Case Problem: Alumni Giving 1. Use methods of descriptive statistics to summariz
ID: 3264338 • Letter: C
Question
Case Problem: Alumni Giving
1. Use methods of descriptive statistics to summarize the data
1a. Complete the table given below for the descriptive statistics for graduation rate, % of classes under 20, student-faculty ratio, and alumni giving
Graduation Rate
% of Classes Under 20
Student-Faculty Ratio
Alumni Giving Rate
Mean
Median
standard deviation
Minimum
Maximum
Range
b. Determine the correlations for each pair of variables are shown in the table below.
Graduation Rate
% of Classes Under 20
Student-Faculty Ratio
Alumni Giving Rate
Graduation Rate
% of Classes Under 20
Student-Faculty Ratio
Alumni Giving Rate
c. What type of relationship does the % of classes under 20 and student-faculty ratio demonstrate? Use data to support your answer.
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.
2a. PASTE the image of your Excel output BELOW that provides the estimated simple linear regression model showing how the alumni giving rate (y) is related to the graduate rate (x)
2b. i. The estimated simple linear regression equation is
ii. The coefficient of determination r2 is
iii. Interpret the meaning of the coefficient of determination r2
iv. Test the hypothesis of relationship between the alumni giving rate (y) and the graduation rate (x), at the 0.05 level of significance throughout this problem. What conclusion can we draw?
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.
3a. PASTE the Excel output that provides the estimated multiple linear regression model showing how the alumni giving rate (y) is related to the graduate rate (x1), % of Classes Under 20 (x2), and Student-Faculty Ratio (x3).
3b. i. The multiple simple linear regression equation is
ii. The coefficient of determination r2 is
iii. Interpret the meaning of the coefficient of determination r2
iv. Test the hypothesis of relationship between the alumni giving rate (y) and the graduation rate (x1), at the 0.05 level of significance throughout this problem. What conclusion can we draw about the effect of graduation rate on alumni giving rate while holding the holding % of classes under 20 and student-faculty ratio constant?
v. Test the hypothesis of relationship between the alumni giving rate (y) and the % of classes under 20 (x2), at the 0.05 level of significance throughout this problem. What conclusion can we draw about the alumni giving rate and when controlling for the graduation rate and the % of classes under 20?
vi. Test the hypothesis of relationship between the alumni giving rate (y) and student-faculty ratio (x3), at the 0.05 level of significance throughout this problem. What conclusion can we draw about the alumni giving rate and student-faculty ratio when controlling for the graduation rate and the % of classes under 20
Graduation Rate
% of Classes Under 20
Student-Faculty Ratio
Alumni Giving Rate
Mean
Median
standard deviation
Minimum
Maximum
Range
Explanation / Answer
Answer:
1. Use methods of descriptive statistics to summarize the data
1a. Complete the table given below for the descriptive statistics for graduation rate, % of classes under 20, student-faculty ratio, and alumni giving
Graduation Rate
% of Classes Under 20
Student-Faculty Ratio
Alumni Giving Rate
n
48
48
48
48
mean
83.04
55.73
11.54
29.27
sample standard deviation
8.61
13.19
4.85
13.44
minimum
66
29
3
7
maximum
97
77
23
67
range
31
48
20
60
median
83.50
59.50
10.50
29.00
mode
92.00
65.00
13.00
13.00
b. Determine the correlations for each pair of variables are shown in the table below.
Correlation Matrix
Graduation Rate
% of Classes Under 20
Student-Faculty Ratio
Alumni Giving Rate
Graduation Rate
1.000
% of Classes Under 20
.583
1.000
Student-Faculty Ratio
-.605
-.786
1.000
Alumni Giving Rate
.756
.646
-.742
1.000
48
sample size
± .285
critical value .05 (two-tail)
± .368
critical value .01 (two-tail)
c. What type of relationship does the % of classes under 20 and student-faculty ratio demonstrate? Use data to support your answer.
The correlation r = -0.786. The relation is negative.
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.
2a. PASTE the image of your Excel output BELOW that provides the estimated simple linear regression model showing how the alumni giving rate (y) is related to the graduate rate (x)
2b. i. The estimated simple linear regression equation is
alumni giving rate =-68.7612+1.1805* graduate rate
ii. The coefficient of determination r2 is 0.571
iii. Interpret the meaning of the coefficient of determination r2
57.1% of variance in alumni giving rate is explained by graduate rate.
iv. Test the hypothesis of relationship between the alumni giving rate (y) and the graduation rate (x), at the 0.05 level of significance throughout this problem. What conclusion can we draw?
