Statistics & Research Methodology (103) Contributors :
Prof. (Gp.Capt.) D.P.Apte Prof. Anagha Gupte Prof. Anjali Mote Prof. S.N.Parasnis
Session Plan Subject: Statistics & Research Methodology (103)
Lecture No. 1
2
3
4 5
6
7 8
9 10 11 12 13 14
Topic
Statistics: Introduction to business statistics, Importance, Definition & classification Collection of data: Primary & secondary data, Sources and Classification of data, Tabular presentation Presentation of data: Frequency distribution tables, Graphical presentation Measures of Central Tendency and Variability: Ungrouped data Case-1
Measures of Central Tendency, Measures of variability : Grouped data Fundamentals of Probability, Laws of probability Case -2
Probability: Conditional probability, Bayes’ theorem Probability distributions: Binomial distribution Case -3 Probability distributions: Poisson and Exponential distribution Probability distribution: Normal distribution Case - 4
Suggested Reading a. Chapter 1 b. Chapter 1
a. Chapter 2 b. Chapter 1
a. Chapter 2 b. Chapter 1 a. Chapter 3, 4 b. Chapter 1 a. Chapters 1, 2, 3, 7. b. Chapter 1 a. Chapter 3, 4. b. Chapter 1. a. Chapter 5. b. Chapter 2 a. Chapter 1, 2, 3, 7 b. Chapter 1. a. Chapter 5 b. Chapter 2 a. Chapter 6 b. Chapter 3 a. Chapter 5, 6 b. Chapters 2 a. Chapter 6 b. Chapter 3 a. Chapter 6 b. Chapter 3,4 a. Chapter 5, 6
Sampling : Importance and sampling techniques Sampling distribution: Concept, sampling distribution of mean Case - 5
b. a. b. a. b. a.
18
Sampling distribution of proportion and variance
b. a. b.
19
Estimation for decision making
a.
20
Case - 6
b. a. b.
21
Concepts of Hypothesis Testing
22
Tests of significance - I
a. b. a.
Case - 7
b. a.
15 16 17
23
b. 24
Tests of significance - II
a. b.
25
Tests of significance - III
a. b.
26
27
28
Case-8
a.
Test of significance - IV
b. a.
Correlation: Simple correlation, Multiple and partial correlation
b. a. b.
Chapter 3 Chapter 7 Chapter 5 Chapter 7 Chapter 5 Chapters 5, 6 Chapters 4 Chapters 7 Chapters 5, 6 Chapter 6,7, 8 Chapter 5, 6 Chapter 6, 7 Chapters 4, 5 Chapter 9 Chapter 7 Chapter 9, 10, 11 Chapter 7, 8 Chapters 9, 10 Chapters 7, 8 Chapters 9, 10, 11 Chapters 7, 8 Chapters 10, 11 Chapters 7, 8 Chapter 9, 10,11 Chapter 8 Chapter 9, 10, 11 Chapter 7, 8 Chapter 12 Chapter 10
29
Case - 9
30
32
Regression analysis – I: Simple Regression Regression analysis – II: Multiple correlation Case - 10
33
Analysis of variance
34 35
Surprise Test - 1 Surprise Test- 2
31
a. b. a. b. a. b. a.
Chapter 14 Chapter 9 Chapter 13 Chapter 10 Chapter 13 Chapter 11 Chapter 12, 13 b. Chapter 10 a. Chapter 14 b. Chapter 9
Suggested Reading
a. Statistical tools for Managers: Using MS Excel – By Prof. (Gp.Capt.) D.P.Apte. b. Complete Business Statistics: By Amir D. Aczel; Jayavel Sounderpandian.
