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# Multiple Regression Analysis

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Q.4

Artificial Data on Two Independent and Four Dependent Variables *

x1 y1 y2 y3 x2 y4
10 8.04 9.14 7.46 8 6.58
8 6.95 8.14 6.77 8 5.76
13 7.58 8.74 12.74 8 7.71
9 8.81 8.77 7.11 8 8.84
11 8.33 9.26 7.81 8 8.47
14 9.96 8.1 8.84 8 7.04
6 7.24 6.13 6.08 8 5.25
4 4.26 3.1 5.39 19 12.5
12 10.84 9.13 8.15 8 5.56
7 4.82 7.26 6.42 8 7.91
5 5.68 4.74 5.73 8 6.89

* F. J. Anscombe "Graphs and Statistical Analysis,"
American Statistician, vol. 27 (1973) pp. 17-21.

Perform the following regressions:

Independent Variable Dependent Variable
a) x1 y1
b) x1 y2
c) x1 y3
d) x2 y4

In each case, plot a scatter diagram of the independent against
dependent variable. What, if anything, could you do to obtain
better regression results in each of the four regressions?

Q.5.

(From Bodily, Carraway, Frey, and Pfeifer, Quantitative Business Analysis Casebook, Richard D. Irwin, 1996.) In a market-segmentation study, data on net income (I) in thousands, family size (F), and expenditures on consumer durable goods (C) were collected from 20 randomly-selected households:

The company sponsoring the study was interested in isolating the effects of income and family size on the dollar amounts spent for consumer durable goods. Can you help them?

