Practice Questions
1. The simple coefficient of correlations in a tri-variate distribution are given below:
r12 = 0.80, r13 = 0.70, r23 = 0.60
Find the partial coefficients of correlation r12.3, r13.2 and r23.1.
[Ans : 0.665, 0.458 and 0.093]
2. The coefficient of simple correlation in a tri-variate distribution with X1, X2 and X3 are given as below:
r12 = 0.59, r13 = 0.46, r23 = 0.77
Calculate the partial correlation coefficients between two variables by keeping (a) X3 as constant, (b) X2 as constant and (c) X1 as constant.
[Ans: 0.4162, 0.0111 and 0.6955]
3 The zero order
correlation coefficients are r12 = 0.98, r13 = 0.44 and
r23 = 0.54. Calculate the partial correlation coefficient between first and third
variables without considering the effect of second variable.
[Ans: -0.5326]
4. On the basis of observations made on 15 rose plants, the total correlations of yield of rose, seed vessels and height of plant are obtained differently. The coefficient of correlation between yield of rose and seed vessels is 0.80, the coefficient of correlation between yield of rose and height of plant is 0.65 and the coefficient of correlation between seed vessels and height of plant is 0.70. Compute the partial correlation coefficient from the given information by eliminating effect of height of plant.
[Ans: 0.6357]
5. If r12 = 0.95, r13 = 0.97 and r23 = 0.92 prove that r23.1 = -0.02.
6. Test the consistency of following data with the reference of following information.
r23 = 0.7, r13 = - 0.4, r12 = 0.6
[Ans: Since r12.3 = 1.344, the given data is inconsistent]
7. In a triraviate distribution it
is obtained that r12 = 0.863, r13 = 0.648
r23 = 0.709. Find r12.3 and R1.23.
[Ans : r12.3 = 0.75, R1.23 = 0.86]
8. The zero order correlation coefficients are given as r12 = 0.5, r13 = 0.6 and r23 = 0.7. Calculate the multiple correlation coefficients R1.23, R2.13 and R3.12.
[Ans : 0.610, 0.707 and 0.757]
9. The simple correlation coefficients in a tri-variate distribution with demand of goods (X1), price of goods (X2) and taste of customer (X3) are r12 = 0.59, r13 = 0.46, and r23 = 0.77. Calculate the multiple correlation coefficient R1.23.
[Ans : R1.23 = 0.589]
10. The zero order correlation coefficients are given as, r12 = 0.98, r13 = 0.44, r23= 0.54. Calculate multiple correlation coefficient treating first variable as dependent and second and third variable as independent.
[Ans : R1.23 = 0.986]
11. Consider the following results obtained from a sample of 10 size.
|
SX1 = 10 SX2 = 20 SX3 = 30 |
SX1X2 = 10 SX1X3 = 15 SX2X3 = 64 |
SX12 = 20 SX22 = 68 SX32 = 170 |
Find:
a. partial correlation coefficient keeping X3 as constant variable
b. multiple correlation coefficient treating X1 as dependent variable
c. coefficient of multiple determination and interpret the result
[Ans : (a) -0.66 (b) 0.775 (c) 0.60]
12. Find R20.12 and interpret the result on the basis of the following values obtained from 10 sample size.
|
SX0 = 272 SX0X1 = 12005 SX12 = 19461 |
SX1 = 441 SX0X2 = 4013 SX22 = 2173 |
SX2 = 147 SX1X2 = 6485 SX02 = 7428 |
[Ans : R20.12 = 0.727 ]