Case 1: Know Your Friends
Miss XYZ joined MITSOB, Pune for two years PGDM program. She had left her hometown first time in her life except of course few holiday trips. On one hand she was excited about new friends, management studies, and other activities and so on. On the other hand she was worried about studies, placement, getting good friends, homesickness etc. Therefore, she wanted to know more about her course mates. Obviously it was difficult to contact all the students from her batch and know about them. So she decides to collect a sample data from about 25-30 students selected by her. She was not sure about random collection of data. So she learnt about random number generator and used student roll numbers to select the random sample. With the help of faculty she prepared questionnaire and collected data. From the data she prepared report for submission. It was quite nice and elaborate. However, when she compared it with friends, she found lot of variation. So she decided to take information from her two friends who had also collected the similar data. Of course she has to take some precautions while using the data from the friends. a) b) c) d) e) f) g) h)
i)
j)
k)
Collect the data. Draw Simple Bar Diagram to represent specialization wise strength of students. Draw Pie diagram to represent the number of students in each of the blood groups. Prepare the gender-wise frequency table for the blood groups. Plot graph of height Vs. weight of the students. Comment on the graph. Plot graph of family income Vs. expenditure of the student. Comment on the graph. Calculate gender-wise mean and standard deviation of marks of 10th and 12th standard. Comment on the basis of the results. Prepare a two way table showing age-wise and gender-wise strength of students. Represent this data with the help of a multiple bar diagram taking age on X axis and gender-wise strength on Y axis. Also comment on the basis of this graph. Prepare a two way table showing father’s education and mother’s education taking education categories as Below Graduate, Graduate and Post graduate. Comment on the basis of this table. Calculate mean and standard deviation of marks obtained by the students in 10th and 12th standard according to having a job experience or not. Comment on the values you obtained. You are free to do additional analysis if you feel so.
Note: Sample quest ionnaire is attached. However, please feel free to add few quest ion based on your purpose.
Developed by: - Prof.(Gp.Capt.) D.P.Apte
MIT School of Business Student’s Profile Batch
Roll No
Specialization for PGDM ______________________ Name _____________________ Date of Birth (MM/DD/YY) -----/-----/------ Blood group
_________
Email address:_________________________ Contact no.: Landline: _________________ Mobile: _________________ Present Address:_____________________________________________ Permanent Address:___________________________________________ Educational Qualification
i)
10th percentage _______%
ii)
12th percentage _______%
iii)
Graduation Percentage ________%
iv)
Graduation Type: Arts / Commerce / Science / Engineering / Management / Pharmacy/ Other (Specify)___________________
v)
Name of the College (for graduation):________________________________
vi)
College Address:________________________________________________
vii)
Work experience: Yes / No
viii)
Whether attended coaching for competitive exams (CAT/MAT/CET etc) Yes / No If Yes, mention the name of the institute:_____________________________
Payment of the current fees by i) Own ii) Loan Future plans: i) Job ii) Family Business iii) Own business iv) First job then business Family Background
Father’s Education: _______________ Father’s occupation:___________________ Mother’s Education: _______________ Mother’s occupation:___________________ Family Type: Nuclear / Joint / Single parent Number of family members staying together:___________
Family income (in Rs. Per year):_____________ Newspaper read: __________________________ How did you come to know about MITSOB i)
Newspaper Advertise (Specify)____________________
ii)
Internet : MITSOB website / Google / Facebook / Shiksha / Other
iii)
Recommendation from friends
Case 2: Better to be roughly right than precisely wrong
After joining the institute Mr. ABC was pondering over through, whether he has made a right decision of joining this institute. Infrastructure was good, he could see that. Faculty was excellent; he could experience that in last three weeks. But he was not sure about academic level of the students. He could observe that few of them were very good, participating in discussions, taking initiative for learning, helping others to learn different subjects. However, few others were hardly participating, timid and not sure about the learning process. So Mr. ABC decides to check past academic performance of the students from his batch. Since the graduation background of students was different and some of them have also not got their graduation results, he decided to take 10 th and 12th standard percentage of marks as a basis for his study. a) Collect 10th and 12th standard percentage of marks from 30 students randomly. b) Based on data collected find mean, median, standard deviation for both the data sets (10th marks & 12th marks) using MS Excel. (Take help of faculty/friends). c) From the mean of your sample, estimate average percentage of 10th and 12th of your entire batch. d) Prepare two frequency distribution tables separately (10 th & 12th std) for the marks of 30 students taking classes as 40-45, 45-50,... e) Draw graphs by taking classes on X-axis and frequency on Y-axis. Comment on the basis of the graphs. f) Collect the sample means (sample of 30) from your 10 friends. Comment on the observations. g) Find the grand mean for entire batch from population data. h) Prepare two frequency distribution tables separately (10 th & 12th std) for the marks of entire batch. i) What have you learnt from your observations? Comment. j) Where do you stand according academic performance in the entire batch? Note: To save you from collecting sample data, we have collected population data.
