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
partial table shows a portion of the data for 48 US 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.
Managerial Report
1. Use methods of descriptive statistics to summarize the data.
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 the Graduation Rate, % of Classes Under 20, and
Student/Faculty Ratio as independent variables. Discuss your findings.
4. Based on the results in parts 2 and 3, do you believe another regression model may be
more appropriate? Estimate this model, and discuss your results.
5. What conclusions and recommendations can you derive from your analysis? What
universities are achieving a substantially higher alumni giving rate than would be
expected, given their Graduation Rate, % of Classes Under 20, and Student/Faculty
Ratio? What universities are achieving a substantially lower alumni giving rate than
would be expected, given their Graduation Rate, % of Classes Under 20, and
Student/Faculty Ratio? What other independent variables could be included in the
model?

University State Graduation Rate % of Classes Under 20 Student-Faculty Ratio Alumni Giving Rate Boston College MA 87 37 13 25 Brandeis University MA 77 69 8 35 Brown University RI 91 58 8 39 California Institute of Technology CA 87 64 3 46 Carnegie Mellon University PA 77 68 10 26 Case Western Reserve Univ. OH 74 50 8 29 College of William and Mary VA 90 47 12 25 Columbia University NY 91 70 7 29 Cornell University NY 93 71 13 35 Dartmouth College NH 93 62 10 54 Duke University NC 93 66 8 44 Emory University GA 82 66 7 36 Georgetown University DC 92 53 10 27 Harvard University MA 95 71 8 45 Johns Hopkins University MD 89 66 9 29 Lehigh University PA 79 56 11 39 Massachusetts Inst. of Technology MA 94 67 6 46 New York University NY 74 65 13 14 Northwestern University IL 90 64 8 29 Pennsylvania State Univ. PA 81 31 19 22 Princeton University NJ 95 67 5 69 Rice University TX 90 64 8 42 Stanford University CA 91 67 7 32 Tufts University MA 86 69 9 30 Tulane University LA 72 57 12 17 U. of California–Berkeley CA 85 60 17 16 U. of California–Davis CA 73 32 19 6 U. of California–Irvine CA 76 40 20 10 U. of California–Los Angeles CA 80 42 18 11 U. of California–San Diego CA 81 48 19 10 U. of California–Santa Barbara CA 70 47 20 13 U. of Chicago IL 83 64 4 38 U. of Florida FL 67 30 23 20 U. of Illinois–Urbana Champaign IL 79 29 15 21 U. of Michigan–Ann Arbor MI 83 49 15 15 U. of North Carolina–Chapel Hill NC 84 41 16 28 U. of Notre Dame IN 96 53 13 47 U. of Pennsylvania PA 88 65 7 40 U. of Rochester NY 77 65 10 24 U. of Southern California CA 70 51 13 22 U. of Texas–Austin TX 67 39 21 12 U. of Virginia VA 93 45 13 27 U. of Washington WA 72 37 12 12 U. of Wisconsin–Madison WI 75 35 13 11 Vanderbilt University TN 82 70 9 30 Wake Forest University NC 82 61 11 40 Washington University–St. Louis MO 86 74 7 33 Yale University CT 95 76 7 48

Explanation / Answer

Regression Analysis

1. Use methods of descriptive statistics to summarize the data.

Solution:

The descriptive statistics for the variables included in the data set are summarised as below:

Descriptive Statistics

N

Minimum

Maximum

Mean

Std. Deviation

Graduation Rate

48

67.00

96.00

83.4792

8.34620

% of Classes Under 20

48

29.00

76.00

55.7917

13.44802

Student-Faculty Ratio

48

3.00

23.00

11.5417

4.85079

Alumni Giving Rate

48

6.00

69.00

29.1250

13.52165

Valid N (listwise)

48

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.

Solution:

The regression model for prediction of alumni giving rate based on graduation rate is given as below:

Model Summary

Model

R

R Square

Adjusted R Square

Std. Error of the Estimate

1

.711a

.505

.494

9.61667

a. Predictors: (Constant), Graduation Rate

Here, the correlation coefficient between the dependent variable alumni giving rate and independent variable graduation rate is given as 0.711 which means there is a high positive linear association or relationship exists between the dependent variable alumni giving rate and independent variable graduation rate. The value for R square or the coefficient of determination is given as 0.505, so we concluded that about 50.5% of the variation in the dependent variable alumni giving rate is explained by the independent variable graduation rate.

The ANOVA table for this regression analysis is given as below:

ANOVAb

Model

Sum of Squares

df

Mean Square

F

Sig.

1

Regression

4339.155

1

4339.155

46.920

.000a

Residual

4254.095

46

92.480

Total

8593.250

47

a. Predictors: (Constant), Graduation Rate

b. Dependent Variable: Alumni Giving Rate

The p-value is given as 0.00, so we reject the null hypothesis that there is no any significant relationship exists. The coefficients for regression equation are summarised in the following table:

Coefficientsa

Model

Unstandardized Coefficients

Standardized Coefficients

t

Sig.

