19. The standard error of the estimate can be decreased for a regression equation by a.

Increasing the number of observations

b.

Using ‘t’ distribution

c.

Using two tails of a ‘t’ distribution

d.

Choosing an observation point close to the estimate required

e.

Increasing the number of independent variables.

20. A person expects a gain by investing in a share. The probability distribution of the gains is given below: Gain (Rs.)

Probability (%)

6

0.15

12

0.25

20

0.30

28

0.30

What is the expected gain from the share? a.

16.0.

b.

16.5.

c.

17.5.

d.

18.3.

e.

24.0.

21. Which of the following statements regarding multicollinearity is false? a.

As degree of correlation between the independent variables increases, the regression coefficients become less reliable.

b.

It reduces the accuracy of the model.

c.

It hurts any sensitivity analysis.

d.

It can happen that the model may be accepted through the F test and ANOVA but the individual coefficients may be rejected through t test.

e.

All of the above.

22. When the base year values are used as the weighted average of relatives price the index is same as a.

The Laspeyres price index

b.

The Paasche’s price index

c.

The unweighted average of relatives price

d.


e.


23. For the data 7, 5, 9, 1 what is the centered moving average of order 3?

434

a.

5.0.

b.

5.5.

c.

6.0.

d.

7.3.

e.

7.5.

Part III

24. The estimated value of a variable according to the trend equation is 84.5 in a particular year. The actual value of the variable is 86.4. The relative cyclical residual is a.

0.84

b.

1.16

c.

1.54

d.

1.96

e.

2.25.

25. In a time series the actual value of the variable plotted against time is 54 for a particular year. The relative cyclical residual is equal to 8. What is the estimated value of the variable? a.

46.

b.

50.

c.

58.

d.

62.

e.

64.

26. Which of the following is/are true? a.

Luxury is necessarily one of the facets of quality.

b.

Generally high quality items are costly compared to items of lower quality.

c.

If the variability in the production process is within the accepted limits, then the process is said to be maintaining quality.

d.

It is sufficient that the quality aspect is maintained for some crucial aspects of the production process.

e.

It is cheaper to check the finished product rather than checking each of the individual parts.

27. In x charts, a process is said to be in control, when the points plotted a.

Are on either side of the center line with small deviations

b.

Follow a random path and are between the control limits

c.

Lie beyond the control limits

d.

Lie near the control limits

e.


28. A contingency table prepared for chi-square test contains two rows and two columns. The degrees of freedom in this case will be a.

0

b.

1

c.

2

d.

3

e.

4.

29. The analysis of variance allows us to test whether the difference(s) among __________ sample means is/are significant or not. a. More than one b. More than two c. Less than three d. Less than four e. Any number of. 435


30. Consider the following contingency table. State of Economy Annual Sales of Product Q High Medium Low Boom 245 200 175 Moderate 195 170 155 Recession 165 140 125 The number of degrees of freedom in the above data is a. 3 b. 4 c. 6 d. 8 e. 9. 31. For conducting a chi-square test for testing a hypothesis the data have been collected and tabulated in 5 rows and 4 columns. The relevant degree of freedom is a. 19 b. 18 c. 16 d. 15 e. 12. 32. Approximately 60% of the observations in a normal distribution fall with in the range of μ±0.52σ a. b.

μ±0.67σ

c.

μ±0.84σ

d.

μ±1.04σ

e.

μ±1.28σ .

33. Simulation is used under a. Conditions of certainty b. Conditions of uncertainty c. Conditions of risk d. Conditions of known probabilities e. Decision tree approach. 34. Mathematical modeling requires the setting up of mathematical relationships which would represent the a. Problem b. System c. Criterion function d. Linear programming problem e. Decision variables. 35. Consider the following probability distribution of a variable X X 5.0 10.00 15.00 20.0 25.0 30.0 35.0 p 0.1 0.05 0.05 0.1 0.3 0.2 0.2 In a Monte Carlo simulation process the number 80 was generated in a particular run from a uniform distribution of 0-99. The value of X in this run will be a. b. c. d. e. 436

5 10 25 30 35.

Part III

36. Multicollinearity is the a.

Existence of correlation among independent variables

b.

Absence of correlation among independent variables

c.

Existence of correlation among dependent variables

d.

Absence of correlation among dependent variables

e.

Predictive power of the constants.

37. Multiple regression analysis is used to a.

Relate changes in more than one dependent variable to one independent variable

b.

Relate changes in more than one dependent variable to more than one independent variable

c.

Relate changes in one dependent variable to more than one independent variable

d.

Relate changes in one dependent variable to one independent variable

e.

Establish a cause-effect relationship between multiple pairs of dependent and independent variables considered at the same time.

38. Which of the following is/are use(s) of index numbers? a.

Establishing trends.

b.

Policy making.

c.

Determining the purchasing power of the currency.

d.

Deflating the time series data.

e.

All of the above.

39. Which of the following statement about runs test is false? a.

The direction of share price changes are considered.

b.

The magnitudes of price changes are considered.

c.

The excessive influence of extreme observation is removed.

d.

A run is said to last as long as the price changes do not change direction.

e.

It support that share price changes are random.

40. If the error sum of squares = 92 and the total sum of squares = 192 then what percentage of the changes in the dependent variable is explained by the changes in the independent variable? a.

72.17%.

b.

69.22%.

c.

52.08%.

d.

47.92%.

e.

27.12%.


(

ˆ For a simple linear regression equation ∑ Y − Y

)

2

= 256 and the standard error of estimate is 4.

What is the number of observations? a. 4. b. 12. c. 16. d. 18. e. 24. (1 point) 437


2.

3.

4.

5.

ˆ = 4 + 5X and, it The regression relationship between the variables X and Y is given by Y explains 64% of the changes in the variable Y. What is the coefficient of correlation between X and Y? a. –0.80. b. 0.80. c. 0.08. d. –0.08. e. 0.64. (1 point) The error sum of squares for a multiple regression equation with 3 independent variables, is 1024. If the number of observations is 68 then, what is the standard error of estimate? a. 3.88. b. 4.00. c. 5.02. d. 15.06. e. 45.18. (1 point) For a basket of commodities, Laspeyres price index is 121 and Paasches price index is 169. What is Fisher’s ideal price index for the same basket of commodities? a. 130. b. 135. c. 143. d. 145. e. 150. (1 point) Zardine India actually sold 72,000 cars in the year 2002 against the estimated 64,000 cars for the same year according to the secular trend equation. What is the percent of trend measure for the year 2002? a. 88.89. b. 112.50. c. 120. d. 125. e. 136. (1 point)

Based on the following information answer the Questions from 6 to 8. Pioneer Bank has observed that 70% of its borrowers pay their loan installments regularly. It is assumed that the loan accounts are independent of each other. The manager of the bank has randomly selected 6 loan accounts. 6. What is the probability that at least four out of the six borrowers are paying their loan installments regularly? a. 74.43%. b. 85.10%. c. 90.20%. d. 92.95%. e. 96.82%. (2 points)

438

Part III

7.

8.

What is the probability that maximum two out of the six borrowers are paying their loan installments regularly? a. 1.02%. b. 1.10%. c. 7.05%. d. 18.52%. e. 25.57%. (2 points) What is the probability that exactly three out of the six borrowers are paying their loan installments regularly? a. 18.52%. b. 25.57%. c. 74.43%. d. 85.10%. e. 96.82%. (1 point)

Based on the following information answer the Questions from 9 to 10. A survey was conducted in a city, on the retail prices of a specific type of mangoes. At a confidence level of 95 percent it was estimated that the mean retail price of the fruit fell between Rs.25.50 per kg and Rs.27.46 per kg. The number of retail fruit merchants sampled was 36 and less than 5 percent of the population of the retail fruit merchants in the city were sampled. 9. The mean price per kg of the fruit for the sample, is a. Rs.25.50 b. Rs.26.48 c. Rs.27.46 d. Rs.28.75 e. Rs.29.50. (2 points) 10. The standard deviation of the price per kg of the fruit for the sample, is a. Rs.0.50 b. Rs.1.00 c. Rs.1.50 d. Rs.3.00 e. Rs.6.00. (2 points) 11. The manager of Food Fun, a fast food shop, is deliberating on the number of pieces of a popular snack, “Double Delight,” to prepare for the evening. Each piece of “Double Delight” costs Rs.10 to prepare and is sold at Rs.16. The item is prepared and sold only during the evenings. Any unsold pieces cannot be stored because the item is highly perishable. The manager of the shop has provided the following data on the sales of the item over the past 200 days: Number of pieces sold No. of days 150 10 151 20 152 30 153 60 154 40 155 24 156 16 439


It is assumed that in future, the demand for the item can assume only one of the aforementioned values. What is the optimal number of pieces of the item that should be prepared for the evening so that the profit is maximum? a. 150. b. 151. c. 152. d. 153. e. 156. (3 points) Based on the following information answer the Questions from 12 to 13. An investment manager has identified three portfolios represented as X, Y and Z for investment. Investment has to be made in only one of the three portfolios. According to his assessment, three states of nature are possible during the period of holding the investment, which will lead to different gains in rupee-terms. The following table provides the information on the monetary gains from each portfolio under the three states of nature along with the probabilities of occurrence of the three states of nature: States of nature Portfolio S1 S2 S3 P(S1) = 0.25 P(S2) = 0.40 P(S3) = 0.35 X Rs.1,00,000 Rs.2,50,000 Rs.–50,000 Y Rs.–40,000 Rs.1,50,000 Rs.2,00,000 Z Rs.4,00,000 Rs.–50,000 Rs.1,00,000 12. According to the expected value criterion, in which portfolio should the investment be made? a. X. b. Y. c. Z. d. Either of the three portfolios. e. None of the portfolios. (1 point) 13. What is the maximum amount that the investment manager may spend in order to obtain further information from extremely reliable sources, which will remove all the uncertainties affecting his investment decision? a. Rs. 25,000. b. Rs. 62,500. c. Rs.1,20,000. d. Rs.1,50,000. e. Rs.2,70,000. (2 points) 14. A management consultant has collected the following data on the sales and advertising expenses of eight firms in the apparell industry: Sales (Rs. in crore) 52 52 57 62 67 58 56 60 Advertising expenses (Rs. in crore) 11 13 16 17 18 14 15 16 Approximately what percentage of the variation in sales is explained by the variation in advertising expenses? a. 40%. b. 50%. c. 64%. d. 81%. e. 90%. (3 points) 440

Part III

15. If a sample of size 38 is collected from a finite population of 600, then the finite population correction factor is equal to a.

