Question: Let \(y\) be the GPA of college students (in 4 points scale), \(x_{1}\) is the percentile in the high school graduating class (for example, \(x_{1}=10\) means the top 10 percent of the class), and \(x_{2}\) is the combined math and verbal scores on the SAT exam. Based on 4137 college students' information, we run a multiple linear regression model and obtained the following best fitted equation:

with \(R^{2}=0.273\).

1. Why does it make sense for the coefficient on \(x_{1}\) to be negative?
2. What is the predicted college GPA for a student who was in the 20 th percentile in his graduating class and received the combined math and verbal scores on the SAT exam to be 1050 ?
3. What is the predicted college GPA for a student who was in the 1 st percentile in his graduating class and received the combined math and verbal scores on the SAT exam to be $1800 ?$ Does the prediction make sense? Why or why not?
4. Suppose that two high school graduates, A and B, graduated in the same percentile from high school, but A's SAT score was 140 points higher than B's SAT score. What is the predicted difference in college GPA for these two students? Is the difference large?
5. Holding \(x_{1}\) fixed, what difference in SAT scores leads to a predicted college GPA difference of $0.5$ ? Comment on your answer.

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