B

Std. Error

Beta

1

(Constant)

-66.979

14.099

-4.751

.000

Graduation Rate

1.151

.168

.711

6.850

.000

a. Dependent Variable: Alumni Giving Rate

The regression equation is given as below:

Alumni giving rate = -66.979 + 1.151*Graduation rate

3. Develop an estimated multiple linear regression model that could be used to predict the alumni giving rate using the Graduation Rate, % of Classes Under 20, and Student/Faculty Ratio as independent variables. Discuss your findings.

Solution:

The multiple linear regression model for prediction of alumni giving rate by using the graduation rate, % of classes under 20 and student faculty ratio is given as below;

Model Summary

Model

R

R Square

Adjusted R Square

Std. Error of the Estimate

1

.816a

.666

.644

8.07216

a. Predictors: (Constant), Student-Faculty Ratio, Graduation Rate, % of Classes Under 20

The multiple correlation coefficient between the dependent variable alumni giving rate and the independent variables is given as 0.816, this means there is a high positive linear relationship exists between the dependent variable alumni giving rate and independent variables such as student faculty ratio, graduation rate and percentage of classes under 20. The coefficient of determination or the value for the R square is given as the 0.666, this means we concluded that about 66.6% of the variation in the dependent variable alumni giving rate is explained by the independent variables student faculty ratio, graduation rate, percentage of classes under 20.

ANOVAb

Model

Sum of Squares

df

Mean Square

F

Sig.

1

Regression

5726.217

3

1908.739

29.293

.000a

Residual

2867.033

44

65.160

Total

8593.250

47

a. Predictors: (Constant), Student-Faculty Ratio, Graduation Rate, % of Classes Under 20

b. Dependent Variable: Alumni Giving Rate

Here, we get the p-value as 0.00 which is less than the given level of significance, so we reject the null hypothesis that there is a no significant relationship exists between the dependent variable and independent variable.

Coefficientsa

Model

Unstandardized Coefficients

Standardized Coefficients

t

Sig.

B

Std. Error

Beta

1

(Constant)

-18.121

18.605

-.974

.335

Graduation Rate

.687

.174

.424

3.943

.000

% of Classes Under 20

.071

.139

.071

.514

.610

Student-Faculty Ratio

-1.218

.399

-.437

-3.054

.004

a. Dependent Variable: Alumni Giving Rate

The regression equation is given as below:

Alumni giving rate = -18.121 + 0.687*graduation rate + 0.071*percentage of classes under twenty – 1.218 * student faculty ratio

4. Based on the results in parts 2 and 3, do you believe another regression model may be more appropriate? Estimate this model, and discuss your results.

Solution:

Based on the results in parts 2 and 3 we do not believe another regression model is more appropriate because in the both regression models, we get the p-values as less than the given level of significance. Also, the explained variation in the dependent variable for the both model is considerable.

5. What conclusions and recommendations can you derive from your analysis? What universities are achieving a substantially higher alumni giving rate than would be expected, given their Graduation Rate, % of Classes Under 20, and Student/Faculty Ratio? What universities are achieving a substantially lower alumni giving rate than would be expected, given their Graduation Rate, % of Classes Under 20, and Student/Faculty Ratio? What other independent variables could be included in the model?

Solution:

For the first regression model, the correlation coefficient between the dependent variable alumni giving rate and independent variable graduation rate is given as 0.711 which means there is a high positive linear association or relationship exists between the dependent variable alumni giving rate and independent variable graduation rate. The value for R square or the coefficient of determination is given as 0.505, so we concluded that about 50.5% of the variation in the dependent variable alumni giving rate is explained by the independent variable graduation rate. The p-value is given as 0.00, so we reject the null hypothesis that there is no any significant relationship exists.

For the second regression model, multiple correlation coefficient between the dependent variable alumni giving rate and the independent variables is given as 0.816, this means there is a high positive linear relationship exists between the dependent variable alumni giving rate and independent variables such as student faculty ratio, graduation rate and percentage of classes under 20. The coefficient of determination or the value for the R square is given as the 0.666; this means we concluded that about 66.6% of the variation in the dependent variable alumni giving rate is explained by the independent variables student faculty ratio, graduation rate, percentage of classes under 20. We get the p-value as 0.00 which is less than the given level of significance, so we reject the null hypothesis that there is a no significant relationship exists between the dependent variable and independent variable.

Descriptive Statistics

N

Minimum

Maximum

Mean

Std. Deviation

Graduation Rate

48

67.00

96.00

83.4792

8.34620

% of Classes Under 20

48

29.00

76.00

55.7917

13.44802

Student-Faculty Ratio

48

3.00

23.00

11.5417

4.85079

Alumni Giving Rate

48

6.00

69.00

29.1250

13.52165

Valid N (listwise)

48

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