0.927

b.

0.936

c.

0.938

d.

0.969

e.

1.000. (2 points)

16. If X is the independent variable and Y is the dependent variable, and the coefficient of correlation between them is –0.36 then approximately what percentage of the variations in Y is explained by the variations in X? (Rounded to the nearest integer.) a.

6%

b.

13%

c.

47%

d.

71%

e.

60%. (1 point)

17. The slope and the intercept on y-axis for a simple regression line are 3 and 5 respectively. If the dependent variable has a value of 8 then the estimated value of the dependent variable is a.

1

b.

3

c.

5

d.

29

e.

43. (2 points)

18. The actual number of cars (Y) sold by an Indian auto major in the year 2001 was 72430 ∧

corresponding to an estimated number (Y) of 68000 according to the equation of secular trend. The percent of trend measure for the year 2001 is a.

49.25

b.

93.92

c.

96.94

d.

103.26

e.

106.51. (1 point)

19. Given that, in a linear regression a = 3.24, b = 0.56, ΣY =104, Y = 13, n = 8, ΣY2 = 1484, ΣXY= 1972, and Y2 = 169. Then, the value of coefficient of determination is a.

0.49

b.

0.56

c.

0.68

d.

0.74

e.

0.82. (1 point)

441


20. Beta value of a stock is 1.2 and co-efficient of determination is 0.9. What is the variance of the security’s return if variance of the market returns is 350? a.

280.

b.

315.

c.

350.

d.

420.

e.

560. (2 points)

21. Demand No. of Days

20

24

26

28

32

34

15

20

30

10

13

12

What is the expected demand for the above data? a.

26.25.

b.

26.50.

c.

26.60.

d.

26.64.

e.

26.70. (2 points)

22. You are given the following paired values of variables X and Y X

10

12

14

15

17

Y

25

27

29

31

33

The slope of the regression line is a.

1.159

b.

1.162

c.

1.164

d.

1.169

e.

All of the above. (3 points)

23. X

Y

Z

4

5

6

4

3

6

3

1

2

6

8

9

What is the Grand Mean of above data? a.

4.75

b.

19.25

c.

5.15

d.

3.25

e.

6.25. (2 points)

442

Part III

24. Given r12 = 0.5, r13 = 0.8 and r23 = 0.7. Then R1.23 will be a. – 0.1400 b. 0.2800 c. 0.5714 d. 0.6567 e. 0.8044 (2 points) 25. A retailer has received a consignment of 500 numbers of 50 grams coffee packs manufactured by a FMCG company. He has randomly checked 10 coffee packs and has found out the weights to be as: 49.1 grams 49.0 grams 49.3 grams 49.2 grams 49.4 grams 49.6 grams 49.8 grams 49.1 grams 49.5 grams 49.4 grams It is assumed that the distribution of weights of the coffee packs is normal. Should he reject the entire consignment keeping the customer’s interest in mind? Use a significance level of 5%. a. The retailer should accept the entire consignment. b. The retailer should reject the entire consignment. c. We cannot infer anything. d. Significance level has to be improved to 10%. e. Both (c) and (d) above. (4 points) 26. Shyam Fabricators purchased a used milling machine from Kakatiya Engineering Works. It was communicated to Shyam Fabricators that there would be a probability of 5% that the output from the machine will be defective. It is expected that this rate of defective output will continue in future. What is the probability that maximum three out of every ten jobs done in that machine will be defective? a. 100%. b. 95%. c. 98.84%. d. 99.89%. e. 97.24%. (4 points) 27. A newly formed company is marketing their product in four districts. The marketing manager of the company is concerned that their market share is unevenly distributed in the four districts. In a survey, a random sample of 100 consumers in each district was taken. The following results were obtained: District District District District A B C D Number of consumers who purchase the product 30 40 45 35 Number of consumers who do not purchase the product 70 60 55 65 Which of the following statements are correct? a. The market share of the product is even in the four districts. b. The market share of the product is not even in the four districts. c. The market share of the product is even in three districts but not in one district. d. We cannot infer anything. e. The sample has to be increased to 200. (4 points) 443


28. The manager of Polar Fast Foods is deliberating on the number of pieces of a popular snack, “Miracle Puff,” to prepare for the evening. Each piece of “Miracle Puff” costs Rs.10 to prepare and is sold at Rs.16. Any unsold pieces cannot be stored because the item is highly perishable. The manager of the shop has arrived at the following probability distribution of sales on the basis of past experience. Number of pieces that can Probability be sold 150 0.05 151 0.10 152 0.15 153 0.30 154 0.20 155 0.12 156 0.08 What is the number of pieces of the item to be prepared for the evening so that the profit in the evening is maximum? a.

150.

b.

156.

c.

155.

d.

154.

e.

153. (3 points)

29. A wholesaler of food grains wants to know the overall change in prices of the commodities he deals in the following information is given. Year Commodity Wheat Rice Pulses Sugar Salt

1 Price (Rs./Kg.) 7.50 10.00 25.00 10.60 2.60

2 Quantity (Kg.) 5000 6000 3000 3000 500

Price (Rs./Kg.) 10.25 14.00 28.00 14.50 4.50

Quantity (Kg.) 4800 5500 3500 2400 600

What is Fisher’s ideal price index using Year 1 as the base year? a.

128.12.

b.

138.12.

c.

148.12.

d.

158.12.

e.

168.12. (4 points)

444

Model Question Paper IV Suggested Answers Part A: Basic Concepts 1.

(c) The t-distribution approximates the normal distribution as the sample size increases. Different sample sizes have different degrees of freedom and the t-distribution changes with the degrees of freedoms. The standard normal distribution has a mean of zero. Hence (a), (b) and (d) are false. The standard normal distribution has a standard deviation of 1; hence its variance is 12 = 1. Thus (c) is true.

2.

(b) In stratified sampling the population is divided into groups so that the elements in each group are homogeneous and the groups differ from each other. Hence (b) is the answer. (a), (c), (d) and (e) are sampling techniques which do not involve formation of groups consisting of homogeneous elements while the groups themselves differ from each other.

3.

(d) A type II error occurs in hypothesis testing when the null hypothesis is false and accepted. Hence (d) is the answer. (a), (b), (c) and (e) are not the instances of type II error.

4.

(d) According to the maximin criterion that alternative is selected which offers the maximum gain among the minimum gains. According to the maximax criterion that alternative is selected which offers the maximum gain among the maximum gains. According to the Hunwicz criterion weighted gains are used for decision making. Expected value criterion is used for decision making under conditions of risk. Hence (a), (b), (c) and (e) are not the answer. Regret criterion is based on the opportunity costs of making particular decisions.

5.

(e) Options (a), (b), (c) and (d) are distributions which are not appropriate for representing experiments which have the characteristics stated in the question. Binomial distribution represents such an experiment. Hence the answer is (e).

6.

(d) Options (a), (b) and (e) are characteristics of the normal distribution. Choice (d) is not a characteristic of the normal distribution because the two tails of the normal distribution curve never touch the horizontal axis.

7.

(a) Option (b) is true with regard to a point estimate. Options (c) and (d) are true with regard to an interval estimate. Option (a) is false because there is no reason why only point estimates should be used for estimating the parameters of large populations.

8. (c) Options (a), (b) and (d) are not expressed as percentage of the estimated secular trend values. Cyclical variations are expressed as a percentage of the estimated secular trend values, hence (c) is the answer. 9. (d) Expected value is not the value which the random variable will assume always. It is used under conditions of risk and it is not based on the degree of optimism of the decision maker. Hence (a), (b) and (c) are not true. It is a weighted average of the values that the random variable can assume. Hence (d) is true. 10. (c) Choice (a) represents the alternative hypothesis for a two tailed test. Choice (b) represents the alternative hypothesis for a left tailed test. Choice (d) represents an incorrect representation of the alternative hypothesis. Choice (c) represents the alternative hypothesis for a right tailed test of mean for one sample. Hence (c) is the answer. 11. (b) F-distribution have to be used for testing the equality of more than two population means, hence (b) is the answer. Choice (a) is used for testing the equality of more than two population proportions. Options (c) and (d) cannot be used for testing the equality of more than two population means. 12. (d) Option (a) indicates that the process mean is increasing. Option (b) indicates that the process mean may be shifting. Option (c) indicates that there may be random variations or there may be the influence of such factors which vary cyclically. Choice (e) indicates that the process variability has been reduced to a great extent. Choice (d) indicates that two distinct populations are being observed. Hence (d) is the answer.

