# 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.

First- Admissions Test Scores

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.

Column F Total score on the admissions test administered by

Educational Testing Service. This test is scored so

that the average score among all those to whom it

was administered in a given year will be about 500,

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

an average of about 28, a standard deviation of about

8, and a range from 0 to 52.

Column H Verbal score on the admissions test, scored the same

way as the quantitative score.

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##### 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.

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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

Adjusted R Square 0.959446645

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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