Alumni donations are an important source of revenue for colleges and universitie

Question

Alumni donations are an important source of revenue for colleges and universities. If administrators could determine the factors that could lead to increases in the percentage of alumni who make a donation, they might be able to implement policies that could lead to increased revenues. Research shows that students who are more satisfied with their contact with teachers are more likely to graduate. As a result, one might suspect that smaller class sizes and lower student/faculty ratios might lead to a higher percentage of satisfied graduates, which in turn might lead to increases in the percentage of alumni who make a donation. The following table shows data for 48 national universities. The Graduation Rate column is the percentage of students who initially enrolled at the university and graduated. The % of Classes Under 20 column shows the percentages of classes offered with fewer than 20 students. The Student/Faculty Ratio column is the number of students enrolled divided by the total number of faculty. Finally, the Alumni Giving Rate column is the percentage of alumni who made a donation to the university.

2. Develop an estimated simple linear regression model that can be used to predict the alumni giving rate, given the graduation rate. Discuss your findings.

3. Develop an estimated multiple linear regression model that could be used to predict the alumni giving rate using Graduation Rate, % of classes under 20, and student/faculty Ratio as independdent variables. Discuss your findings.

University State Graduation Rate % of Classes Under 20 Student-Faculty Ratio Alumni Giving Rate Boston College MA 85 39 13 25 Brandeis University MA 79 68 8 33 Brown University RI 93 60 8 40 California Institute of Technology CA 85 65 3 46 Carnegie Mellon University PA 75 67 10 28 Case Western Reserve Univ. OH 72 52 8 31 College of William and Mary VA 89 45 12 27 Columbia University NY 90 69 7 31 Cornell University NY 91 72 13 35 Dartmouth College NH 94 61 10 53 Duke University NC 92 68 8 45 Emory University GA 84 65 7 37 Georgetown University DC 91 54 10 29 Harvard University MA 97 73 8 46 Johns Hopkins University MD 89 64 9 27 Lehigh University PA 81 55 11 40 Massachusetts Inst. of Technology MA 92 65 6 44 New York University NY 72 63 13 13 Northwestern University IL 90 66 8 30 Pennsylvania State Univ. PA 80 32 19 21 Princeton University NJ 95 68 5 67 Rice University TX 92 62 8 40 Stanford University CA 92 69 7 34 Tufts University MA 87 67 9 29 Tulane University LA 72 56 12 17 U. of California–Berkeley CA 83 58 17 18 U. of California–Davis CA 74 32 19 7 U. of California–Irvine CA 74 42 20 9 U. of California–Los Angeles CA 78 41 18 13 U. of California–San Diego CA 80 48 19 8 U. of California–Santa Barbara CA 70 45 20 12 U. of Chicago IL 84 65 4 36 U. of Florida FL 67 31 23 19 U. of Illinois–Urbana Champaign IL 77 29 15 23 U. of Michigan–Ann Arbor MI 83 51 15 13 U. of North Carolina–Chapel Hill NC 82 40 16 26 U. of Notre Dame IN 94 53 13 49 U. of Pennsylvania PA 90 65 7 41 U. of Rochester NY 76 63 10 23 U. of Southern California CA 70 53 13 22 U. of Texas–Austin TX 66 39 21 13 U. of Virginia VA 92 44 13 28 U. of Washington WA 70 37 12 12 U. of Wisconsin–Madison WI 73 37 13 13 Vanderbilt University TN 82 68 9 31 Wake Forest University NC 82 59 11 38 Washington University–St. Louis MO 86 73 7 33 Yale University CT 94 77 7 50

Explanation / Answer

The data table as per required values :