13. (d) Multicollinearity reduces the effects of the individual variables in the model; however the variables may together explain the dependent variable. It does not reduce the predictive power of the model. It reduces the effectiveness of sensitivity analysis of the model. Hence options (a), (b) and (c) are not true, (d) is true. 14. (b) R chart is plotted using process variability. x chart is plotted using process mean. Scatter diagram is the plot of points in regression analysis. Histogram is the graphical representation of frequency distribution. p chart is plotted using qualitative factors of the output. Hence (b) is the answer. 15. (a) Coefficient of correlation between x and y (r) =

Cov (x, y) σx .σy

Hence, r = 0 ⇒ Cov (x, y) = 0 x (σ x . σ y ) = 0 Hence (a) is the answer. 16. (a) Laspeyre’s price index has an upward bias. Paasches price index has a downward bias. Fisher’s ideal price index is free from bias. Marshall-Edgeworth price index closely approximates Fisher’s ideal price index, hence this can also be said to be nearly free from bias. Hence (a) is the answer. 17. (b) The method of least squares minimizes the sum of the squared differences between the values of the dependent variable and their estimated values as per the regression equation. Hence (c) is true. Options (a), (b), (d) and (e) are not true with regard to the method of least squares. 18. (d) The mean of a sampling distribution of mean is equal to the population mean. The sampling distribution of mean approaches normality as the sample size increases. As the standard error of mean increases, the value of any sample mean will be farther from the value of population mean. As the standard error of mean decreases the precision of the sample mean as an estimator increases because the sample will have lesser variability from the population mean. Hence options (a), (b), (c) and (e) are not the answer, and (d) is the answer. 19. (a) The standard error of the estimate can be decreased for a regression equation by increasing the number of observations. 20. (d) The expected gain from the share is = (0.15 x 6) + (0.25 x 12) + (0.30 x 20) + (0.30 x 28) = 18.3. 21. (b) As the degree of correlation between the independent variables increases, the regression coefficients become less reliable. That is, although the independent variables may together explain the dependent variable, but because of Multicollinearity the coefficients of the explanatory variables may be rejected. (It can happen that the model may be accepted through ANOVA and the F test), but the individual coefficients may be rejected (through the t test). This is because the interplay among the independent variables reduces the influence of the individual variables in the model. In the extreme case, if two variables were identical, then the influence of each one in the model would be reduced. 22. (a) Weighted average of relative price index (when the base year values are used as weights) = Σ(P1/P0 x 100)(P0Q0)]/ΣP0Q0 = ΣP1/P0 x P0Q0 x 100/ΣP0Q0 = ΣP1Q0/ΣP0Q0 x 100. which incidentally is the weighted aggregate price index using Laspeyres method. The weighted average of price relatives using base values is the same as the weighted aggregate of actual prices (Laspeyres method) only when the arithmetic mean is used for averaging the relatives and the base year values are used as weights. If either of these conditions are not satisfied, such a comparison would not be possible. 446

Part III

23. (c) The centered moving average of order 3 is calculated using 3 datas at a time. 7

We have Moving average

9

5 7

1 5

6

Centered moving average

24. (e) Relative Cyclical Residual Measure =

ˆ (Y − Y) (86.4 − 84.5) x 100 x 100 = ˆ Y 84.5

= 2.25. 25. (b) Relative Cyclical Residual Measure

ˆ ˆ (Y − Y) (54 − Y) ˆ = 50 x 100 ⇒ Y x 100 = ˆ ˆ Y Y

=

26. (c) The production process is consistent and within the accepted limits then the process is said to be maintaining quality. 27. (e) In x charts, a process is said to be in control, when all the points are within the control limits (i.e., upper and lower limits). A small deviation implies that the point is very much within the control limits. 28. (b) DOF = (Number of rows – 1) (Number of columns – 1) = (2 – 1) (2 – 1) = 1. 29. (b) ANOVA allows us to test whether the difference(s) among more than two sample means are significant or not. 30. (b) DOF = (Number of rows – 1) (Number of columns – 1) = (3 – 1) (3 – 1) = 4. 31. (e) DOF = (Number of rows – 1) (Number of columns – 1) = (5 – 1) (4 – 1) = 12. 32. (c) 60% of the observation means that 40% of the observations lie on either side of the mean. 33. (b) Simulation is performed under the condition of uncertainity. 34. (b) In simulation mathematical modeling requires the setting up of mathematical relationship which would represent the system. Relationship can expressed as equations as well as inequalities. 35. (e)

X

Probability

Cumulative Probability

Random numbers allotted

5

0.10

0.10

00-09

10

0.05

0.15

10-14

15

0.05

0.20

15-19

20

0.10

0.30

20-29

25

0.30

0.60

30-59

30

0.20

0.80

60-79

35

0.20

1.00

80-99

Since the random number generated was 80 (which fall in between 80-99), the value of X in this run is 35. 36. (a) Multicollinearity refers to existence or presence of correlation among independent variables. 37. (c) Multiple regression analysis is used to relate changes in one dependent variable to more than one independent variable. 447

38. (e) All the alternatives are stating the different uses of an index number. 39. (b) In runs test the magnitude of price changes are ignored. All other statements are correct. 40. (c) Regression sum of squares (RSS) = 192 – 92 = 100, so, the percentage of changes in the dependent variable, which is =

RSS 100 = = 0.5208 ≅ 52.08% . TSS 192

Part B: Problems 1.

(d) For any simple linear regression equation, the standard error of estimate =

ˆ 2 ∑ (y − y) = 4.00 n−2 So,

ˆ 2 ∑ (y − y) 256 =16 i.e. n – 2 = 16 n−2

Or, n = 18. 2.

(b) Coefficient of determination = r2 =

64 = 0.64 100

∴ Coefficient of correlation = r 2 = 0.64 = 0.8 Since, the regression equation has a positive slope the coefficient of correlation is positive. 3.

(b) Coefficient of determination = r2 =

64 = 0.64 100

∴ Coefficient of correlation = r 2 = 0.64 = 0.8 Since the regression equation has a positive slope the coefficient of correlation is positive. =

1024 = 16 = 4.00 . 64

4.

(c) Laspeyres price index (L) = 121 and Paaseshes price index (P) = 169. ∴ Fisher’s ideal price index will be = L × P = 121× 169 = 143 .

5.

(b) The percent of trend measure =

6.

(a) The probability that a client is regularly paying the loan installments is 0.70.

72, 000 y ×100 = 112.50 . × 100 = ˆy 64, 000

The probability that at least 4 out of the six borrowers pay their loan installments regularly =

7.

8.

4

6

4

2

5

6

=

15 × (0.7) .(0.3) + 6 × (0.7) × 0.3 + (0.7)

=

0.7443 i.e. 74.43%.

(c) The probability that maximum 2 out of the six borrowers pay their loan installments regularly

=

6C 0 (0.7)0 ⋅ (0.3)6 + 6C 1 (0.7)1 x (0.3)5 + 6C 2 x (0.7)2 x (0.3)4

=

1 × 1 × (0.3)6 + 6 × 0.7 × (0.3)5 + 15 × (0.7)2 × (0.3)4

=

0.07047 i.e. 7.05%.

(a)The probability that exactly three out of the six borrowers pay their loan installments regularly

= 448

6C (0.7)4 ⋅ (0.3)2 + 6C 5 (0.7)5 ⋅ (0.3) + 6C (0.7)6

6C 3 x (0.7)3 x (0.3)3 = 0.18522 i.e. 18.52%.

Part III

9.

(b) The sample size is greater than 30 and the standard deviation of population is not known. Hence the appropriate distribution applicable is the normal distribution. A confidence interval of 95 per cent implies that 0.475 of the area of the normal curve is covered on either side of the sample mean. From the normal distribution table it is found that the appropriate z-value is 1.96. x =? σˆ x = ? s=?

For the given confidence interval : Lower limit

=

Rs.25.50 per kg

Upper limit

=

Rs.27.46 per kg.

∴ x – 1.96 σˆ x

=

25.50

…….. (1)

x + 1.96 σˆ x

=

27.46

………(2)

Adding equations (1) and (2) we get :

2x =

52.96

or x =

26.48 (Rs. per kg.)

∴ Mean price of the sample =Rs.26.48 per kg. 10. (d) Subtracting equation (A) from equation (B) we get :

3.92 σˆ x = 1.96 ∴

σˆ x =

Now, σˆ x = ∴ 0.50

=

or s = 0.50

1.96 = 0.50 (Rs. per kg) 3.92

s n s 36

36 = 3 (Rs. per kg)

∴ Standard deviation of the sample = Rs.3.00 per kg. 11. (d) Marginal Profit (MP) =16 – 10 = Rs.6 Marginal Loss (ML) = Rs.10 Minimum required probability of selling one or more additional pieces to justify their preparation, ML 10 p* = = = 0.625 10 + 6 ML + MP

Cumulative probability that sales will be at this level or more 150 10 / 200 = 0.05 1.00 151 20 / 200 = 0.10 0.95 152 30 / 200 = 0.15 0.85 153 60 / 200 = 0.30 0.70 154 40 / 200 = 0.20 0.40 155 24 / 200 = 0.12 0.20 156 16 / 200 = 0.08 0.08 The number of pieces prepared must be such that the probability of selling three or more number of pieces (i.e. cumulative probability) is at least equal to the minimum required probability (p*) of 0.625. From the above table it can be seen that – For 153 pieces, cumulative probability = 0.70 > p* For 154 pieces, cumulative probability = 0.40 < p* Hence, the optimal number of pieces to be prepared is 153. No. of pieces

Probability

449

12. (b) The expected value criterion can be applied as follows:

Portfolio X : 100000 (0.25) + 250000 (0.40) – 50000 (0.35) = Rs.1,07,500 ∴Expected gain from portfolio X = Rs.1,07,500 Portfolio Y : – 40000 (0.25) + 150000 (0.40) + 200000 (0.35) = Rs.1,20,000 ∴Expected gain from portfolio Y = Rs.1,20,000 Portfolio Z : 400000 (0.25) – 50000 (0.40) + 100000 (0.35) = Rs.1,15,000 ∴Expected gain from portfolio Z = Rs.1,15,000 From above, it can be seen that the maximum expected gain (Rs.1,20,000) is associated with Portfolio Y. ∴ Investment must be made in portfolio Y. 13. (d) The question actually asks for Expected Value of Perfect Information (EVPI). According to this concept, under conditions of complete certainty, the decision maker will always make that decision which will lead to the maximum gain under any state of nature. In the current, context with the information from extremely reliable sources, the investment manager will make his investment decision under completely certain conditions. Thus, if he gets the perfect information that the state of nature S1 will be occurring he will invest in portfolio X which provides the highest gain under S1. Similarly with the perfect information on the occurrence of any other state of nature he will invest in that portfolio which will provide the highest gain under that state of nature. This is shown below:

State of Nature Portfolio

S1 P(S1) = 0.25

S2 P(S2) = 0.40

S3 P(S3) = 0.35

Z

Rs.400,000

–

–

X

–

Rs. 250,000

–

Y

–

–

Rs.200,000

Expected gain under certainty (or expected gain with perfect information) = 4,00,000 (0.25) + 2,50,000 (0.40) + 2,00,000 (0.35) = Rs.2,70,000 EVPI =

Expected gain under certainty – Maximum expected gain without perfect information. = 2,70,000 – 1,20,000 = Rs.1,50,000

∴The maximum amount that the investment manager may pay to obtain further information from reliable sources (i.e. expected value of perfect information) is Rs.1,50,000. 14. (d) Let the following notations be used:

X:

Advertising expenses (Rs. in crore)

Y:

Sales (Rs. in crore)

Percentage of variation in Y that is explained by variation in X = Coefficient of determination = r2 r=

450

∑ ( X − X )( Y − Y ) 2 2 ∑ (X − X) ∑ (Y − Y)

Part III

Y

52

52

57

62

67

58

56

60

X

11

13

16

17

18

14

15

16

Y –Y

–6

–6

–1

4

9

0

–2

2

X –X

–4

–2

1

2

3

–1

0

1

(Y – Y )2

36

36

1

16

81

0

4

4

178

2

(X – X )

16

4

1

4

9

1

0

1

36

(X – X )(Y – Y )

24

12

–1

8

27

0

0

2

72

Y=

464 ∑X ∑ Y 120 = = 58 X = = = 15 8 8 n n

72

∴r=

36 × 178

= 0.8994.

