Using the data on $4,137$ college students, the following equation was estimated by OLS:

$$

colGP Ai $=1\mathrm{.}3920\mathrm{.}0135$hsperci $+0\mathrm{.}00148$SATi, R$2=0\mathrm{.}273$

where colGP Ai is college GPA measured on a four$-$point scale, hsperci is the student$$s

percentile in their high school graduating class $($defined so that, for example, hsperci

$=5$ means the top $5\%$ of the class$),$ and SATi is the combined math and verbal scores

on the student achievement test.

$($a$)$ Why does it make sense for the coefficient on hsperci to be negative?

$($b$)$ If we instead estimated the model without SATi, would you expect the bias in

the coefficient on hsperci to be positive or negative?

$($c$)$ Returning to the full model, what is the predicted college GPA when hsperci $=20$

and SATi $=1050$

$($d$)$ Suppose that two high school graduates, A and B$,$ graduated in the same per$-$

centile from high school, but Student A$$s SAT score was $140$ points higher $($about

one standard deviation in the sample$).$ What is the predicted difference in college

GPA for these two students? Is the difference large?

$($e$)$ Holding hsperci fixed, what difference in SAT scores leads to a predicted colGP Ai

difference of $\mathrm{.}5,$ or one$-$half of a grade point? Comment on your answer.