Case 3: Job Applications
A business graduate very much wants to get a job in any one of the top 10 accounting firms. Applying to any one of these companies requires at lot of effort and paperwork and is therefore costly. She estimates the cost of applying to each of the 10 companies and the probability of getting a job offer there. These data are tabulated below. The tabulation is in the decreasing order of cost. 1.
If the graduate applies to all 10 companies, what is the probability that she will get at least one offer?
2.
If she can apply to only one company, based on cost and success probability criteria alone, should she apply to company 5? Why or why not"?
3.
If she applies to companies 2, 5, 8, and 9, what is the total cost? What is the probability that she will get at least one offer?
4.
If she wants to be at least one 75% confident of getting at least one offer, to which companies should she apply to minimize the total cost? (This is a trial-and-error problem.)
5.
If she is willing to spend $1,500, to which companies should she apply to maximize her chances of getting at least one job? (This is a trial-and-error problem.)
Company Cost
1
2
3
4
5
6
7
8
9
10
$870 $600 $540 $500 $400 $320 $300 $230 $200 $170
Probability
0.38
035
0.28
0.20
0.18
0.18
0.17
0.14
0.14
0.08
* Case taken from Complete Business Statistics by Amir Aczel & J. Sounderpandian, TATA Mc Graw Hill.
Case 4: Microchip Contract
A company receives an order for five custom-made microchips at a price of $7,500 each. The company will produce the chips one by one using a complex process which has only a 67% chance of producing a defect-free chip at each trial. After five defect-free chips are produced the process will be stopped. A cost accountant at the company has prepared the following cost report: The cost of production includes a $14,800 fixed cost and a $2,700X unit variable cost. Thus if X number of chips are produced, the total cost of production would be 14,800 + 2.700X dollars. The revenue minus the cost of production will be the profit. After some analysis the finance manager of the company says that the risk may be too high and thinks the order should not be accepted. 1.
What distribution will the number of chips produced, X, follow?
2.
What is the expected value and standard deviation of X?
3. What is the expected value and standard deviation of the profit? 4 What is the break-even X (allow fractional values for X)? 5. 6.
What is the probability that accepting the o rder will result in a loss? A popular measure of risk in a venture is value at risk, which is the loss suffered at the 5 th percentile of the return from the venture. In this problem, find an integer x such that P[X >x] is approximately 5%.
7.
For the x value found in part 6, calculate the loss, and thus the value at risk.
8.
Express the value at risk as a percentage of the expected value of the profit.
9.
What is your assessment of the risk and reward in the order? Should the company accept the order? The sales manager of the company says that the customer is very likely to agree to increase the order quantity from five to eight chips. But he is not sure whether the matter should be pursued with the customer.
10. "If accepting an order of five itself is risky, will it not be even more risky to accept an order for eight?" asks the sales manager. How would you answer him? 11. Calculate
eight.
the expected value and standard deviation of the profit for an order quantity of
12. What
is the value at risk for an order quantity of eight, computed in a manner similar to parts 6 and 7 above? Express the value at risk as a percentage o f the expected profit.
13. Looking
at the answer to parts 3,8,11, and 12, would you say the risk and reward have become more favorable, compared to an order quantity of five?
14. Should
the company pursue the matter of increasing the order quantity to eight with the customer?
* Case taken from Complete Business Statistics by Amir Aczel & J. Sounderpandian, TATA Mc Graw Hill.
Case 5: Acceptable Pins
A company supplies pins in bulk to a customer. The company uses an automatic lathe to produce the pins. Due to many causes-vibrations, temperature, wear and tear, and the like-the lengths of the pins made by the machine are normally distributed with a mean of 1.012 inches and a standard deviation of 0.018 inch. The customer will buy only those pins with lengths in the interval 1.00 ± 0.02 inch. In other words, the customer wants the length to be 1.00 inch but will accept up to 0.02 inch deviation on either side. This 0.02 inch is known as the tolerance. 1. What percentage of the pins will be acceptable to the consumer? In order to improve percentage accepted, the production manager and the engineers discuss adjusting the population mean and standard deviation of the length of the pins. 2.