∴ r2 = 0.8089 = 0.81 ∴ The variation in advertising expenses explain 81% of the variation in sales. 15. (d) Finite population correction factor

=

N−n = N −1

600 − 38 = 0.969. 600 − 1

16. (b) The percentage of variations in the dependent variable that is explained by the variations in the independent variable is known as coefficient of determination, which is the square of coefficient of correlation. In this case coefficient of determination is (– 0.36)2 = 0.1296 ≅ 13%. 17. (a) If y is dependent on variable x then we can write that the estimated value of y, yˆ = 3x + 5.

When y = 8 the estimated value of x would be x =

y−5 8−5 = =1. 3 3

18. (e) The percent of trend measure ∧

= (Y / Y ) x 100 = 19. (c) R 2 =

7243 x 100 = 106.51. 6800

aΣy + bΣ x y − n(y)2 Σy 2 − n(y)2

= 3.24 (104) + (0.56) (1972) – 8(13)2 / (1484 – 8(13)2) = [336.96 + 1104.32 – 1352] /(132) = 0.68 20. (e) Systematic risk = β2 x Variance of the market return

= (1.2)2 x 350 = 504 Systematic risk = R2 x variance of the security’s return 504 = 0.9 x variance of the security’s return

⇒

Variance of the security’s return =

504 = 560. 0.9

21. (d) Expected Demand

451

= 20 x (15/100) + 24 x (20/100) + 26 x (30/100) + 28 x (10/100) + 32 x (13/100) + 34 x (12/100) = 3 + 4.8 + 7.8 + 2.8 + 4.16 + 4.08 = 26.64. 22. (c) In a regression line, Y = a + bX, b represents the slope of the regression line.

The value of b=

[ ∑ XY − n(X)(Y)] 2

68 = 13.6 5

X = Y=

145 5

∑XY 2

∑X

b =

=

2

[ ∑ X − n(X) ]

= 29

= 2006 = 954

[2006 − (5)(13.6)(29)] [954 − 5(13.6) 2 ]

34 = 1.1643. 29.2

( )

23. (a) Grand Mean x

=

(5 + 3 + 1 + 8) + (6 + 6 + 2 + 9) + (4 + 4 + 3 + 6) 17 + 23 + 17 = 12 12

=

57 = 4.75 12

24. (e) R123 = =

(r122 + r132 − 2r12 r23 r13 ) 1 − r232

(0.5) 2 + (0.8) 2 − 2(0.5) (0.8) (0.7) 1 − (0.7)2

= 0.647 = 0.804 25. (b) Mean weight of the sample (x) =

493.4 ∑x = = 49.34 gms n 10

Sample standard deviation (s) =

=

∑(x − x)

2

n −1

0.564 = 0.2503 ≅ 0.25 gms 10 − 1

10 = 0.02 < 0.5 500 Hence the standard error applicable to infinite populations has to be calculated.

Sampling fraction =

452

Part III ∧

∴ Estimated standard error of mean, σ x = =

0.25 10

∧

σ n

≅ 0.079 gms.

The consignment should be rejected if it can be inferred at a significance level of 5% that the entire consignment is underweight i.e., the mean weight of the coffee packs in the consignment is less than 50 gms. So a left tailed test of hypothesis has to be conducted. The null and the alternative hypotheses are given below: H0: μ = 50 H1: μ < 50 Since the sample size is less than 30 and the standard deviation of population is unknown, the ‘t’ distribution has to be used.

∴Standardized ‘t’ statistic for the sample =

x − 50 σˆ x

49.34 − 50 = – 8.35 0.079 Degrees of freedom = n – 1 = 10 – 1 = 9 The t-value from the t-distribution table for 9 degrees of freedom and 10% of the area in both tails combined i.e., 5% of the area in each tail = 1.833. So the cut-off t-value on the left side of the sampling distribution will be –1.833. ∴t=

0.05

–1.833

Since –8.35 < –1.833 (and it falls in the rejection region i.e., the shaded region in left tail) we find that the sample mean is significantly less than 50. Hence we reject the null hypothesis (μ = 50) and accept the alterative hypothesis (μ <50). So we infer that the mean weight of the consignment is less than 50 gms. Hence the retailer should reject the consignment. 26. (d) The binomial distribution should be used.

According to the binomial distribution formula, probability of ‘r’ success in ‘n’ trials, p(r) = nCr . pr qn–r p

= Probability of being defective = 5% = 0.05

q

= Probability of not being defective = 1– p = 1 – 0.05 = 0.95

n

= Number of trials = 10

r

= Number of successes i.e., defective jobs in this case = 3

Probability of maximum 3 out of 10 jobs being defective is P(<3) = P(0) + P(1) + P(2) + P(3)

453

=

10

C0 (0.05)0 (0.95)10 + 10C1 (0.05)1 (0.95)9 + 10C2 (0.05)2 (0.95)8 + 10C3 (0.05)3 (0.95)7

= 1 x 1 x 0.5987 + 10 x 0.05 x 0.6302 + 45 x 0.0025 x 0.6634 + 120 x 0.000125 x 0.6983 = 0.5987 + 0.3151 + 0.0746 + 0.0105 = 0.9989 i.e., 99.89%. 27. (a) Contingency Table

District A

District B

District C

Number of persons who purchase the product (x)

30

40

45

35

150

Number of persons who do not purchase the product (y)

70

60

55

65

250

100

100

100

100

400

Total

Combined proportion of people who purchase this product (p) =

District D

Total

150 = 0.375 400

Combined proportion of people who do not purchase this product = 1 – 0.375 = 0.625 Frequency of persons, who purchase this product District A

District B

District C

District D

30

40

45

35

100 x 0.375 = 37.50

100 x 0.375 = 37.50

100 x 0.375 = 37.50

100 x 0.375 = 37.50

Observed (actual) frequency (fo) Expected (theoretical) frequency (fe)

Frequency of persons who do not purchase this product District A

District B

District C

District D

Observed (actual) frequency (fo)

70

60

55

65

Expected (theoretical) frequency (fe)

100 x 0.625 = 62.5

100 x 0.625 = 62.5

100 x 0.625 = 62.5

100 x 0.625 = 62.5

Finding the value of χ 2 : 2

fe

f0 – fe

(f0 − fe )

30

37.5

–7.5

56.25

1.500

40

37.5

2.5

6.25

0.167

45

37.5

7.5

56.25

1.500

35

37.5

–2.5

6.25

0.167

70

62.5

7.5

56.25

0.90

60

62.5

–2.5

6.25

0.10

55

62.5

–7.5

56.25

0.90

65

62.5

2.5

6.25

0.10

χ2 = ∑

fe

(f 0 − f e ) 2 = 5.334 fe

Null Hypothesis

454

(f0 − fe )

2

f0

: H0 : PA = PB = PC = PD (Proportion of consumers from each of the four districts are equal) B

Part III

Alternative Hypothesis:

H1: PA, PB, PC and PD are not at all equal (Proportion of consumers from each of the four districts are not equal) B

Since our contingency table for this problem has two rows and four columns, the number of degrees of freedom is (2 – 1) (4 – 1) = 1 x 3 = 3 In the chi-square table we find that the value, corresponding to 10% significance level and 3 degrees of freedom, is 6.251. We can interpret this to mean that with 3 degrees of freedom, the region to the right of the chi-square value 6.251 contains 0.10 of the area under curve. Since 5.334 is less than 6.251, we accept the null hypothesis. Hence we infer that the market share of this product is even in the four districts. 28. (e)

No. of pieces Probability

Cumulative probability that sales will be at this level or more

150

0.05

1.00

151

0.10

0.95

152

0.15

0.85

153

0.30

0.70

154

0.20

0.40

155

0.12

0.20

156

0.08

0.08

The number of pieces prepared must be such that the probability of selling three or more number of pieces (i.e., cumulative probability) is at least equal to the minimum required probability (p*) of 0.625. From the above table it can be seen that– For 153 pieces, cumulative probability = 0.70 > p* For 154 pieces, cumulative probability = 0.40 < p* Hence, the optimal number of pieces to be prepared is 153. 29. (a) Fisher’s ideal price index = x 100

ΣP1Q0 = (10.25 x 5000) + (14.00 x 6000) + (28.00 x 3000) + (14.50 x 3000) + (4.50 x 500) = 265000 ΣP1Q1 = (10.25 x 4800) + (14.00 x 5500) + (28.00 x 3500) + (14.50 x 2400) + (4.50 x 600) = 261700 ΣP0Q0 = (7.5 x 5000) + (10.00 x 6000) + (25.00 x 3000) + (10.60 x 3000) + (2.60 x 500) = 205600 ΣP0Q1 = (7.5 x 4800) + (10.00 x 5500) + (25.00 x 3500) + (10.60 x 2400) + (2.60 x 600) = 205500 ∴ Fisher’s ideal price index x 100 = 128.12 (approx.)