13. You are given the following information relating to a tri-variate distribution.
|
Production |
9 |
12 |
10 |
7 |
17 |
|
Productivity |
2 |
5 |
4 |
3 |
6 |
|
Land |
4 |
5 |
5 |
3 |
8 |
Determine:
a. partial correlation coefficient keeping land as constant variable
b. multiple correlation coefficient treating production as dependent variable
c. coefficient of multiple determination
[Ans: (a) 0.42 (b) 0.986 (c) 0.973]
14. Find the multiple regression equation of X1 on X2 and X3 from the data relating to three variables given below:
|
x1 |
8 |
5 |
4 |
9 |
7 |
|
x2 |
4 |
2 |
3 |
4 |
3 |
|
x3 |
3 |
2 |
1 |
2 |
0 |
[Ans: (i) : X1 = 0.429 + 1.873X2+ 0.111X3]
15. The table shows the corresponding values of the three variables X1, X2 and X3.
|
X0 |
9 |
5 |
7 |
8 |
6 |
10 |
|
X1 |
25 |
12 |
20 |
30 |
40 |
33 |
|
X2 |
66 |
51 |
55 |
58 |
60 |
70 |
Find the regression equation of X0 on X1 and X2 and estimate the value of X0 when X1 is 50 and X2 is 100.
[Ans: 0 = -7.862 - 0.049X1 + 0.278X2; and 0 = 17.488]
16. Following data reveals the sales of a company due to the number of sales persons and the years of experience.
|
Sales (000) Rs: |
20 |
30 |
25 |
20 |
40 |
60 |
15 |
|
No. of sales person involved: |
2 |
3 |
5 |
4 |
2 |
1 |
4 |
|
Average years of experience: |
5 |
7 |
11 |
10 |
8 |
7 |
8 |
a. Estimate the best line of fit.
b. Estimate the sales of the company using one sales person with experience of nine years.
[Ans: (a) 1 =21.22- 15.34 X2 +6.85 X3, (b) 1 = Rs. 67530]
17. A survey on income and expenditure of few families resulted in following data.
|
Expenditure on food (Rs. '000') |
5 |
7 |
8 |
9 |
11 |
|
Annual income (Rs. '000') |
25 |
40 |
30 |
50 |
25 |
|
Family size (number) |
3 |
2 |
4 |
5 |
1 |
a. Calculate the least square equation that best relates the three variables.
b. Estimate the expenditure on food of a family with annual income of 50,000 and having 4 family members.
[Ans: (i) 1 = 7.34 + 0.07X2 - 0.58X3 (ii) Rs. 8520]
18. The National planning commission is perferming preliminary study to determine the relationship between certain economic indicators and annual percentage change in gross national product (GNP). Two such indicators being examined are government deficit (in Lakhs Rs.) and industrial average. Data for 6 years are given below
|
Percentage change in GNP (Y) |
2.5 |
1.0 |
4.0 |
1.0 |
1.5 |
3.0 |
|
Government deficit (X1) |
50 |
200 |
60 |
100 |
90 |
40 |
|
Industrial average (X2) |
950 |
700 |
1100 |
800 |
850 |
900 |
a. Calculate the least square regression equation that best fit the given data.
b. What percentage change in GNP would be expected in a year with Government deficit of Rs. 120 Lakhs and industrial average of 1000.
[Ans (a) = -5.35 + 0.00067X1 + 0.0084X2 (b) 3.13]
19. The information given below has been gathered from a random sample of apartment renters in a city. We are trying to predict rent in rupees per month based on the size of the apartment (number of rooms) & the distance from the down town (in km.)
|
Rent in Rs. (Y) |
360 |
1,000 |
450 |
540 |
350 |
300 |
|
No. of rooms (X1 ) |
2 |
6 |
3 |
4 |
2 |
1 |
|
Distance from down town (X2) |
1 |
1 |
2 |
3 |
10 |
4 |
Calculate the least square equation that best relates those three variables. If someone looking for a 2 bed room apartment and two kilometers from down town, what rent should be expected to pay?
[Ans : = 95.55 + 137.50 X: - 2.30X2 , = 365.95]
20. Fit the best regression line to the following data
|
Productivity |
25 |
28 |
26 |
30 |
34 |
37 |
|
Labor |
50 |
48 |
50 |
55 |
60 |
65 |
|
Inputs |
5 |
10 |
9 |
8 |
12 |
11 |
a. What will be the productivity if input is 15 and labor is 70?
b. Compute coefficient of multiple determination.