X1

X2

X3

Y

X12

X22

X32

X1*X2

X2*X3

X1*X3

X1*Y

X2*Y

X3*Y

85

39

13

25

7225

1521

169

3315

507

1105

2125

975

325

79

68

8

33

6241

4624

64

5372

544

632

2607

2244

264

93

60

8

40

8649

3600

64

5580

480

744

3720

2400

320

85

65

3

46

7225

4225

9

5525

195

255

3910

2990

138

75

67

10

28

5625

4489

100

5025

670

750

2100

1876

280

72

52

8

31

5184

2704

64

3744

416

576

2232

1612

248

89

45

12

27

7921

2025

144

4005

540

1068

2403

1215

324

90

69

7

31

8100

4761

49

6210

483

630

2790

2139

217

91

72

13

35

8281

5184

169

6552

936

1183

3185

2520

455

94

61

10

53

8836

3721

100

5734

610

940

4982

3233

530

92

68

8

45

8464

4624

64

6256

544

736

4140

3060

360

84

65

7

37

7056

4225

49

5460

455

588

3108

2405

259

91

54

10

29

8281

2916

100

4914

540

910

2639

1566

290

97

73

8

46

9409

5329

64

7081

584

776

4462

3358

368

89

64

9

27

7921

4096

81

5696

576

801

2403

1728

243

81

55

11

40

6561

3025

121

4455

605

891

3240

2200

440

92

65

6

44

8464

4225

36

5980

390

552

4048

2860

264

72

63

13

13

5184

3969

169

4536

819

936

936

819

169

90

66

8

30

8100

4356

64

5940

528

720

2700

1980

240

80

32

19

21

6400

1024

361

2560

608

1520

1680

672

399

95

68

5

67

9025

4624

25

6460

340

475

6365

4556

335

92

62

8

40

8464

3844

64

5704

496

736

3680

2480

320

92

69

7

34

8464

4761

49

6348

483

644

3128

2346

238

87

67

9

29

7569

4489

81

5829

603

783

2523

1943

261

72

56

12

17

5184

3136

144

4032

672

864

1224

952

204

83

58

17

18

6889

3364

289

4814

986

1411

1494

1044

306

74

32

19

7

5476

1024

361

2368

608

1406

518

224

133

74

42

20

9

5476

1764

400

3108

840

1480

666

378

180

78

41

18

13

6084

1681

324

3198

738

1404

1014

533

234

80

48

19

8

6400

2304

361

3840

912

1520

640

384

152

70

45

20

12

4900

2025

400

3150

900

1400

840

540

240

84

65

4

36

7056

4225

16

5460

260

336

3024

2340

144

67

31

23

19

4489

961

529

2077

713

1541

1273

589

437

77

29

15

23

5929

841

225

2233

435

1155

1771

667

345

83

51

15

13

6889

2601

225

4233

765

1245

1079

663

195

82

40

16

26

6724

1600

256

3280

640

1312

2132

1040

416

94

53

13

49

8836

2809

169

4982

689

1222

4606

2597

637

90

65

7

41

8100

4225

49

5850

455

630

3690

2665

287

76

63

10

23

5776

3969

100

4788

630

760

1748

1449

230

70

53

13

22

4900

2809

169

3710

689

910

1540

1166

286

66

39

21

13

4356

1521

441

2574

819

1386

858

507

273

92

44

13

28

8464

1936

169

4048

572

1196

2576

1232

364

70

37

12

12

4900

1369

144

2590

444

840

840

444

144

73

37

13

13

5329

1369

169

2701

481

949

949

481

169

82

68

9

31

6724

4624

81

5576

612

738

2542

2108

279

82

59

11

38

6724

3481

121

4838

649

902

3116

2242

418

86

73

7

33

7396

5329

49

6278

511

602

2838

2409

231

94

77

7

50

8836

5929

49

7238

539

658

4700

3850

350

3986

2675

554

1405

334486

157257

7500

225247

28511

44818

120784

83681

13941

Thus,

X1 = 3986,         X2 = 2675,      X3 = 554,      X12 = 334486,      X22 = 157257,      X32 = 7500,

X1 X2 = 225247,    X2 X3 = 28511,    X1 X3 = 44818,  

Y = 1405,          X1 Y = 120784,    X2 Y = 83681,    X3 Y= 13941,

(2)

An estimated simple linear regression model that can be used to predict the alumni giving rate Y, given the graduation rate X1, i.e., Y on X1.

We know that, simple linear regression equation is

                                        Y = a + b X1   ____________________(1)

Where,

                     Y= N*a + b X1   ____________________(2)

                     X1 Y = a* X1+ b X12           ____________________(3)

Then, equation (2) and (3) is

                     1405= 48*a + 3986 b     ____________________(4)

                     120784 = 3986*a+ 334486 b   ____________________(5)

We get, by equation (4) and (5) to solution of non homogeneous linear equation system

a = -68.761

b = 1.181

Hence, by equation (1), multiple regression line equation is

Y = -68.761 + 1.181 X1

(3)

An estimated multiple linear regression model that could be used to predict the alumni giving rate Y, given the Graduation Rate X1, % of classes under 20 X2, and student/faculty Ratio X3 as independdent variables i.e., Y on X1, X2  and X3.