455

Model Question Paper V Time: 3 Hours


Answer all the questions. Each question carries one point. 1. The number of Degrees of Freedom (DOF) in a 3×4 contingency table is a. 2 b. 3 c. 6 d. 8 e. 2.

If we want to compare five sample proportions then, we have to use the a. Normal distribution b. t-distribution c. Chi-square distribution d. Binomial distribution e.

3.

All of the above.

Which of the following patterns of x chart is generally desirable to any quality control manager? a. Increasing trend. b. Decreasing trend. c. Cycles. d. Hugging the center line. e.

6.

Decision under conditions of uncertainty.

The control chart plotted using the variability of the process results is known as x chart a. b. R chart c. P chart d. K chart e.

5.

Standard normal distribution.

If sufficient knowledge is available about the state of nature to assign probabilities to their likelihood of occurrence then decision are a. Decisions under conditions of certainty b. Decision under conditions of optimism c. Decision under conditions of pessimism d. Decision under conditions of risk e.

4.

9.


Which of the following statement about simulation is true? a. It produces optimum solutions. b. Reliability of results is independent of mathematical model. c. It is flexible. d. With increasing parameters simulations become easier. e.

It is not -so- computer intensive.

Part III

7.

The relative cyclical residual is expressed as Y−Y x 100 a. Y ∧

b.

Y−Y ∧

x 100

Y ∧

c.

Y−Y x 100 Y

d.

Y−Y x 100 Y

∧

∧

Y−Y x 100. Y 8. Which of the following statements is/are true? a. The secular trend allows us to describe the historical pattern of the data. b. Secular trend represents the short to medium term variation in time series data. c. The component of a time series that oscillates predictably above and below the trend line for a period less than one year is known as irregular variation. d. Seasonal variations are periodic and repetitive in nature. e. Both (a) and (d) above. 9. Which of the following price indices has a downward bias? a. Laspeyres Index. b. Paasche’s Index. c. Fisher’s Index. d. Marshall Edegeworth Index. e. All of the above. 10. Unweighted aggregates price index is equal to P1 a. x 100 P0

e.

P1

b.

∑

c.

∑ P1

P0

∑ P0

d.

∑ P1

e.

nP1

nP0 ∑ P0

x 100 x 100 x 100 x 100.

11. The regression equation of ‘y’ with respect to two independent variables, ‘x1’ and ‘x2’, is y = a + b1x1 + b2x2. In which of the following situations will the coefficient of determination be equal to zero? a. F = 0. b. F = ∞ . c. ESS = 0. d. TSS = 1. e. Both (a) and (d) above. 12. Which of the following is not true in the context of multiple regression? a. Multicollinearity does not reduce the accuracy of the predictive power of the model. b. Multicollinearity does not reduce the effectiveness of the sensitivity analysis. c. Multicollinearity reduces the reliability of the regression co-efficients. d. Multicollinearity reduces the influence of the individual variables in the model. e. Independent variables may together explain the dependent variable in case of a multicollinearity problem. 457


13. Which of the following is the correct expression for the co-efficient of correlation between two variables x and y? Cov (x, y) a. . σ 2x σ 2y

b. c. d. e.

Cov (x, y) . σx σy Cov (x, y) σ 2x

Cov (x, y) σ 2y

. .

σx σy . Cov (x, y)

14. Which of the following is false? a. Only point estimates are used for estimating the parameters of large populations. b. A point estimate provides a single number as the estimated value of the population parameter. c. An interval estimate provides a range of values with which the population parameter may fall. d. An interval estimate also indicates the probability of the true population parameter falling within the range of values specified. e. None of the above. 15. Which of the following is the correct representation of alternative hypothesis for a right tailed test of mean using one sample? (μo = Hypothesized value of the population mean) a. H1: μ = μo. b. H1: μ ≠ μo. c. H1: μ > μo. d. H1: μ < μo. e. H1: x < μo.. 16. The sales of Mehta Industries Ltd. in the year 2001 is Rs.7.50 crore. The estimated sales for the same year as per the secular trend equation is Rs.6.25 crore. The relative cyclical residual for the year 2001 is a. 15.0 % b. 20.0 % c. 45.0 % d. 83.3 % e. 120.0 %. 17. The standard normal distribution has a a. Mean = 1 and standard deviation = 0 b. Mean = 1 and standard deviation = 1 c. Mean = 0 and standard deviation = 1 d. Mean = 0 and standard deviation = 0 e. Mean = –1 and standard deviation = 0. 18. Which of the following is not a type of random sampling? a. Simple Random sampling. b. Stratified sampling. c. Systematic sampling. d. Judgmental sampling. e. Cluster sampling.

458

Part III

19. Which of the following elements is/are commonly found in problems of decision-making under uncertain conditions? a. The decision maker has an objective to be achieved. b. There are several courses of action. c. There is a calculable measure of benefit or worth of the various courses of action. d. There are events which are not controllable by the decision maker. e. All of the above. 20. Which of the following is a measure of variability of one variable (or data set) in relation to another variable (or data set)? a. Variance. b. Semi-variance. c. Covariance. d. Standard deviation. e. Range. 21. X and Y are two random variables. The variance and covariance are denoted by ‘V’ and ‘Cov’ respectively. Which of the following equations is true? a. V(X) + V(Y) = Cov (X, Y). b. V(X) + V(Y) = V(X+Y) + Cov (X, Y). c. V(X) + V(Y) = V(X+Y) – Cov (X,Y). d. V(X) + V(Y) = V(X+Y) + 2Cov (X, Y). e. V(X) + V(Y) = V(X+Y) – 2Cov (X,Y). 22. Which of the following is not true? a. The standard normal distribution is a normal distribution. b. The standard normal distribution has a variance of 1. c. The normal distribution is a continuous probability distribution. d. The t-distribution remains the same for all sample sizes. e. The t-distribution approximates the normal distribution as the sample size increases. 23. In which of the following methods of sampling the population is divided into groups so that each group has a wide variation within itself and is representative of the population as a whole? a. Random sampling. b. Judgmental sampling. c. Stratified sampling. d. Systematic sampling. e. Cluster sampling. 24. Which of the following is true according to the central limit theorem? a. The distribution of the sample is always symmetric. b. The distribution of the sample is symmetric only for small sample sizes. c. The sampling distribution of the mean approaches normality as the sample size increases. d. The sampling distribution of the mean approaches normality as the sample size decreases. e. The sampling distribution of the mean never approaches normality. 25. For which of the following distributions, the z-values and the observed values are same? a. Binomial distribution. b. Standard normal distribution. c. Normal distribution. d. Lognormal distribution. e. F-distribution.

459


26. Which of the following represents the alternative hypothesis for a two tailed test on population mean? (μ0 is the hypothesized value of the population mean)

a.

H0 : μ = μ0.

b.

H0 : μ < μ0.

c.

H1 : μ > μ0.

d.

H1 : μ < μ0.

e.

H1 : μ ≠ μ0.

27. Which of the following is not true with regard to the concept of expected value?

a.

It is applied for taking decisions under uncertain conditions.

b.

It is a weighted average of the values that a random variable can assume.

c.

Every value that may be assumed by the random variable is weighted by its probability of occurrence.

d.

It is the predicted value that the random variable will assume at any time.

e.

All of the above.

28. The standard error of estimate

a.

Is a measure of central tendency for the distribution of the values estimated by the regression equation. b. Measures the variability of the observed values of the dependent variable around their mean. c. Measures the variability of the observed values of the dependent variable around the regression line. d. Measures the variability of the observed values of the independent variable around their mean. e. Measures the variability of the observed values of the independent variable around the regression line. 29. Which of the following is true with regard to multicollinearity in the context of multiple regression? a. It increases the reliability of the regression coefficients. b. It reduces the predictive ability of the model. c. It increases the accuracy of the model. d. It hurts the sensitivity analysis of the model e. All of the above. 30. A regression model cannot explain any variations in the values of the dependent variable if

a.

ESS = 0

b.

R2 = 1

c.

RSS = 0

d.

RSS > 0

e.

TSS > 0.

31. Which of the following is true with regard to index numbers?

460

a.


b.

Paasche’s price index tends to overestimate the rise in prices.

c.

Unweighted aggregates price index numbers do not link the price changes to the usage or consumption levels.

d.

Laspeyres price index tends to underestimate the rise in prices.

e.

All of the above.

Part III

32. Suppose, that a hypothesis test is being performed for a process in which a Type-I error will be very costly, but a Type II error will be relatively inexpensive and unimportant. The best choice for α in this test will be

a.

0.1

b.

0.25

c.

0.50

d.

0.75

e.

0.90.

33. Which among the following is Sampling Fraction?

a.

n/N.

b.

N/n.

c.

N/(n – 1).

d.

n/(N – 1).

e.

(n – 1)/n.

34. The weighted average of price relatives is the same as the weighted aggregate of actual prices, only when?

a.

Arithmetic mean is used for averaging the relatives.

b.

Current year values are used as weights.

c.

Given year values are used as weights.

d.

Base year values are used as weights.

e.


35. Which of the following are the tests for the consistency of the index numbers?

a.

Circular test.

b.

F-test.

c.

t-test.

d.

Time reversal test.

e.


36. When a problem has a large number of possible actions, we would normally use

a.

Conditional table

b.

Marginal table

c.

Utility table

d.

Marginal analysis

e.

None of the above.

37. The coefficient of correlation between two variables ‘X’ and ‘Y’ is –0.70. ‘X’ is the independent variable and ‘Y’ is the dependent variable in the simple regression relationship between them. What percentage of the variations in the value of Y is not explained by the regression relationship?

a.

30%.

b.

49%.

c.

51%.

d.

60%.

e.

70%.

461


38. The mean of the sampling distribution of the mean will be _____ to/than the population mean regardless of the sample size and the population.

a.

Greater

b.

Lesser

c.

Equal

d.

