Question

Using the data in GPA2 on 4,137 college students, the following equation was estimated by OLS: $$ \begin{aligned} \widehat{\text {colgpa}} &=1.392-.0135 \text { hsperc }+. .00148 \text { sat } \\ n &=4.137, R^{2}=.273 \end{aligned} $$ where colgpa is measured on a four-point scale, hsperc is the percentile in the high school graduating class (defined so that, for example, hsperc $=5$ means the top $5 \%$ of the class), and sat is the combined math and verbal scores on the student achievement test. i. Why does it make sense for the coefficient on $h s p e r c$ to be negative? ii. What is the predicted college GPA when hsperc $=20$ and $s a t=1,050 ?$ iii. Suppose that two high school graduates, A and B, graduated in the same percentile 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? iv. Holding hsperc fixed, what difference in SAT scores leads to a predicted colgpa difference of $.50,$ or one-half of a grade point? Comment on your answer.

          Using the data in GPA2 on 4,137 college students, the following equation was estimated by OLS:
$$
\begin{aligned}
\widehat{\text {colgpa}} &=1.392-.0135 \text { hsperc }+. .00148 \text { sat } \\
n &=4.137, R^{2}=.273
\end{aligned}
$$
where colgpa is measured on a four-point scale, hsperc is the percentile in the high school graduating class (defined so that, for example, hsperc $=5$ means the top $5 \%$ of the class), and sat is the combined math and verbal scores on the student achievement test.
i. Why does it make sense for the coefficient on $h s p e r c$ to be negative?
ii. What is the predicted college GPA when hsperc $=20$ and $s a t=1,050 ?$
iii. Suppose that two high school graduates, A and B, graduated in the same percentile 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?
iv. Holding hsperc fixed, what difference in SAT scores leads to a predicted colgpa difference of
$.50,$ or one-half of a grade point? Comment on your answer.

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Added by Nate F.

A Modern Approach

Jeffrey M. Wooldridge 7th Edition

Chapter 3

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Solved by Expert Juan Nicolás

09/09/2021

Step 1

The coefficient on hsperc is negative because it is defined such that a higher value means a lower ranking in the class. For example, a student with hsperc = 5 is in the top 5% of their class, while a student with hsperc = 95 is in the bottom 5%. Show more…

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Using the data in GPA2 on 4,137 college students, the following equation was estimated by OLS: $$ \begin{aligned} \widehat{\text {colgpa}} &=1.392-.0135 \text { hsperc }+. .00148 \text { sat } \\ n &=4.137, R^{2}=.273 \end{aligned} $$ where colgpa is measured on a four-point scale, hsperc is the percentile in the high school graduating class (defined so that, for example, hsperc $=5$ means the top $5 \%$ of the class), and sat is the combined math and verbal scores on the student achievement test. i. Why does it make sense for the coefficient on $h s p e r c$ to be negative? ii. What is the predicted college GPA when hsperc $=20$ and $s a t=1,050 ?$ iii. Suppose that two high school graduates, A and B, graduated in the same percentile 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? iv. Holding hsperc fixed, what difference in SAT scores leads to a predicted colgpa difference of $.50,$ or one-half of a grade point? Comment on your answer.

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00:01 Hello, so today i'm going to answer your question about the gpa and the influence of specific parameters the first question that you have is why does it make sense for the coefficient of the high percentile to be negative? it makes sense because it have the opposite relationship so when you have a better grade you will be in a smaller percent you will be the best one percent the best two percent so when you are in 50 you will have the best 50 percent of 50 percent of the other students are on top of you.

00:34 So people that have better result in the exam, they will also have better gpa, so the idea is that you understand that this variable behave that better outcome have a smaller value, so it's just that you keep it in mind.

00:46 Then what is the predicted value when you have specific parameters? then what i did is just replace the coefficients.

00:53 Of course, this is beta 0 and the value that they give you.

00:56 This is 3 .216, and it's just to keep it in mind.

01:01 Then they said that suppose two high schoolers, they will have 140 points higher.

01:09 So it is that you will have to multiply this difference of sit of the coefficient.

01:15 In general, you just have to make the derivative, but the straightforward solution would be this.

01:20 Because when you do the derivative, the only coefficient that is going to be, there is going to is this one and the difference is going to be this 140 points.

01:27 So just to keep you mind, this is the effect.

01:29 And then they ask you what is the predicted difference in college, holding the everything content, what is the amount of values that you need to have these values...

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