6.
Develop a regression model that will predict students' first-year grade-point averages.

Year College
Average Section Age Average Total Quantitative Verbal
1st Yr. Avg. Section Age Coll. Avg. Total Quantitative Verbal
73.76 3 24 2.7 568 34 35
76.59 7 27 2.9 602 37 37
85.41 8 23 3.3 674 39 45
71.65 3 25 2.3 579 38 33
75.18 5 24 2.3 713 47 43
79.24 2 22 2.4 548 36 30
80.12 1 22 3.1 616 39 36
84.88 5 24 2.9 674 36 46
75.88 6 24 3.5 629 37 40
85.41 4 26 2.3 682 43 42
79.59 2 28 3.0 746 47 48
74.29 5 25 2.5 568 33 36
77.65 6 22 2.9 572 33 36
77.65 4 29 3.6 635 37 41
73.24 2 30 1.7 581 30 40
79.24 2 24 2.2 602 32 42
76.76 3 27 2.9 546 26 39
79.94 7 28 3.2 747 48 46
77.12 3 25 3.8 516 32 29
83.47 4 23 3.8 702 48 40
72.53 8 21 3.0 571 38 31
75.35 6 23 3.6 582 30 41
78.00 7 28 2.5 550 33 33
73.41 2 22 3.0 499 27 31
75.18 1 33 2.6 491 37 22
70.24 3 34 2.5 463 25 29
73.41 1 25 3.4 402 29 16
76.76 6 23 2.9 662 42 40
62.56 1 32 1.8 576 32 37
74.47 3 34 2.7 645 42 37
77.82 3 24 3.3 625 45 32
72.35 4 25 3.0 424 17 29
72.00 1 26 2.7 568 30 38
80.82 7 23 3.8 546 35 30
75.18 1 22 3.1 503 27 31
79.41 2 23 2.7 598 33 39
76.24 8 22 2.4 651 41 39
82.59 4 30 1.9 652 44 37
79.41 8 23 3.3 605 38 36
76.76 1 28 3.2 593 32 40
71.12 8 27 2.4 628 28 48
74.47 4 24 2.5 530 25 38
73.41 3 33 2.9 704 40 49
74.47 3 28 2.3 580 34 36
73.76 8 20 2.7 475 32 24
75.18 3 26 2.1 510 23 37
78.18 8 31 2.8 564 29 38
78.18 1 27 3.0 626 31 45
78.53 1 31 2.7 583 26 44
75.35 4 22 2.8 602 42 32
77.47 7 26 2.3 613 30 44
80.12 3 22 3.5 625 37 40
73.94 4 22 2.7 631 34 44
72.35 2 23 3.2 564 40 28
78.18 4 22 3.5 644 39 40
70.76 3 22 3.1 430 23 26
77.47 1 25 2.8 615 35 40
73.59 4 23 3.2 552 39 27
75.71 4 26 2.7 587 31 39
79.59 8 26 2.9 594 35 37
77.82 1 24 2.7 568 37 31
71.12 1 23 2.8 543 34 31
80.82 4 24 3.3 624 39 37
76.41 3 22 2.5 556 34 33
79.59 4 22 3.7 605 34 39
71.29 3 23 2.8 646 32 47
75.35 4 24 2.6 550 32 35
68.65 3 26 2.3 498 23 35
74.82 1 22 3.2 676 46 38
75.00 4 28 2.7 659 41 40
77.82 8 23 3.2 502 20 39
75.18 6 29 2.9 588 31 40
81.53 5 25 2.0 613 39 36
79.24 2 23 3.1 650 40 40
76.06 8 22 2.5 591 31 40
77.12 4 27 2.6 550 30 35
72.35 1 24 2.7 552 28 38
79.94 3 22 3.8 651 33 47
79.94 8 22 2.4 609 34 40
73.59 1 27 2.7 552 33 32
74.65 4 22 3.4 586 41 30
74.29 8 22 2.9 553 23 41
76.41 7 33 2.3 503 28 31
78.88 1 29 2.2 557 31 36
77.82 5 21 3.0 625 42 36
75.00 6 23 2.6 569 29 40
73.24 7 22 2.6 527 22 40
78.53 7 26 2.3 527 31 31
74.12 8 24 2.4 496 33 25
74.82 7 22 2.7 620 34 42
74.47 7 23 3.5 552 37 29
73.76 2 22 2.3 598 33 40
75.53 3 24 3.9 635 45 34
78.35 2 22 3.6 726 43 48
78.35 5 22 3.0 702 44 44
71.12 6 25 2.4 564 36 33
73.24 8 29 2.1 613 38 37
77.82 8 28 2.6 598 34 39
72.88 5 28 2.7 501 29 30
75.71 5 27 2.8 590 37 35
73.59 3 25 3.3 738 40 52
79.76 4 32 3.0 689 37 48
78.00 8 25 2.3 576 44 26
70.94 6 26 2.3 596 43 30
71.12 1 28 1.9 534 28 36
76.76 2 22 2.9 634 41 38
76.76 1 30 2.9 520 30 32
76.24 7 22 3.6 568 30 38
73.41 4 22 2.6 602 38 36
84.00 6 27 2.9 662 46 37
79.06 3 23 3.7 556 46 22
75.71 8 21 2.6 483 25 32
74.29 7 29 3.3 434 16 31
76.76 2 24 2.5 609 36 39
74.29 6 27 2.6 618 36 40
78.71 2 33 2.3 615 40 35
76.24 3 28 2.1 578 44 26
77.65 7 25 2.3 668 40 43
74.65 5 25 2.7 587 34 37
77.12 6 22 2.8 515 34 28
79.06 6 22 2.2 613 38 37
75.88 3 33 3.2 670 34 47
79.59 8 22 2.6 775 50 49
79.94 3 30 2.5 624 42 34
72.53 3 22 3.4 651 41 40
70.24 6 23 3.1 632 42 37
73.59 2 28 2.1 479 26 30
71.29 8 25 2.6 577 37 33
77.82 6 25 2.5 621 33 42
72.71 5 31 2.6 516 29 32
76.24 3 28 2.4 503 27 32
76.76 5 27 2.6 573 38 32
69.88 2 25 3.2 613 34 41
80.47 4 27 2.2 613 43 32
71.29 8 26 2.5 616 42 34
78.71 1 28 2.5 641 37 41
84.35 2 24 2.9 725 43 48
77.82 6 23 3.3 631 37 41
78.00 4 32 2.5 579 35 35
76.59 8 34 2.1 651 37 43
70.24 3 25 2.8 533 34 29
75.35 7 23 2.9 506 33 26
68.29 4 28 2.7 468 18 35
75.88 2 26 2.8 645 45 34
73.41 4 24 2.5 467 25 29
72.88 3 29 3.4 542 26 38
77.82 3 28 2.8 559 31 36
79.41 7 28 2.5 502 30 28
76.94 4 24 2.9 673 41 43
72.88 8 22 2.7 580 25 43