Cannot be ascertained

e.


39. A stock, A is consistently giving better results than other stock B for the past ten years. Which of the following is true?

a.

The average returns of stock A is greater than the average return of stock B.

b.

the risk associated with stock A (as measured by standard deviation) is greater than the risk associated with stock B.

c.

The risk associated with stock B is greater than that of stock A.

d.


e.


40. Which of the following is not true with regard to a value index?

a.

It measures changes in net monetary worth.

b.

It combines price changes and quantity changes.

c.

It is not as useful as price indices or quantity indices.

d.

It cannot distinguish between the price changes and quantity changes.

e.

An increase in the value index indicates with certainty that the prices have increased.

Part B: Problems (60 Points) Solve all the problems. Points are indicated against each problem. Based on the following information answer the Questions from 1 to 7. The manager of a movie theatre in Hyderabad, Dreams, has observed that a certain hit movie has been shown for an average number of 49 days in various movie theatres in the rest of the state with a standard deviation of 6 days. The manager wants to check the popularity of the movie at Hyderabad and, has taken a random sample of 12 movie theatres out of 50 movie theatres in which that movie has been shown. He has found out that the movie has been shown for an average number of 46 days. It is to be tested whether the popularity of the movie at Hyderabad is significantly less than its popularity in the rest of the state, at a 1% significance level. 1.

Which of the following correctly represent the null and alternative hypotheses for testing whether the mean observed period of showing the movie at Hyderabad is significantly less than the mean observed period in the rest of the state? i.

H0

:

μ = 46

ii.

H1

:

μ = 46

iii.

H0

:

μ = 49

iv.

H1

:

μ < 49

a.


b.


c.

Both (iii) and (iv) above.

d.

Both (i) and (iv) above.

e.

Both (ii) and (iv) above. (1 point)

462

Part III

2.

The standard error of mean is a.

0.500.

b.

1.525.

c.

1.615.

d.

1.732.

e.

1.850. (1 point)

3.

The probability distribution which is required for the hypothesis test is a.

t-distribution.

b.


c.

F distribution.

d.

Chi-square distribution.

e.

Binomial distribution. (1 point)

4.

The standardized value of the sample mean is approximately a.

– 6.00.

b.

– 1.97.

c.

– 1.86.

d.

– 1.732.

e.

1.97. (1 point)

5.

The critical value for the test at 1% significance level is a.

– 2.32.

b.

– 2.15.

c.

2.20.

d.

2.32.

e.

2.45. (1 point)

6.

What is the location of the rejection region in the probability distribution? a.

The entire area to the left of the mean.

b.

The entire area to the right of the mean.

c.

Under both the tails.

d.

Under the left tail.

e.

Under the right tail. (1 point)

7.

At a 1% significance level, it can be said that the a.

Sample statistic is incorrect.

b.

Population standard deviation is incorrect.

c.

Null hypothesis is accepted.

d.

Null hypothesis is rejected.

e.

Null hypothesis is true. (1 point) 463


Based on the following information answer the questions 8 to 16

Fortune Industries Ltd. used three different methods to train its new employees. The number of units of output produced per hour by different employees trained by one of the three training methods are given below: Method A

50

45

55

44

Method B

64

48

52

56

44

Method C

46

42

48

45

57

42

It has to be tested whether the three different training methods lead to different levels of productivity. The significance level of the test is 5 percent. 8.

Which of the following is/are the null and alternative hypotheses for testing whether the mean number of units produced per hour is different for the three different training methods? i.

H0: μA = μB = μC.

ii.

H0: μA ≠ μA ≠ μC.

iii.

H1: μA, μB and μC are not all equal.

iv.

H1: μA = μB = μC.

a.


b.


c.

Both (i) and (iv) above.

B

B

B

d.


e.

Both (iii) and (iv) above. (1 point)

9.

The estimate of the population variance on the basis of the variance among the sample means is a.

45.25

b.

48.36

c.

52.58

d.

55.65

e.

57.25.

(2 points) 10. The variance of the number of units produced per hour for employees who have been trained by method A is a. 20.25 b. 25.67 c. 28.26 d. 30.75 e. 33.65. (1 point) 11. The variance of the number of units produced per hour for employees who have been trained by method B is a.

25.67.

b.

31.07.

c.

45.28.

d.

59.20.

e.

63.24. (1 point)

464

Part III

12. The variance of the number of units produced per hour for employees who have been trained by method C is a. 20.25 b. 25.67 c. 31.07 d. 45.28 e. 59.20.

(1 point) 13. The estimate of the population variance on the basis of the variance within the samples is a. 25.67 b. 31.07 c. 38.646 d. 39.097 e. 59.20. (1 point) 14. The F ratio for the samples is a. 0.80 b. 1.345 c. 1.54 d. 1.75 e. 2.01.

(1 point) 15. For a significance level of 5 percent, the critical value of F ratio is a. 1.194 b. 2.025 c. 2.512 d. 3.525 e. 3.89.

(1 point) 16. At a significance level of 5 percent it can be said that the a. Sample data are incorrect b. Sample means are incorrect c. Null hypothesis is accepted d. Null hypothesis is rejected e. Sample F ratio is incorrect.

(1 point) 17. Given r12 = the correlation coefficient between Y and X1 = 0.9 r13 = the correlation coefficient between Y and X2 = 0.8 r23 = the correlation coefficient between X1 and X2 = 0.5 What is partial correlation coefficient between Y and X1? a. 0.360. b. 0.444. c. 0.720. d. 0.864. e. 0.962.

(2 points) 465


18. An investment manager has identified three portfolios represented as X, Y and Z for making investment. According to his assessment three states of nature are possible during the period of holding the investment which will lead to different gains in rupee-terms. The following table provides the information on the monetary gains from each portfolio under the three states of nature along with the probabilities of occurrence of the three states of nature:

States of nature Portfolio

S1

S2

S3

P(S1) = 0.25

P(S2) = 0.40

P(S3) = 0.35

X

Rs.1,00,000

Rs.2,50,000

Rs.–50,000

Y

Rs.–40,000

Rs.1,50,000

Rs.2,00,000

Z

Rs.4,00,000

Rs.–50,000

Rs.1,00,000

In which portfolio investment should be made on the basis of the expected value criterion? a. Portfolio X. b. Portfolio Y. c. Portfolio Z. d. Indifferent. e. Equal amount in all the three. (3 points) 19. X is a normal distribution with Mean 1 and Standard Deviation 1.5. What is the width of a 95% confidence interval for the population Mean?

a.

5.88.

b.

3.92.

c.

1.00.

d.

0.73.

e.

0.09. (2 points)

20. A fruit vendor who retails apples buys them for Rs.17 a dozen and sells them for Rs.25. If they are not sold within a week, the retailer can recoup only Rs.8 of the original investment because of spoilage. The value of MP and p* for the given situation is

a.

8 and 0.529

b.

17 and 0.68

c.

25 and 0.264

d.

17 and 0.264

e.

25 and 0.5. (3 points)

21. For a normally distributed variable x, μ = σ. Hence, the probability that x takes a value between 3μ and 4μ is

a.

0.0215

b.

0.0315

c.

0.0517

d.

0.0619

e.

0.0717. (2 points)

466

Part III

22. In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is a. 10–5 b. (1/2)5 c. (1/5)2 d. (9/10) e. (9/10)5.

(3 points)

∧

23. For estimating the regression equation Y = a + bX, the data on 5 samples is given below: (X, Y) = (2,4), (3,5), (5,7), (7,9), (8,11). The value of ‘a’ is given by a. 1.62 b. 2.36 c. 3.96 d. 4.85 e. 5.76.

(3 points) 24. A newly formed company is marketing their product in four districts. The marketing manager of the company is concerned that their market share is unevenly distributed in the four districts. In a survey, a random sample of 100 consumers in each district was taken. The following results were obtained: District A District B District C District D Number of consumers who 30 40 45 35 purchase the product Number of consumers who 70 60 55 65 don’t purchase the product

What is the expected ( theoretical) Frequency of persons, who purchase this product in district D? a. 25.0. b. 35.0. c. 37.5. d. 53.8. e. 65.0. (3 points) 25. Classic Cosmetics Ltd., manufactures and sells cosmetic items. The following data has been collected from the company’s finance department. The finance manager wants to know the degree of relationship between the sales and operating expenses of the company.

(Rs. in lakh) Sales Operating expenses

92 63

98 68

80 56

107 72

112 76

128 83

What is the standard error of estimate, which may occur if a regression line is plotted using the observations? Coefficient of correlation, r = 0.9955 and Total Sum of Squares = 457.33 a. 0.6845. b. 0.827. c. 0.983. d. 1.013. e. 1.027. (3 points) 467


26. For binomial distribution with n = 7 and p = 0.2 the value of P(r ≥ 4) is a. 0.030 b. 0.0305 c. 0.0312 d. 0.0341 e. All of the above.