If the lathe can be adjusted to have the mean of the lengths to any desired value, what should it be adjusted to? Why?
3.
Suppose the mean cannot be adjusted, but the standard deviation can be reduced. What maximum value of the standard deviation would make 90% of the parts acceptable to the consumer? (Assume the mean to be 1.012.)
4.
Repeat question 3, with 95% and 99% of the pins acceptable.
5.
In practice, which one do you think is easier to adjust, the mean or the standard deviation? Why?
The production manager then considers the costs involved. The cost of resetting the machine to adjust the population mean involves the engineers' time and the cost of production time lost. The cost of reducing the population standard deviation involves, in addition to these costs, the cost of overhauling the machine and reengineering the process. 6.
Assume it costs $150x2 to decrease the standard deviation by (x /1000) inch. Find the cost of reducing the standard deviation to the values found in question 3 and 4.
7.
Now assume that the mean has been adjusted to the best value found in question 2 at a cost of $80. Calculate the reduction in standard deviation necessary to have 90%, 95%, and 99% of the parts acceptable. Calculate the respective costs, as in quesation6.
8.
Based on your answers to questions 6 and 7, what are your recommended mean and standard deviation?
* Case taken from Complete Business Statistics by Amir Aczel & J. Sounderpandian, TATA Mc Graw Hill.
Case 6: Acceptance sampling of pins
A company supplies pins in bulk to a customer. The company uses an automatic lathe to produce the pins. Factors such as vibration, temperature, and wear and tear affect the pins, so that the lengths of the pins made by the machine are normally distributed with mean of 1.008 inches and a standard deviation of 0.045 inch. The company supplies the pins in large batches to a customer. The customer will take a random sample of 50 pins from the batch and compute the mean. If the sample mean is within the interval 1.00 inch ±0.010inch, then the customer will buy the whole batch. 1. What is the probability that a batch will be acceptable to the consumer? Is the probability large enough to be an acceptable level of performance? To improve the probability of acceptance, the production manager and the engineers discuss adjusting the population mean and standard deviation of the lengths of the pins. 2. If the lathe can be adjusted to have the mean of the lengths at any desired value, what should it be adjusted to? Why? 3. Suppose the mean cannot be adjusted, but the standard deviation can be reduced. What maximum value of the standard deviation would make 90% of the parts acceptable to the consumer? (Assume the mean continues to be 1.008 inches.) 4. Repeat part 3 with 95% and 99%of the pins acceptable. 5. In practice, which one do you think is easier to adjust, the standard deviation? Why? 6. The production manager then considers the costs involved. The cost of resetting the machine to adjust the population mean involves the engineers' time and the cost of production time lost. The cost of reducing the population standard deviation involves in addition to these costs, the cost of overhauling the machine and reengineering the process. 7. Assume it costs $150x2 to decrease the standard deviation by (x/1,000) inch. Find the cost of reducing the standard deviation to the values found in parts 3 and 4. 8. Now assume that the mean has been adjusted to the best value found in part 2 at a cost of $80. Calculate the reduction in standard deviation necessary to have 90%, 95% and 99% of the parts acceptable. Calculate the respective costs, as in part6. 9. Based on your answers to parts 6 and 7, what are your recommended mean and standard deviation to which the machine should be adjusted? * Case taken from Complete Business Statistics by Amir Aczel & J. Sounderpandian, TATA Mc Graw Hill.
Case 7: Tiresome Tire I
When a tire is constructed of more than one ply, the interply shear strength is an important property to check. The specification for a particular type of tire calls for a strength of 2,800 pounds per square inch (psi). The tire manufacturer tests the tires using the null hypothesis H0 :
µ > 2,800 psi
µ is the mean strength of a large batch of tires. From past experience, it is known that the population standard deviation is 10 psi.
