AST-01 ASSIGNMENT BOOKLET
Bachelor’s Degree Programme STATISTICAL TECHNIQUES st
st
(Valid from 1 July, 2013 to 31 March, 2014) It is compulsory to submit the assignment before filling in the exam form.
School of Sciences Indira Gandhi National Open University Maidan Garhi New Delhi-110068 2013-2014
Dear Student, Please read the section on assignments in the Programme Guide for Elective courses that we sent you after your enrolment. A weightage of 30 per cent, as you are aware, has been earmarked for continuous evaluation, which would consist of one tutor-marked assignment for this course. The assignment is in this booklet. Instructions for Formatting Your Assignments
Before attempting the assignment please read the following instructions carefully. 1) On top of the first page of your answer sheet, please write the details exactly in the following format:
ROLL NO: …………………………………………… NAME: …………………………………………… ADDRESS: ……………………………………………
…………………………………………… …………………………………………… COURSE CODE:
…………………………….
COURSE TITLE:
…………………………….
ASSIGNMENT NO. ………………………….… STUDY CENTRE: ………………………..…..
DATE: ……………………….………………...
PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND TO AVOID DELAY.
2) Use only foolscap size writing paper (but not of very thin variety) for writing your answers. 3) Leave 4 cm margin on the left, top and bottom of your answer sheet. 4) Your answers should be precise. 5) While solving problems, clearly indicate which part of which question is being solved. 6) This assignment is valid only upto March, 2014. If you have failed in this assignment or fail to submit it by 31st March, 2014, then you need to get the assignment for the January, 2014 cycle and submit it as per the instructions given in that assignment. 7) It is compulsory to submit the assignment before filling in the exam form. We strongly suggest that you retain a copy of your answer sheets.
We wish you good luck.
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Assignment (To be done after studying the course material) Course Code: AST-01 Assignment Code: AST-01/TMA/2013-14 Maximum Marks: 100
1. a) A lot contains 50 defective and 50 non defective bulbs. Two bulbs are drawn at random one at a time with replacement. The events A, B, C are defined as follows: A = first bulb is defective, B = second bulb is non defective, C = The two bulbs are both defective or both non defective. Determine whether A, B a nd C are independent. Also calculate P A (A ∩ C ), P(B ∩ C ) and P (A ∩ B). (5) b) From a population of 20,000 observations, a sample of 500 observations is selected. Calculate the standard error of sample mean if the population standard deviation equals 20. (3) c) A random sample of 700 units from a large consignment showed that 200 were (2) damaged. Find 95 % confidence interval for the proportion of damaged unit in t he consignment. 2. (a) Two floppies are selected at random without replacement from a box containing 7 good and 3 defective floppies. Let A be the event that the first floppy drawn is defective, and let B be the event that the second floppy drawn is defective. (i) (ii)
(5)
Find the conditional probabilities P(B/A) and P(B/AC) C C Show that P(B) = P(B/A). P(A) + P(B/A ) P(A ) = P(A).
(b) A company produces one-kilogram sugar packets. The specifications on the net content are 1000 ± 5 grams. Assuming that the net content follows normal distribution with mean weight as 1005 grams and the process capability equal to 30 grams, find out the proportion of packets that have weight less than lower specification limit. What should be the mean if t his proportion is to be reduced to 0.01?
(5)
3. a) Two new types of petrol, called premium and super, are introduced in the market, and their manufactures claim that they give extra mileage. Following data were obtained on extra mileage which is defined as actual mileage minus 10. (10) Data on extra mileage Ordinary Petrol
1
2
2
1
Premium Petrol
2
2
1
3
Super Petrol
4
1
2
3
(i)
Using ANOVA, test whether premium or super give an extra mileage.
(ii)
What is your estimate for the error variance?
(iii)
Assuming that the error variance is known and is equal to 1, obtain the 95 % confidence interval for the mean extra mileage of super. 3
4. a) The number of total accidents during the first half of 2008 are as given in the Table below.
(7)
Data on Fatal Accidents Month No. of fatal accidents
January 2
February 4
March 3
April 4
May 3
June 2
i) Draw the graph and comment. ii) Prepare the table of forecasts and errors by applying simple exponential smoothing for the data given in (i) (take the exponential smoothing constant as 0.1). b) Among a group of 50 students of class 12, five students are 16 years old, thirty students are 17 years old, five students are 18 years old and ten students are 19 years old. Let a student be drawn from this group and let X be the age of the selected students ( in complete years).