(3 points) 27. The manager of Modern Electricals Co. wants to formulate a relationship between total cost and sales revenues which may be used to estimate the total cost for a given level of sales. The following information is collected by him from the past records. Sales (Rs. in lakh) Total cost (Rs. in lakh)

20 23 38 17 22 35

28 22

45 40 39 36

What is the coefficient of determination? a. 24.33. b. 0.00123. c. 0.0595. d. 0.967. e. 1.42. (4 points) 28. The prices and quantities of some commodities consumed by a family in the years 1997 and 2001 are given below: Year Commodity

1

2

Price Quantity Price Quantity (Rs./kg) (kgs) (Rs./kg) (kgs) Tea 400 8 650 6 Sugar 14.50 120 16.50 100 Milk Powder 70 10 80 10 Rice 12.50 480 14.00 500 You are required to find out Fisher’s ideal price index for the year 2 using the year 1 as the base year. a. 124.94 b. 133.50 c. 147.00 d. 156.10 e. 178.45. (3 points) 29. The new chairman of Hindustan Soaps Ltd. wants to know the estimated value of the accumulated stock of their products with their dealers. He has asked the general manager to get the information within a week. The general manager has sent a fax message to each of their 12,000 dealers in different parts of India. Only 750 responses were received by the evening of the sixth day. The mean value of the accumulated stocks with these dealers is Rs.1.25 crore with a standard deviation of Rs.20.05 lakh. It is assumed that the distribution of the values of accumulated stocks of their products with all the dealers is normal. Which of the following represent a 90 percent confidence interval for the mean value of the accumulated stocks with all their dealers? a. 123.837-126.163. b. 123.36-126.64. c. 99.211-100.789. d. 100.211-101.211. e. 98.46-100.54. (3 points) 468

Part III

30. The Traffic Police Department in a city is studying the number of fatal accidents that have taken place over the last six years. The following data are collected from their records:

Year 2001 2000 1999 1998 1997 1996 Number of fatal accidents 200 185 180 190 175 185 What is the slope of the linear trend equation which can be used to predict the number of fatal accidents in a year? a. – 0.737. b. + 0.737. c. – 1.357. d. + 1.357. e. – 1.858. (3 points) 31. A marketing firm has the following data on number of people contacted and number of people purchased a product during 5 months: No. of people contacted (1,000)

200

180

400

450 240

No. of people purchased the product (1,000)

30

18

50

60

28

What is the percentage of variations in number of people who purchased the product that is explained by the variation in number of people contacted. a.

11.8%

b.

60.32%

c.

82.72%

d.

95.5%

e.

97.7%. (3 points)

469

Model Question Paper V Suggested Answers Part A: Basic Concepts 1. 2. 3. 4. 5.

6. 7.

(c) Degrees of freedom in 3 x 4 contingency table = (3 – 1)(4 – 1) = 2 x 3 = 6. (c) The chi-square test needs to be used for comparing more than two sample proportions and the appropriate distribution for that is the chi-square distribution. (d) Decision making under condition of risk has sufficient knowledge about the states of nature to assign probabilities to this likelihood of occurrence. (b) The control chart that is plotted using the variability of the process results is known as R chart. (d) The pattern “Hugging the center line” indicates that the deviations are minor and uniform in nature. This indicates that the variation has been reduced to a great extent. So for any quality control manager this pattern is generally desirable. (c) Advantage of simulation approach is its flexibility. Extremely complex system with uncertainty can be studied with a reasonable degree of accuracy. ˆ (b) It is ratio of the difference between the r and the corresponding Y values and the Y values. Mathematically it is

ˆ Y−Y x100 . ˆ Y

(e) Both statements (a) and (d) are true. Secular trend represents long-term behavior of a variable; hence statement (b) is false. Seasonal variations are repetitive and predictable movements above and below the trend line occurring within one year or less; hence statement (c) is false. 9. (b) Because people tend to spend less on goods when their prices are rising, Paasches index which bases on current weights, produces an index which has a downward bias. 10. (c) The unweighted aggregate index is calculated by totaling the current year/given year’s element and then dividing the result by the sum of the same elements during the base period. So, unweighted Aggregates Price Index 8.

=

∑ P1 ∑ P0

X100 .

11. (a) F =

RSS/k ESS/(n − (k + 1))

When, F = 0, RSS / k = 0 i.e., RSS = 0

RSS = 0. TSS 12. (b) Multicollinearity reduces the effect of individual variables on the model. So, It reduces the effect of any sensitivity analysis. So, coefficient of determination =

13. (b) Correlation between two variables x and y is given by ρ=

Cov(x, y) σx σy

.

14. (a) Option (b) is true with regard to appoint estimate. Options (c) and (d) are with regard to an interval estimate. Option (a) is false because there is no reason why only point estimates should be used for estimating the parameters of large population. 15. (c) In a right tailed test the critical region lies to the right of mean and the values that fall to the right of the mean are higher than the mean. Hence the alternative hypothesis for a right tailed test is H1 : μ > μ0.

Part III

16. (b) Relative cyclical residual =

ˆ (Y - Y) x 100 ˆ Y

= (7.50 – 6.25) x 100/6.25 = 20% 17. (c) The standard normal distribution has zero Mean and unit standard deviation. 18. (d) In judgmental sampling the sample is selected according to the judgment of the investigator. Hence some subjectivity creeps into the sampling process and the truly random nature of the sample is lost to some extent. 19. (e) All the elements stated in the alternatives are commonly found in problems of decision making under uncertain conditions. 20. (c) Co-variance is the measure of the variability of one variable in relation to another variable. 21. (e) V(x + y) = V(x) + V(y) + 2 Cov (x, y) or, V(x) + V(y) = V(x + y) – 2 Cov (x, y). 22. (d) t-distribution depends on the degrees of freedom (df). For each ‘df’ there will be a particular t-distribution. ‘df’ is dependent on the sample size. We know that as sample size increases the t-distribution approaches the normal distribution. Alternatives (a), (b), (c) and (e) are true. Alternative (d) is false. 23. (e) In case of cluster sampling the population is divided into clusters or groups. The elements of each cluster are not homogeneous. Each cluster is representative of the population as a whole. 24. (c) If be the mean of a random sample y size n drawn from a population having mean μ and standard deviation ( σ ), then the sampling distribution of the sample mean x is approximately a normal distribution provided the sample size n is sufficiently large. 25. (b) For standard normal distribution mean = 0 and standard deviation = 1. If the observed value is ‘x’ then the corresponding z-value would be (x – 0)/1 = x. 26. (e) Alternative hypothesis for a two-tailed test on population mean is represented as H1 : μ ≠ μ 0 .

27. (d) Expected value does not predict the value of the random variable. Hence, alternative (d) is false; alternatives (a), (b) and (c) are true. 28. (c) Standard error of estimate measures the variability, or scatter of the observed values of the dependent variable around the regression line. 29. (d) Multicollinearity does not reduce the accuracy of the model or predictive powers but it hurts any sensitivity analysis. 30. (c) If RSS = 0, then R2 = 0; hence in such a case the regression model cannot explain the variations in the dependent variable. 31. (c) A weighted index attaches weights according to their significance. The unweighted index does not replete the reality since the price changes are not linked to any usage/consumption level. 32. (a) Since the type-I error (α) is very costly, we have to select α that is very low. 33. (a) Sampling Fraction = n/N, where n and N represent the size of the sample and population respectively. 34. (e) The weighted average of price relatives using base values is the same as the weighted aggregate of actual prices (Laspeyers method). When the Arithmetic Mean is used for averaging the relatives and the base year values are used as weights. 35. (e) The consistency of the index numbers has been tested over the years and the most important of these tests are (i) the time reversal test, (ii) the factor reversal test, and (iii) the circular test.

471


36. (d) When a problem has a large number of possible actions, it is very difficult to construct a conditional profit table and hence marginal analysis should be done. 37. (c) Proportion of variations in the value of the dependent variable that is explained by the regression relationship = Coefficient of determination = r2 = (–0.70)2 = 0.49 ∴ Proportion of variations in the value of the dependent variable that is not explained by the regression relationship = 1 – r2 = 1 – 0.49 = 0.51 i.e., 51%. 38. (c) When the population is normally distributed, the sampling distribution has a mean equal to the population mean. Symbolically, μ = x. 39. (a) If a stock is consistently giving better results, it can be concluded that average returns of the stock is greater than the average returns of other stock. 40. (e) A value index cannot distinguish between the price changes and quantity changes. Hence an increase in the value index cannot indicate with certainty that the prices have increased.

Part B: Problems (60 Points) 1.

(c) H0 : μ = 49 H1 : μ < 49.

2.

(b) Standard error of mean, σ x = =

6

x

σ n

N−n N −1

50 − 12 = 1.525 . 50 − 1

3.

12 (b) The standard deviation of the population is known. Hence, the normal distribution is the appropriate distribution.

4.

(b) The standardized value of the sample mean, z =

5.

(a) From the standard normal table it can be seen that the critical z value for an area approximately 0.49 to the left of the mean is –2.32. (d)

6.

=

46 − 49 = − 1.97 . 1.525

Acceptance region

0.01 Rejection region

x −μ σx

-2.32

μ

-1.97

7.

(c) At a 1% significance level it can be said that the null hypothesis is accepted because the standardized value of the sample mean falls in the acceptance region.

8.

(a) The statements of the hypothesis are as follows: H0 : μA = μB = μC H1 : μA, μB and μC are not all equal. B

9.

472

(c) The observations are stated in the following table: Methods A B C

50 64

Observations 45 55 44 48 52 56

44

46

42

57

48

45

42

Sum 194 264

Mean 48.5 52.80

280

46.67

Part III

Grand mean =

194 + 264 + 280 = 49.20 = X 15

The estimate of the population variance on the basis of the variance among the sample means

{

2 1 = $σ 1 = 4x(48.5 − 49.2) 2 + 5(52.8 − 49.2) 2 + 6x(46.67 − 49.2) 2 2

}

$σ 2 = 1 {4x0.49 + 12.96x5 + 6.4009x6} = 52.58 1 2

10. (b) Variance of the number of units produced per hour by employees who have been trained by method A,

(

1 σ2 = x −x A n − 1 ∑ Ai A A

)

2

nA = 4 and x A = 48.5 Now, ∑ ( x Ai − x A ) = (50 – 48.5)2 + (45 – 48.5)2 + (55 – 48.5)2 + (44 – 48.5)2 2

= (2.25 + 12.25 + 42.25 + 20.25) = 77. So, σ 2A =

77 = 25.67. 4 −1

11. (d) Variance of the number of units produced per hour by employees who have been trained by method B, σ2B =

2 1 ∑ ( x Bi - x B ) nB −1

nB = 5 and x B = 52.8 B

∑ ( x Bi − x B )

2

= (64 – 52.8)2 + (48 – 52.8)2 + (52 – 52.8)2 + (56 – 52.8)2 + (44 – 52.8)2

= 125.44 + 23.04 + 0.64 + 10.24 + 77.44 = 236.80 So, σ 2B =

236.80 = 59.20. 4

The rejection region lies under the left tail. 12. (c) Variance of the number of units produced per hour by employees who have been trained by method C, 2 1 σ2 = x − xC ) C n −1 ∑ ( Ci C

nC = 6 and x C = 280 = 46.67 6 Now, ∑ ( x C i − x C ) = (46 – 46.67)2 + (42 – 46.67)2 + (48 – 46.67)2 + (45 – 46.67)2 2