We know that, multiple regression line equation is

                                        Y = a + b1 X1+ b2 X2 + b3 X3 ____________________(6)

Where,

                     Y= N*a + b1 X1+ b2 X2 + b3 X3 ____________________(7)

                     X1 Y = a* X1+ b1 X12 + b2 X1X2 + b3 X1X3 ____________________(8)

                     X2 Y = a* X2+ b1 X1 X2 + b2 X22 + b3 X2X3 ____________________(9)

                     X3 Y = a* X3+ b1 X1 X3 + b2 X2 X3 + b3 X32 ____________________(10)

Simlarly as part (2)

We get,

a = -20.720

b1 = 0.748

b2 = 0.029

b3 = -1.192

Hence, by equation (6), multiple regression line equation is

                Y = -20.720 + 0.748 X1+ 0.029 X2 -1.192 X3

X1

X2

X3

Y

X12

X22

X32

X1*X2

X2*X3

X1*X3

X1*Y

X2*Y

X3*Y

85

39

13

25

7225

1521

169

3315

507

1105

2125

975

325

79

68

8

33

6241

4624

64

5372

544

632

2607

2244

264

93

60

8

40

8649

3600

64

5580

480

744

3720

2400

320

85

65

3

46

7225

4225

9

5525

195

255

3910

2990

138

75

67

10

28

5625

4489

100

5025

670

750

2100

1876

280

72

52

8

31

5184

2704

64

3744

416

576

2232

1612

248

89

45

12

27

7921

2025

144

4005

540

1068

2403

1215

324

90

69

7

31

8100

4761

49

6210

483

630

2790

2139

217

91

72

13

35

8281

5184

169

6552

936

1183

3185

2520

455

94

61

10

53

8836

3721

100

5734

610

940

4982

3233

530

92

68

8

45

8464

4624

64

6256

544

736

4140

3060

360

84

65

7

37

7056

4225

49

5460

455

588

3108

2405

259

91

54

10

29

8281

2916

100

4914

540

910

2639

1566

290

97

73

8

46

9409

5329

64

7081

584

776

4462

3358

368

89

64

9

27

7921

4096

81

5696

576

801

2403

1728

243

81

55

11

40

6561

3025

121

4455

605

891

3240

2200

440

92

65

6

44

8464

4225

36

5980

390

552

4048

2860

264

72

63

13

13

5184

3969

169

4536

819

936

936

819

169

90

66

8

30

8100

4356

64

5940

528

720

2700

1980

240

80

32

19

21

6400

1024

361

2560

608

1520

1680

672

399

95

68

5

67

9025

4624

25

6460

340

475

6365

4556

335

92

62

8

40

8464

3844

64

5704

496

736

3680

2480

320

92

69

7

34

8464

4761

49

6348

483

644

3128

2346

238

87

67

9

29

7569

4489

81

5829

603

783

2523

1943

261

72

56

12

17

5184

3136

144

4032

672

864

1224

952

204

83

58

17

18

6889

3364

289

4814

986

1411

1494

1044

306

74

32

19

7

5476

1024

361

2368

608

1406

518

224

133

74

42

20

9

5476

1764

400

3108

840

1480

666

378

180

78

41

18

13

6084

1681

324

3198

738

1404

1014

533

234

80

48

19

8

6400

2304

361

3840

912

1520

640

384

152

70

45

20

12

4900

2025

400

3150

900

1400

840

540

240

84

65

4

36

7056

4225

16

5460

260

336

3024

2340

144

67

31

23

19

4489

961

529

2077

713

1541

1273

589

437

77

29

15

23

5929

841

225

2233

435

1155

1771

667

345

83

51

15

13

6889

2601

225

4233

765

1245

1079

663

195

82

40

16

26

6724

1600

256

3280

640

1312

2132

1040

416

94

53

13

49

8836

2809

169

4982

689

1222

4606

2597

637

90

65

7

41

8100

4225

49

5850

455

630

3690

2665

287

76

63

10

23

5776

3969

100

4788

630

760

1748

1449

230

70

53

13

22

4900

2809

169

3710

689

910

1540

1166

286

66

39

21

13

4356

1521

441

2574

819

1386

858

507

273

92

44

13

28

8464

1936

169

4048

572

1196

2576

1232

364

70

37

12

12

4900

1369

144

2590

444

840

840

444

144

73

37

13

13

5329

1369

169

2701

481

949

949

481

169

82

68

9

31

6724

4624

81

5576

612

738

2542

2108

279

82

59

11

38

6724

3481

121

4838

649

902

3116

2242

418

86

73

7

33

7396

5329

49

6278

511

602

2838

2409

231

94

77

7

50

8836

5929

49

7238

539

658

4700

3850

350

3986

2675

554

1405

334486

157257

7500

225247

28511

44818

120784

83681

13941

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