Testing the shear strength requires a costly destructive test and therefore the sample size needs to be kept at a minimum. A type I error will result in the rejection of a large number of good tires and is therefore costly. A type II error of passing a faulty batch of tires can result in fatal accidents on the roads, and therefore is extremely costly. (For purposes of this case, the probability of type II error, 3, is always calculated at µ =2,790 psi.) It is believed that β should be at most 1%. Currently, the company conducts the test with a sample size of 40 and an α of 5% 1.
To help the manufacturer get a clear picture of type I and type II error probabilities, draw a β versus α chart for sample sizes of 30, 40, 60, and 80. If β is to be at most 1 % with α = 5% which sample size among these four values is suitable?
2.
Calculate the exact sample size required for α = 5% and β = 1%. Construct a sensitivity analysis table for the required sample size for u ranging from 2,788 to 2,794 psi and β ranging from 1% to 5%.
3.
For the current practice of n = 40 and α = 5% plot the power curve of the test. Can this chart be used to convince the manufacturer about the high probability of passing batches that have strength of less than 2,800 psi?
4.
To present the manufacturer with a comparison of a sample size of 80 versus 40, plot the OC curve for those two sample sizes. Keep an α of 5%.
5.
The manufacturer is hesitant to increase the sample size beyond 40 due to the concomitant increase in testing costs and, more important, due to the increased time required for the tests. The production process needs to wait until the tests are completed, and that means loss of production time. A suggestion is made by the production manager to increase α to 10% as a means of reducing β. Give an account of the benefits and the drawbacks of that move. Provide supporting numerical results wherever possible.
* Case taken from Complete Business Statistics by Amir Aczel & J. Sounderpandian, TATA Mc Graw Hill.
Case 8: Tiresome Tire II
A tire manufacturing company invests a new, cheaper method for carrying out one of the steps in the manufacturing process. The company wants to test the new method before adopting it, because the method could alter the interply shear strength of the tires produced. To test the acceptability of the new method, the company formulates the null and alternative hypothesis as H0 : µ1 - µ 2 < = 0 H1: µ1 -µ2 >0
Where µ1 is the population mean of the interply shear strength of the tires produced by the old method and µ2 that of the tires produced by the new method. The evidence is gathered through a destructive test of 40 randomly selected tires from each method. Following are the dat a No. Sample 1 Sample 2 No. Sample1 Sample2 1 2792 2713 21 2693 2683 2 2755 2741 22 2740 2664 3 2745 2701 23 2731 2757 4 2731 2731 24 2707 2736 5 2799 2747 25 2754 2741 6 2793 2679 26 2690 2767 7 2705 2773 27 2797 2751 8 2729 2676 28 2761 2723 9 2747 2677 29 2760 2763 10 2725 2721 30 2777 2750 11 2515 2742 31 2774 2686 12 2782 2775 32 2713 2727 13 2718 2680 33 2741 2757 14 2719 2786 34 2789 2788 15 2751 2732 35 2723 2676 16 2755 2740 36 2713 2779 17 2685 2760 37 2781 2676 18 2700 2748 38 2706 2690 19 2712 2660 39 2776 2764 20 2778 2789 40 2738 2720
1. Test the null hypothesis at α = 0.5.
Later it was found that quite a few tires failed on the road. As a part of the investigation, the above hypothesis test is reviewed. Considering the high cost of type II error, the value of 5% for α is questioned. The response was that the cost of type I error is also high because the new method could save millions of dollars. What value for α would you say is appropriate? Will the null hypothesis be rejected at that α? 2.
A review of the tests conducted on the samples reveals that 40 otherwise identical pairs of tires were randomly selected and used. The two tires in each pair underwent the two different methods, and all other steps in the manufacturing process were identically carried out on the two tires. By virtue of this fact, it is argued that a paired difference test is more appropriate. Conduct a paired difference test at α = 0.5.
3.
There is a move to reduce the variance of the strength by improving the process. Will the reduction in the variance of the process increase or decrease the chances of type I and type I errors?
* Case taken from Complete Business Statistics by Amir Aczel & J. Sounderpandian, TATA Mc Graw Hill.