+ (57 - 46.67)2 + (42 – 46.67)2 = 0.44 + 21.81 + 1.77 + 2.79 + 106.71 + 21.81 = 155.33 So, σc2 =

155.33 = 31.07 5

13. (d) The estimate of the population variance on the basis of the variance within the samples = ⎛ n −1 ⎞ 1 $σ22 = ∑ ⎜ j ⎟s 2 = {25.67x3 + 59.20x4 + 31.07x5} = 39.097. ⎜ n T − k ⎟ j 15 − 3 ⎝ ⎠

473


14. (b) The F ratio for the sample =

$σ 2 52.58 1 = = 1.345 where the degrees of freedom are 2 $σ 2 39.097 1

and 12 for the numerator and denominator respectively. 15. (e) The number of degrees of freedom for the numerator of the F ratio = Number of samples – 1 = 3 –1 = 2. The number of degrees of freedom for the denominator of the F ratio = nT – k = 15 – 3 = 12. The corresponding F2,12 value at 5% level of significance obtained from the F table (critical value) = 3.89. 16. (c) Since 3.89>1.345 the F ratio for the samples falls in the acceptance region. Hence the null hypothesis is accepted. 17. (e) We have R12, 3 =

r12

− r13 , r23 2

2

(1 − r13 )(1 − r23 )

= =

0.9 − 0.8 x 0.5 2

2

(1 − 0.8 )(1 − 0.5 ) 0.5 0.27

= 0.962

18. (b) The expected value criterion can be applied as: Portfolio X:

100000 (0.25) + 250000 (0.40) – 50000 (0.35) = Rs.1,07,500

∴Expected gain from portfolio X = Rs.1,07,500 Portfolio Y:

− 40000 (0.25)+ 150000 (0.40) + 200000 (0.35) = Rs.1,20,000

∴Expected gain from portfolio Y = Rs.1,20,000 Portfolio Z :

400000 (0.25) – 50000 (0.40) + 100000 (0.35) = Rs.1,15,000

∴Expected gain from portfolio Z = Rs.1,15,000 From above, it can be seen that the maximum expected gain (Rs.1,20,000) is associated with Portfolio Y. ∴ Investment must be made in Portfolio Y. 19. (a) The area of one side is equal to 1.5 x 1.96 = 2.94. Thus, the width of a 95% confidence interval for the population mean is equal to 2.94 2.94 x 2 = 5.88. 20. (a) MP = 25 – 17 = 8 P*

=

ML (MP + ML)

=

(17 − 8) (8 + 17 − 8)

=

9 = 0.529. 17

21. (a) The probability that x takes a value between 3μ (or μ + 2σ) is equal to 0.4821 and the probability that x takes a value between 4μ (or μ + 3σ) is 0.4987. Thus, the probability that x takes a value between 3μ and 4μ is equal to 0.4987 – 0.4821 = 0.0215. 474

Part III

22. (e) Probability that a bulb is defective =

10 1 = 100 10

Probability that a bulb is not defective = 1 − =

1 10

9 10 5

⎛9 ⎞ Probability that out of a sample of 5 bulbs no bulb is defective = ⎜ ⎟ . ⎝ 10 ⎠

23. (a) Mean of X =

[2 + 3 + 5 + 7 + 8] =5 5

Mean of Y =

[4 + 5 + 7 + 9 + 11] = 7.2 5

b=

[∑ XY − n(Mean of X) (Mean Y)] [∑ X 2 − n(X) 2 ]

= 1.11538 a = (7.2 – 1.11538 x 5) = 1.62. 24. (c) District A District B

District C District D

Total

Number of persons who purchase the product (x)

30

40

45

35

150

Number of persons who do not purchase the product (y)

70

60

55

65

250

Total

100

100

100

100

400

Combined proportion of people who purchase this product (p) = 150/400 = 0.375 Combined proportion of people who do not purchase this product = 1 – 0.375 = 0.625 Frequency of persons, who purchase this product District A District B District C District D Observed (actual) frequency (fo) 30 40 45 35 Expected (theoretical) frequency (fe) 100 × 0.375 100 × 0.375 100 × 0.375 100 × 0.375 = 37.50 = 37.50 = 37.50 = 37.50 25. (d) Coefficient of determination = Total Sum of Squares − Error Sum of Squares Total Sum of Squares

(0.9955)2 =

457.33 − ESS

457.33 r = 1 – ESS = 4.107 2

Standard error estimate, Se =

ESS n−2

=

4.107 6−2

= 1.013. 26. (d) P(r > 4) = P(r = 4) + P(r = 5) + P(r = 6) + P(r = 7) = P(0.0287) + P(0.0043) + P(0.00036) + P(0.0000128) = 0.034. [Using Binomial Distribution] 475


27. (d) Finding the values of a and b: Y = a + bX Where, X = Sales (Rs. in lakh) Y = Total costs (Rs. in lakh) b = 194 a = 171 n=6 = 5987 = 6782 = 28.5 b = 0.899 a = –0.568 = –0.568 + 0.899 x Calculation of coefficient of determination: Coefficient of determination (r2) = =

a ∑ Y + b ∑ XY − nY 2 ∑ Y 2 − nY 2 (−0.568)(171) + (0.899)(5987) − 6(28.5) 2 5299 − 6(28.5) 2

411.685 = 0.967. 425.5 28. (a) Let the following notations be used: =

P0

=

Price of commodity in base year (1997)

Q0

=

Quantity of commodity command in base year (1997)

P1

=

Price of commodity in the year 2001

Q0

=

Quantity of commodity consumed in the year 2001.

Commodity Tea Sugar Milk Powder Rice

P0 400 14.50 70 12.50

Q0 8 120 10 480

P1 650 16.50 80 14.00

Q1 6 100 10 500

P0Q0 3200 1740 700 6000 11640

P1Q0 5200 1980 800 6720 14700

P0Q1 2400 1450 700 6250 10800

P1Q1 3900 1650 800 7000 13350

Fisher’s ideal price index = square root of

∑ P1Q 0 ∑ P1Q1 x 100 = x ∑ P0 Q 0 ∑ P0 Q1

14700 13350 x 100 = 124.94. x 11640 10800

29. (a) Sample size = n = 750 Sample mean = x = Rs.125.0 lakh Sample s.d. = s = Rs. 20.05 lakh The problem asked is to calculate an interval estimate of the mean value of stocks lying with all 12,000 dealers of Hindustan Soaps Ltd., so that it can be 90% confident that the population mean falls within that interval. The sample size is over 30, so central limit theorem enables us to use the normal distribution as the sampling distribution. Here we do not know the population standard deviation, and so we will use the sample standard deviation as the estimate of the population standard deviation. Since we have a finite population size of 12,000 and since our sample size is (750/12000) i.e., 6.25% more than 5 percent of the population, we will use the formula for deriving the standard error of mean for finite populations:

σ xˆ =

476

σˆ n

x

N − n N − 1

Part III

Where σˆ = estimate of population standard deviation N = size of the population n = size of the sample. ∴The value of standard error of mean is 20.05 750

(12000 − 750) = 0.7089 (12000 − 1)

x

We consider 90% confidence level, which would include 45% of the area on either side of the mean of the sampling distribution. Looking at the normal table for 0.45 value, we find that about 0.45 of the area under the normal curve is located between the mean and a point 1.64 standard errors from the mean. Therefore, 90% of the area is located between ± 1.64 standard errors from the mean and the confidence limits are Upper confidence limit = x + 1.64 σ x = 125 + 1.64 x 0.7089 = 126.163 Lower confidence limit = x – 1.64 σ x = 125 – 1.64 x 0.7089 = 123.837 ∴90% confidence interval for the mean value of stocks with all the 12,000 dealers of Hindustan Soaps Ltd. is Rs.123.837 lakh to Rs.126.163 lakh. 30. (d) Let the following notations be used: X = Year Y = Number of accidents x = Coded value for time variable (year) X =

∑X 1191 = = 1998.5 6 6

X

x = (X − X) x 2 2.5 x 2 = 5 1.5 x 2 = 3 0.5 x 2 = 1 –0.5 x 2 = –1 –1.5 x 2 = –3 –2.5 x 2 = –5

X − X

2001 2000 1999 1998 1997 1996 ΣX = 11991

2001 – 1998.5 2000 – 1998.5 1999 – 1998.5 1998 – 1998.5 1997 – 1998.5 1996 – 1998.5

Y

xY

x2

200 185 180 190 175 185 ΣY = 1115

1000 555 180 –190 –525 –925 ΣxY = 95

25 9 1 1 9 25 Σx2 = 70

∧

∴ Y = a + bx where, ∧

b= Y

∑ XY ∑X2

and a = Y

95 = 1.357 70 ∑Y 1115 = = 185.833 a = n 6

∴b =

∧

∴The estimating equation is: Y = 185.833 + 1.357x. 477


31. (d) Let X : No. of persons contacted (in ’000)

Y : No. of persons who purchased the product (in ’000) Coefficient of correlation, Σ(X − X)(Y − Y) r= Σ(X − X)2 (Y − Y) 2 A. B. C. D.

X Y

E.

CxD

676.8 2188.8 1356.8 3556.8 496.8

F.

C2

8836 12996 11236 24336

2916

∑ (x-x)2

G.

D2

51.84 368.64 163.84 519.84 84.64

∑ (y-y)2

x−x

y−y

8276

200 30 –94 –7.2

180 18 –114 –19.2

400 50 106 12.8

450 60 156 22.8

240 28 –54 –9.2 ∑ (x-x) (y-y) = 8276

= 60320 = 1188.8

= 0.977 60320 x 1188.8 r2 = 0.955 ∴95.5% of the variations in number of persons who purchased the product is explained by variation in number of persons contacted. r=

478

Quantitative Methods 2 ICMR Workbook

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