A college Admissions Officer is interested in determining the extent to which a prospective student's high school GPA and/or SAT score can be used as a basis for predicting his or her college freshman GPA. He believes that prospective students who have higher high school GPAs and SAT scores will have a higher college freshman GPA. He randomly selected forty students who recently completed their freshman year and collected the information reflected in the following table.
|
Student No.
|
College Freshman GPA
|
High School GPA
|
SAT Score
|
|
1
|
3.56
|
3.95
|
1383
|
|
2
|
2.59
|
2.82
|
1014
|
|
3
|
3.09
|
3.29
|
1217
|
|
4
|
3.68
|
3.84
|
1458
|
|
5
|
2.91
|
2.97
|
1157
|
|
6
|
2.48
|
3.09
|
1238
|
|
7
|
2.34
|
2.85
|
856
|
|
8
|
2.38
|
2.84
|
880
|
|
9
|
2.58
|
3.00
|
959
|
|
10
|
2.70
|
3.06
|
1011
|
|
11
|
2.54
|
2.82
|
959
|
|
12
|
2.89
|
3.14
|
1100
|
|
13
|
3.02
|
3.21
|
1156
|
|
14
|
3.33
|
3.46
|
1282
|
|
15
|
3.16
|
3.23
|
1227
|
|
16
|
2.60
|
3.25
|
1266
|
|
17
|
2.65
|
3.23
|
1291
|
|
18
|
3.27
|
3.89
|
1168
|
|
19
|
2.89
|
3.36
|
1041
|
|
20
|
2.91
|
3.31
|
1060
|
|
21
|
2.85
|
3.16
|
1044
|
|
22
|
3.65
|
3.97
|
1350
|
|
23
|
3.54
|
3.77
|
1319
|
|
24
|
3.83
|
3.99
|
1436
|
|
25
|
3.05
|
3.11
|
1150
|
|
26
|
3.15
|
3.94
|
1498
|
|
27
|
2.35
|
2.86
|
1117
|
|
28
|
3.06
|
3.65
|
1458
|
|
29
|
3.25
|
3.78
|
1134
|
|
30
|
3.49
|
3.97
|
1230
|
|
31
|
3.02
|
3.36
|
1075
|
|
32
|
2.91
|
3.16
|
1043
|
|
33
|
3.48
|
3.70
|
1258
|
|
34
|
3.09
|
3.22
|
1128
|
|
35
|
3.66
|
3.73
|
1343
|
|
36
|
2.31
|
2.89
|
1069
|
|
37
|
2.49
|
3.04
|
1154
|
|
38
|
2.42
|
2.88
|
1122
|
|
39
|
2.78
|
3.23
|
1292
|
|
40
|
2.98
|
3.38
|
1015
|
1. Perform a simple linear regression with a 95% confidence level using college freshman GPA as the dependent variable and high school GPA as the independent variable, and evaluate the statistical significance of the regression model.
2. Perform a simple linear regression with a 95% confidence using college freshman GPA as the dependent variable and SAT score as the independent variable, and evaluate the statistical significance of the regression model.
3. Compare the two simple linear regression models, select your preferred simple regression model, and explain the basis for selecting your preferred model.
4. Perform a multiple linear regression with a 95% confidence using college freshman GPA as the dependent variable and high school GPA and SAT score as the independent variables.
a. Evaluate the statistical significance of the regression model as a whole.
b. Evaluate the statistical significance of the linear relationship between the dependent variable and each independent variable.
c. Discuss the extent to which there is evidence of multicollinearity between the independent variables.
5. Compare your preferred simple linear regression model (i.e., the regression model you selected in step 3) to the multiple linear regression model. Discuss whether the simple regression model or multiple regression model would be your overall preferred regression model, including explaining the basis for selecting your preferred model.
6. Discuss the contribution of each independent variable for your overall preferred regression model (i.e., the model you selected in step 5) to predicting the value of the dependent variable. Round the coefficients to four decimal places.
7. Discuss the range of values for the independent variable(s) for your preferred regression model (i.e., the regression model you selected in step 5) for which the regression model is valid.
8. Discuss the p-value for the coefficient for the y-intercept for your overall preferred regression model, including explaining why a p-value that is not less than or equal to α = 0.05 would not be cause for rejecting the regression model. (Hint: Consider the range of values for the independent variables associated with the given data set.)
9. Identify the regression equation associated with your overall preferred regression model and associated degree of error associated with using the model to predict a student's college freshman GPA. Round the coefficients and degree of error to four decimal places.
10. Calculate the predicted college freshman GPA for a student with a high school GPA of 3.25 and an SAT score of 1115 using your overall preferred regression model. Round your answer to four decimal places.
11. Identify the lower and upper limits associated with a 95% confidence level interval estimate for the predicted college freshman GPA for a student with a high school GPA of 3.25 and an SAT score of 1115 using your overall preferred regression model. Round the coefficients and your final answers to four decimal places.