(3)
(i) Is the distribution of X continuous? (ii) Obtain the probability density function of X. (iii) What is the probability that t he student selected is atleast 18 years old.
5. a) An x − R chart is being set up for monitoring the twist angle of a certain paper board. Twenty subgroups (each of size 5) of data have been collected and their summary is furnished in the table below. Sample No
1
2
X
−1
−3
R Sample No
5 11 −2 6
12 12 4
X
R
25
3 1
4
10 13 −2 16
6 14 −1 9
−3
5 7
6 0
7 4
8
6 15 −3
6 16 −1 6
9 17 1
10 18 −1 11
4
17
−2
9 4
10 1
13 19 0
6 20 0
9
6
(4)
(i)
Scrutinize the data (without drawing charts) and check if there are any increasing or decreasing trends in process average.
(ii)
Assuming that there are no systematic patterns in the p rocess, compute the control limits for x − R charts.
(iii)
Do you suspect any assignable causes? If so, at what sample numbers?
b) Define stratified sampling and explain the basic principle that forms the strata.
(3)
c) Suppose three small towns are under study, having population N1 = 60000, (3) N2 = 30000, N 3 = 30000, respectively. A stratified random sample is to be taken with a total sample size of n = 500. Using proportional allocation, determine the sample size of each stratum.
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6. a) The frequency distribution of the daily cost (in Rs.) of commuting back and forth to work by 100 employees of a steel plant is: Cost (in Rs.) Number of employees
0-50 29
50-100 32
100-150 29
(5)
150-200 10
Find the mean daily cost, median daily cost and standard deviation of commuting. b) A study was conducted to see whether the natural perception having equal number (5) of boys and girls was followed by families or not. 160 families were considered as sample. Each family had 5 children. Frequency of families having a particular combination was recorded as follows.
Boy Girl Frequency
5 0 20
4 1 25
3 2 25
2 3 20
1 4 30
0 5 40
Test whether the standard norm is followed or not at 95 % level of significance.
7. a) An economist want to estimate a multiple regression equation in which the (3) amount saved by the ith family depends on the family’s income, whether the family is graduated or non graduate, and whether the family is headed by a male or female. Explain how a regression equation of this sort can be estimated. What is the dependent variable ? What are the independent variables? What assumption must be made ?
b) A population of size 20 is sampled without replacement. The standard deviation of the population is 0.35. We require the standard error of the mean to be not more than 0.15. What is the minimum sample size required for this?
(3)
c) In a population of size N = 5, the values of an attribute are 8, 3, 1, 11 and 4 corresponding to population members 1 to 5.
(4)
i)
Write all different samples of size 2 when simple random sampling (without replacement) is practiced ?
ii)
Verify that y (sample mean) is an unbiased estimate of y (population mean) , using the above data.
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8. a) For the following series of observations, compute the moving average of length 4 and place them in line with the corresponding year. (5)
Year Annual sales (Rs. Crores) Year: Annual sales (Rs. Crores)
1995 2 2000 7
1996 6 2001 2
1997 1 2002 6
1998 5
1999 3
2003 4
2004 2005 8 3
b) The weight of ghee obtained from a tin of milk is uniformly distributed with a mean of 8 kg and range of 1.5 kg. Calculate (5) i) ii)
the probability that a tin of milk will yield ghee weighing between 7 kg and 8.5 kg. the largest and the smallest weights of ghee obtained from a tin of ghee.
9. a) A sample of size of 3 is to be selected from a population of 15 households. List all the Possible samples obtained by: (4) i)
Linear systematic sampling
ii)
Circular systematic sampling
b) A random sample of 10 observations is taken from a normal population having the variance 42.5. Find the probability of obtaining a sample having standard d eviation between 2.4 and 4.9. (4) c) Give two examples from day to day life where cluster sampling can be used. Justify your your choice of example. (2)
10. State whether the following statements are true or false. Give brief justification for each case. a) Pie chart is more appropriate than bar chart for graphically presenting the profits of five plants of a manufacturing company during the financial year 2007-08. b) If an event A implies another event B, then A contains B. c) In a standard normal distribution, the area u nder the curve between and 0 is 1.
−∞
d) A sampling plan with AQL = 0.08 and LTPD = 0.05 is a good sampling plan. e) The number of possible samples in drawing 3 items from a population of 7 items without replacement is equal to 21.
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(10)