Calculated t=7.832, P=0.000 which is < 0.05 level. Regression coefficient is significant.
The relationship between the alumni giving rate (y) and the graduation rate (x) is significant.
Regression Analysis
r²
0.571
n
48
r
0.756
k
1
Std. Error
8.894
Dep. Var.
Alumni Giving Rate
ANOVA table
Source
SS
df
MS
F
p-value
Regression
4,852.4618
1
4,852.4618
61.34
5.24E-10
Residual
3,639.0173
46
79.1091
Total
8,491.4792
47
Regression output
confidence interval
variables
coefficients
std. error
t (df=46)
p-value
95% lower
95% upper
Intercept
-68.7612
12.5827
-5.465
1.82E-06
-94.0888
-43.4336
Graduation Rate
1.1805
0.1507
7.832
5.24E-10
0.8771
1.4839
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.
3a. PASTE the Excel output that provides the estimated multiple linear regression model showing how the alumni giving rate (y) is related to the graduate rate (x1), % of Classes Under 20 (x2), and Student-Faculty Ratio (x3).
3b. i. The multiple simple linear regression equation is
alumni giving rate =-20.7201 +0.7482 x1+0.0290 x2-1.1920 x3
ii. The coefficient of determination r2 is 0.70
iii. Interpret the meaning of the coefficient of determination r2
70.0 % of variance in alumni giving rate is explained by regression model.
iv. Test the hypothesis of relationship between the alumni giving rate (y) and the graduation rate (x1), at the 0.05 level of significance throughout this problem. What conclusion can we draw about the effect of graduation rate on alumni giving rate while holding the holding % of classes under 20 and student-faculty ratio constant?
Calculated t=4.508, P=0.000 which is < 0.05 level. Regression coefficient is significant.
The relationship between the alumni giving rate (y) and the graduation rate (x1) is significant.
v. Test the hypothesis of relationship between the alumni giving rate (y) and the % of classes under 20 (x2), at the 0.05 level of significance throughout this problem. What conclusion can we draw about the alumni giving rate and when controlling for the graduation rate and the % of classes under 20?
Calculated t=0.208, P=0.8358 which is > 0.05 level. Regression coefficient is not significant.
The relationship between the alumni giving rate (y) and the % of classes under 20 (x2), is not significant.
vi. Test the hypothesis of relationship between the alumni giving rate (y) and student-faculty ratio (x3), at the 0.05 level of significance throughout this problem. What conclusion can we draw about the alumni giving rate and student-faculty ratio when controlling for the graduation rate and the % of classes under 20
Calculated t=-3.082, P=0.0035 which is < 0.05 level. Regression coefficient is significant.
The relationship between the alumni giving rate (y) and the student-faculty ratio (x3) is significant.
Regression Analysis
R²
0.700
Adjusted R²
0.679
n
48
R
0.837
k
3
Std. Error
7.610
Dep. Var.
Alumni Giving Rate
ANOVA table
Source
SS
df
MS
F
p-value
Regression
5,943.5311
3
1,981.1770
34.21
1.43E-11
Residual
2,547.9481
44
57.9079
Total
8,491.4792
47
Regression output
confidence interval
variables
coefficients
std. error
t (df=44)
p-value
95% lower
95% upper
Intercept
-20.7201
17.5214
-1.183
.2433
-56.0321
14.5919
Graduation Rate
0.7482
0.1660
4.508
4.80E-05
0.4137
1.0827
% of Classes Under 20
0.0290
0.1393
0.208
.8358
-0.2517
0.3098
Student-Faculty Ratio
-1.1920
0.3867
-3.082
.0035
-1.9714
-0.4126
Graduation Rate
% of Classes Under 20
Student-Faculty Ratio
Alumni Giving Rate
n
48
48
48
48
mean
83.04
55.73
11.54
29.27
sample standard deviation
8.61
13.19
4.85
13.44
minimum
66
29
3
7
maximum
97
77
23
67
range
31
48
20
60
median
83.50
59.50
10.50
29.00
mode
92.00
65.00
13.00
13.00