[Ans: (a) 1 = – 4.93 + 0.54X2 + 0.59X3; 41.72; (b) R2 0.9647]
21. A household survey on monthly expenditure on food yield following data.
|
Monthly expenditure (Rs. '00') |
10 |
15 |
20 |
25 |
30 |
35 |
40 |
|
Monthly income (Rs. '000') |
2 |
4 |
5 |
7 |
6 |
6 |
5 |
|
Size of family (Number) |
4 |
5 |
7 |
10 |
8 |
11 |
4 |
a. Estimate the line of best fit.
b. If the monthly income is Rs. 10,000 with 6 family members, then how much expenditure should the family expect to pay?
c. Obtain the multiple coefficients of determination and correlation coefficient.
[Ans: (a) 1 = 3.345 + 6.796 X2 - 1.761 X3 (b) Rs. 6,074 (c) 0.5158, 0.7182]
22. The internal revenue service is trying to estimate the monthly amount of unpaid taxes discovered in last 6 months by its auditing division with the support of field audit (in terms of labour hour) and the computer hours.
|
Unpaid tax ('000' Rs.) |
10 |
17 |
18 |
26 |
35 |
8 |
|
Labour hour (Field-audit) |
8 |
21 |
14 |
17 |
36 |
9 |
|
Computer hour |
4 |
9 |
11 |
20 |
13 |
28 |
a. Obtain the line of best fit
b. If field-audit hour is 30 and computer hour is also 30 what will be expected unpaid tax discovered?
c. Find the multiple correlation coefficients.
[Ans : (a) 1 = 2.592 + 0.89X2 + 0.06X3 (b) 31092 (c) R1.23 = 0.90]
23. Find the multiple regression equation of X 1 on X2 and X3 from the data relating to three variables given below:
|
Y |
5 |
7 |
8 |
4 |
9 |
|
X1 |
2 |
3 |
4 |
3 |
4 |
|
X2 |
2 |
0 |
3 |
1 |
2 |
a. Estimate the probable value of Y when X1 = 2.5 and X2 =2.
b. Find the standard error of estimate of Y.
c. Calculate the coefficient of multiple determinations.
[Ans : (a) 5.334 (b) 1.83 (c) 0.609]
24. The following are the sales of the company due to the number of sales persons and the years of experience.
|
Sales (Rs. '000' ) |
11 |
9 |
8 |
7 |
5 |
|
No. of Sales Persons |
25 |
50 |
30 |
40 |
25 |
|
Years of Experience |
1 |
5 |
4 |
2 |
3 |
a. Calculate the least square equation that best describe the data.
b. Calculate the standard error of estimate of sales.
c. Find the coefficient of multiple determinations.
d. Find the coefficient of multiple correlations.
[Ans: (a) 1 = 7.33 + 0.07X2 - 0.57X3 (b) 2.966 (c) 0.120 (d) 0.346]
25. The data collected from the random sample of five Genera; Motors salespeople. The annual sales (in 10,000 dollars) depend on the years of education of school completed and the motivation as measured by Higgins motivation scale.
|
Annual sales |
35 |
39.9 |
42.9 |
43.5 |
43.3 |
|
Years of education |
12 |
14 |
15 |
W16 |
18 |
|
Motivation |
32 |
35 |
45 |
50 |
65 |
a. Let use assume you interviewed a potential salesperson and found that he had 13 years of education and he Second 49 on the Higgins motivation scale. What would be your prediction of money in sales this person would bring in an annual basis?
b. Computer the standard error of estimate.
c. Calculate the coefficient of multiple determination and interpret the result.
[Ans: (a) 1 = 5.87 + 3.36X2 – 0.34X3; 32.89 (b) 3.49 (c) 0.53]
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