Regression analysis

1. You are a college admissions officer, and you have the following data:
HS GPA College GPA
2.00 2.50
2.50 1.80
2.70 1.50
3.00 2.00
3.10 2.50
3.20 2.50
3.50 3.00
3.70 2.80
3.75 3.00
3.80 3.90
3.85 4.00
4.00 3.75

These, of course, are High School GPAs and College GPAs (at the end of First Year) for 12 students.

At your esteemed institution, a student is expelled from the college if, at the end of first year, the GPA is not AT LEAST 2.00. You cannot admit students and take their tuition money if they will not "make the grade"-lawyers being what they can be.

Use regression analysis to develop an admission rule based on High School GPA. In other words, your rule is "If a student has a GPA of X, he or she will earn 2.00 or better at the end of the first year at our school." (Alternatively, your rule could be "Unless a student has a High School GPA of X, he or she cannot be admitted.")

Who has more fun than us?

2. I have run a (simple linear) regression analysis, and the SS for Regression is equal to the SS for Residuals. Which of the following is true?
A) My F statistic will be large and statistically significant
B) My F statistic will be equal to 1.00
C) Significance depends on how many participants there were
D) My analysis can never be significant if SS Regression is equal to SS Residuals

EXPLAIN YOUR ANSWER AND SHOW AN EXAMPLE

3. Which of the following makes me happier?
A) My SS about Mean (Y) is very large
B) My SS between ŷ and Mean (Y) is very small
C) My SS between ŷ and actual Y is very small
D) A Model H, 12-Fret, Froggy Bottom Guitar

EXPLAIN YOUR ANSWER AND SHOW AN EXAMPLE

4. What variable, other than High School GPA, could be included in a multiple regression analysis to devise a college admissions rule. This CANNOT include ACT or SAT. Add this variable to your data set (with careful planning) and show how your variable improves the prediction.