Chapter 8, Problem 38E

(a)

To determine

To write the equation of the regression line.

Answer to Problem 38E

  $\text{<mover accent="true"><mo>G</mo><mo>^</mo></mover> PA}=-1.262+0.0021(\text{SAT)}$ .

Explanation of Solution

In the question shows the relationship between the GPA scores and the combined SAT scores. The scatterplot for the two variables showed the association to be reasonably linear. And it is given that:

  $\begin{array}{l}r=0.47\\ {\mu }_1=2.66\\ {\mu }_2=1833\\ {\sigma }_1=0.56\\ {\sigma }_2=123\end{array}$

Thus, the regression equation will be calculated as:

  $\begin{array}{l}\beta =r\times \frac{{\sigma }_1}{{\sigma }_2}\\ =0.47\times \frac{0.56}{123}\\ =0.0021\\ \alpha ={\mu }_1-\beta \times {\mu }_2\\ =2.66-0.0021\times 1833\\ =-1.262\end{array}$

Thus, the regression line will be:

  $\begin{array}{l}\text{<mover accent="true"><mo>G</mo><mo>^</mo></mover> PA}=\alpha +\beta (\text{SAT)}\\ =-1.262+0.0021(\text{SAT)}\end{array}$

(b)

To determine

To explain what the y -intercept of the regression line indicates.

Explanation of Solution

In the question shows the relationship between the GPA scores and the combined SAT scores. The scatterplot for the two variables showed the association to be reasonably linear. And it is given that:

  $\begin{array}{l}r=0.47\\ {\mu }_1=2.66\\ {\mu }_2=1833\\ {\sigma }_1=0.56\\ {\sigma }_2=123\end{array}$

Thus, the regression line will be:

  $\begin{array}{l}\text{<mover accent="true"><mo>G</mo><mo>^</mo></mover> PA}=\alpha +\beta (\text{SAT)}\\ =-1.262+0.0021(\text{SAT)}\end{array}$

Therefore, the y -intercept of the regression line indicates that the average GPA of someone who's SAT score was zero which is technically not possible as we know that the y -intercept is the y -value for when x is zero.

(c)

To determine

To interpret the slope of the regression line.

Explanation of Solution

In the question shows the relationship between the GPA scores and the combined SAT scores. The scatterplot for the two variables showed the association to be reasonably linear. And it is given that:

  $\begin{array}{l}r=0.47\\ {\mu }_1=2.66\\ {\mu }_2=1833\\ {\sigma }_1=0.56\\ {\sigma }_2=123\end{array}$

Thus, the regression line will be:

  $\begin{array}{l}\text{<mover accent="true"><mo>G</mo><mo>^</mo></mover> PA}=\alpha +\beta (\text{SAT)}\\ =-1.262+0.0021(\text{SAT)}\end{array}$

Thus, the slope of a line is the rise divided by the run. So, the slope of the regression line represents that as SAT score increases by one point then GPA score increases by $0.0021$ .

(d)

To determine

To predict the GPA of a freshman who scored a combined $2100$ .

Answer to Problem 38E

As a student who scored a combined $2100$ would have a GPA $3.15$ .

Explanation of Solution

In the question shows the relationship between the GPA scores and the combined SAT scores. The scatterplot for the two variables showed the association to be reasonably linear. And it is given that:

  $\begin{array}{l}r=0.47\\ {\mu }_1=2.66\\ {\mu }_2=1833\\ {\sigma }_1=0.56\\ {\sigma }_2=123\end{array}$

Thus, the regression line will be:

  $\begin{array}{l}\text{<mover accent="true"><mo>G</mo><mo>^</mo></mover> PA}=\alpha +\beta (\text{SAT)}\\ =-1.262+0.0021(\text{SAT)}\end{array}$

Therefore, the predict GPA of a freshman who scored a combined $2100$ can be calculated as:

  $\begin{array}{l}\text{<mover accent="true"><mo>G</mo><mo>^</mo></mover> PA}=-1.262+0.0021(\text{SAT)}\\ =-1.262+0.0021\times 2100\\ =3.15\end{array}$

Thus, as a student who scored a combined

  $2100$ would have a GPA $3.15$ .

(e)

To determine

To explain how effective do you think SAT score would be in predicting academic success during the first semester of the freshman year at this college.

Answer to Problem 38E

It will not be very successful.

Explanation of Solution

In the question shows the relationship between the GPA scores and the combined SAT scores. The scatterplot for the two variables showed the association to be reasonably linear. And it is given that:

  $\begin{array}{l}r=0.47\\ {\mu }_1=2.66\\ {\mu }_2=1833\\ {\sigma }_1=0.56\\ {\sigma }_2=123\end{array}$

Thus, the regression line will be:

  $\begin{array}{l}\text{<mover accent="true"><mo>G</mo><mo>^</mo></mover> PA}=\alpha +\beta (\text{SAT)}\\ =-1.262+0.0021(\text{SAT)}\end{array}$

Thus, we think SAT score would not be very successful in predicting academic success during the first semester of the freshman year at this college as the correlation for this relationship is quite low and would imply that the linear model is not a great predictor of freshmen GPA.

(f)

To determine

To explain as a student would you rather have a positive or a negative residual in this context.

Answer to Problem 38E

A student would you rather have a positive residual in this context.

Explanation of Solution

In the question shows the relationship between the GPA scores and the combined SAT scores. The scatterplot for the two variables showed the association to be reasonably linear. And it is given that:

  $\begin{array}{l}r=0.47\\ {\mu }_1=2.66\\ {\mu }_2=1833\\ {\sigma }_1=0.56\\ {\sigma }_2=123\end{array}$

Thus, the regression line will be:

  $\begin{array}{l}\text{<mover accent="true"><mo>G</mo><mo>^</mo></mover> PA}=\alpha +\beta (\text{SAT)}\\ =-1.262+0.0021(\text{SAT)}\end{array}$

Thus, a student would you rather have a positive residual in this context because a positive residual would mean that your GPA is higher than the one predicted by the linear model.