If administrators could determine the factors that influence increases in the pe

Question

If administrators could determine the factors that influence increases in the percentage of alumni who make a donation (alumni giving rate), they might be able to implement policies that could lead to increased revenue. 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.

1. Run the simple linear regression analysis to develop an estimated equation that could be used to predict the alumni-giving rate by choosing graduation rate, % of classes under 20 students, or student/faculty ratio as the independent variable. Indicate the one with the highest R2.

2. Use the multiple linear regression analysis to develop an estimated regression equation that could be used to predict the alumni-giving rate given the graduation rate, % of classes under 20 students, and the student-faculty ratio. Is it a better fit compared with your answer in (1)? Any multicollinearity exists? Please show the correlation coefficients to comment.

3. What conclusion and recommendation can you derive from F and t tests in (2)?

4. With the multiple linear regression equation in (2), what will be the alumni-giving rate with the graduation rate as 85%, 60% of classes with fewer than 20 students, and student-faculty ratio as 12?

5. Any possible modification you suggest for a better-fit multiple linear regression model? Adding or dropping independent variable(s)? Change to any nonlinear model? Explain your suggestion.

School 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 University 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 John 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 University 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-Berleley 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 Vanderbuilt 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

1)

Graduation Rate: -

SUMMARY OUTPUT

Regression Statistics

Multiple R

0.76

R Square

0.57

Adjusted R Square

0.56

Standard Error

8.89

Observations

48

ANOVA

df

SS

MS

F

Significance F

Regression

1

4852.46

4852.46

61.34

0.00

Residual

46

3639.02

79.11

Total

47

8491.48

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

-68.76

12.58

-5.46

0.00

-94.09

-43.43

Graduation Rate (%)

1.18

0.15

7.83

0.00

0.88

1.48

% of Classes Under 20: -

SUMMARY OUTPUT

Regression Statistics

Multiple R

0.65

R Square

0.42

Adjusted R Square

0.40

Standard Error

10.38

Observations

48

ANOVA

df

SS

MS

F

Significance F

Regression

1

3539.80

3539.80

32.88

0.00

Residual

46

4951.68

107.65

Total

47

8491.48

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

-7.39

6.57

-1.12

0.27

-20.60

5.83

% of Classes Under 20

0.66

0.11

5.73

0.00

0.43

0.89

Student/Faculty Ratio: -

SUMMARY OUTPUT

Regression Statistics

Multiple R

0.74

R Square

0.55

Adjusted R Square

0.54

Standard Error

9.10

Observations

48

ANOVA

df

SS

MS

F

Significance F

Regression

1

4680.11

4680.11

56.49

0.00

Residual

46

3811.37

82.86

Total

47

8491.48

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

53.01

3.42

15.49

0.00

46.13

59.90

Student/Faculty Ratio

-2.06

0.27

-7.52

0.00

-2.61

-1.51

Model with the highest r^2 is Graduation Rate (%)

2)

Multiple Regression: -

SUMMARY OUTPUT

Regression Statistics

Multiple R

0.84

R Square

0.70

Adjusted R Square

0.68

Standard Error

7.61

Observations

48

ANOVA

df

SS

MS

F

Significance F

Regression

3

5943.53

1981.18

34.21

0.00

Residual

44

2547.95

57.91

Total

47

8491.48

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

-20.72

17.52

-1.18

0.24

-56.03

14.59

Graduation Rate (%)

0.75

0.17

4.51

0.00

0.41

1.08

% of Classes Under 20

0.03

0.14

0.21

0.84

-0.25

0.31

Student/Faculty Ratio

-1.19

0.39

-3.08

0.00

-1.97

-0.41

Correlation Matrix

Graduation Rate (%)

% of Classes Under 20

Student/Faculty Ratio

Graduation Rate (%)

1

0.582788432

-0.604937935

% of Classes Under 20

0.582788432

1

-0.785559252

Student/Faculty Ratio

-0.604937935

-0.785559252

1

Looking at the table, there are no strong indications of multicollinearity. Also, it is a better option than 1 because it shows higher r^2

3)

Looking at the FSTAT, we reject the null hypothesis and can conclude that at least one of the variables is having an impact on the dependent variable.

Also, both the independent variables except % of Classes Under 20 are significant.

4)

y=-20.72+0.75* Graduation Rate (%)+0.03*% of Classes Under 20-1.19* Student/Faculty Ratio

y=-20.72+0.75*85+0.03*60-1.19*12

y=-20.72+79.83

y=59.11

5)

Remove % of Classes Under 20 and keep the other two as independent variables for better fit.

SUMMARY OUTPUT

Regression Statistics

Multiple R

0.76

R Square

0.57

Adjusted R Square

0.56

Standard Error

8.89

Observations

48

ANOVA

df

SS

MS

F

Significance F

Regression

1

4852.46

4852.46

61.34

0.00

Residual

46

3639.02

79.11

Total

47

8491.48

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

-68.76

12.58

-5.46

0.00

-94.09

-43.43

Graduation Rate (%)

1.18

0.15

7.83

0.00

0.88

1.48

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