Case 9: Uniform Uniforms
A textile manufacturer has a large order for a cloth meant for making uniforms. The cloth is dyed using four different dyeing lines, which produce approximately equal amounts of cloth each day. Usually not more than one line is used for one product, because no matter how well the process is controlled, there will always be perceptible differences in the shade of the dye from one line to another. But because the volume of the order is large, four lines are being used. It is important to maintain the shade as uniform as possible by minimizing the variance of the brightness of the shade on all cloth produced. Lately, the customer has been complaining about too much variance in the brightness. It was decided to conduct an ANOVA test of the brightness of the cloth from the four lines. Random samples were taken from each line and were measured for brightness. The measurement is on a 0 to 100 scale. The sample data are Line1
Line2
Line3
Line4
1.
66.55
66.16
68.36
72.32
2.
71.91
65.94
66.81
66.69
3.
67.61
68.62
66.50
72.36
4.
66.13
63.86
65.22
70.88
5.
71.31
69.38
65.06
71.05
6.
68.99
64.55
65.42
71.05
7.
71.83
66.82
66.50
68.78
8.
68.99
65.56
64.82
74.40
9.
69.81
63.66
68.31
73.58
10.
72.49
64.71
68.17
73.58
11.
69.99
67.32
65.50
66.72
12.
73.44
71.39
70.39
70.37
13.
70.39
63.78
75.72
14.
68.42
70.42
74.65
15.
71.66
16.
65.14
1. Conduct the test at the 5% significant level, and report your conclusion. 2. Which pairs of lines have significant differences in their average brightness? 3. Stopping a line to adjust its average brightness is costly. If only one line can be stopped and adjusted, which one should it be? To what average brightness value should it be adjusted to minimize the variance in all the cloth produced? 4. If two lines can be stopped and adjusted, which ones should be? To what average brightness value should they be adjusted to minimize the total variance in all the cloth produced? * Case taken from Complete Business Statistics by Amir Aczel & J. Sounderpandian, TATA Mc Graw Hill.
Case 9: Risk and Return
According to the Capital Asset Pricing Model (CAPM) the risk associated with a capital asset is proportional to the slope b obtained by regressing the asset‘s past returns with the corresponding returns of the average portfolio called the market portfolio . (The return of the market portfolio represents the return earned by the average investor. It is weighted average of the returns from all the assets in the market.) The larger the slope b of an asset, the larger is the risk associated with that asset. A slope b of 1 represents average risk. The Returns from an electronic Firm’s stock and the corresponding returns from the market portfolio for the past 15 years are given below. Market Return (%) 16.02 12.17 11.48 17.62 20.01 14 13.22 17.79 15.46 8.09 11 18.52 14.05 8.79 11.6
Stocks Return (%) 21.05 17.25 13.1 18.23 21.52 13.26 15.84 22.18 16.26 5.64 10.55 17.86 12.75 9.13 13.87
1. Carry out the regression and find the slope b for the stock. What is the regression equation? 2. Does the value of the slope indicate that the stock has above-average risk? (For the purposes of this case assume that the risk is average if the slope is in the range 1 ± 0.1, below average if it is less than 0.9, and above average if it is more than 1.1.) 3. Give a 95% confidence interval for this slope b. Can we say the risk is above average with 95% confidence? 4. If the market portfolio return for the current year is 10%, what is the stock's return predicted by the regression equation? Give a 95% confidence interval for this prediction. 5. Construct a residual plot. Do the residuals appear random? 6. Construct a normal probability plot. Do the residuals appear to be normally distributed?
7.
The risk-free rate of return is the rate associated with an investment that has no risk at all, such as lending money to the government. Assume that for the current year the riskfree rate is 6%. According to the CAPM, when the return from the market portfolio is equal to the risk-free rate, the return from every asset must also be equal to the riskfree rate. In other words, if the market portfolio return is 6%, then the stock's return should also be 6%. It implies that the regression line must pass through the point (6, 6). Repeat the regression forcing this constraint. Comment on the risk based on the new regression equation.
* Case taken from Complete Business Statistics by Amir Aczel & J. Sounderpandian, TATA Mc Graw Hill.