Column B A weighted average of final grades in ten first-year
courses. Each course was graded on a 15-point scale, with
possible values from 54 through 96 in steps of 3.

Column C There were eight sections, coded 1 through 8.

Column D Age in years as of the day of matriculation.

Column E Four-year average of college grades, converted, if necessary,
to a four-point scale.

Educational Testing Service. This test is scored so
that the average score among all those to whom it
with a standard deviation of about 100. Scores can
range from 200 to 800.

Column G Quantitative score on the admissions test, scored to have
8, and a range from 0 to 52.

Column H Verbal score on the admissions test, scored the same
way as the quantitative score.

##### Solution Summary

Multiple Regression analysis on weighted average of final grades, Age in years as of the day of matriculation, Four-year average of college grades, Total score on the admissions test administered by Educational Testing Service, Quantitative score on the admissions test, Verbal score on the admissions test.

##### Solution Preview

Calculations are done in MS Excel

Q.4

Artificial Data on Two Independent and Four Dependent Variables *

x1 y1 y2 y3 x2 y4
10 8.04 9.14 7.46 8 6.58
8 6.95 8.14 6.77 8 5.76
13 7.58 8.74 12.74 8 7.71
9 8.81 8.77 7.11 8 8.84
11 8.33 9.26 7.81 8 8.47
14 9.96 8.1 8.84 8 7.04
6 7.24 6.13 6.08 8 5.25
4 4.26 3.1 5.39 19 12.5
12 10.84 9.13 8.15 8 5.56
7 4.82 7.26 6.42 8 7.91
5 5.68 4.74 5.73 8 6.89

* F. J. Anscombe "Graphs and Statistical Analysis,"
American Statistician, vol. 27 (1973) pp. 17-21.

Perform the following regressions:

Independent Variable Dependent Variable
a) x1 y1
b) x1 y2
c) x1 y3
d) x2 y4

In each case, plot a scatter diagram of the independent against
dependent variable. What, if anything, could you do to obtain
better regression results in each of the four regressions?

Ans.
The scatter diagrams are given in the Excel sheet

The regression equation are

a) Y1 = 0.5001X1 + 3.0001
b) Y2 = 0.5X1 + 3.0009 Linear
Y2= -0.1267X12 + 2.7808X1 - 5.9957 Polynomial of degree 2
c) Y3 = 0.4997X1 + 3.0025

For the second model there is a nonlinear relation can be observed between Y2 and X1 and polynomial of order 2 gives a better fit.

Since X2 takes only two distinct value ( 8, 19 ) regression analysis is not suitable for this data.

Q.5.

(From Bodily, Carraway, Frey, and Pfeifer, Quantitative Business Analysis Casebook, Richard D. Irwin, 1996.) In a market-segmentation study, data on net income (I) in thousands, family size (F), and expenditures on consumer durable goods (C) were collected from 20 randomly-selected households:

The company sponsoring the study was interested in isolating the effects of income and family size on the dollar amounts spent for consumer durable goods. Can you help them?

Ans
A regression model can be suggested as

C = B0+B1 I + B2 F

Here B0 ,B1 ,B2 represents the regression coefficients

Regression Statistics
Multiple R 0.984102892
R Square 0.968458502
Standard Error 262.0523394
Observations 10

ANOVA
df SS MS F Significance F
Regression 2 14759540 7379770 107.4649 5.57E-06
Residual 7 480700 68671.43
Total 9 15240240

Coefficients Standard Error t ...

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