Chapter 8, Problem 39E

(a)

To determine

To find out what is the correlation.

Answer to Problem 39E

  $r=0.685$ .

Explanation of Solution

From the previous exercise $37$ , we have the following information as:

The SAT scores of the students are examined and the scatterplot is given for the Math SAT versus the Verbal SAT scores. The researchers wants to study that is the Verbal SAT scores helps in the scores of the Math SAT scores and vice versa. Now, we are supposing that we use math score to estimate verbal scores. Also, it is given that,

  $\begin{array}{l}r=0.685\\ {\mu }_1=596.3\\ {\mu }_2=612.20\\ {\sigma }_1=99.5\\ {\sigma }_2=96.1\end{array}$

Thus, the correlation cannot be affected by which is going to estimate which score whether math score estimate verbal scores or verbal score estimate math score. So, the correlation will remain same for this transformation in the math and verbal score. Thus, $r=0.685$ .

(b)

To determine

To write the equation of the line of regression predicting verbal scores from math scores.

Answer to Problem 39E

  $\text{<mover accent="true"><mo>V</mo><mo>^</mo></mover> erbal}=162.1+0.71(\text{Math)}$ .

Explanation of Solution

From the previous exercise $37$ , we have the following information as:

The SAT scores of the students are examined and the scatterplot is given for the Math SAT versus the Verbal SAT scores. The researchers wants to study that is the Verbal SAT scores helps in the scores of the Math SAT scores and vice versa. Now, we are supposing that we use math score to estimate verbal scores. Also, it is given that,

  $\begin{array}{l}r=0.685\\ {\mu }_1=596.3\\ {\mu }_2=612.20\\ {\sigma }_1=99.5\\ {\sigma }_2=96.1\end{array}$

Thus, the regression equation for the new transformation is calculated as:

  $\begin{array}{l}\beta =r\times \frac{{\sigma }_1}{{\sigma }_2}\\ =0.685\times \frac{99.5}{96.1}\\ =0.71\\ \alpha ={\mu }_1-\beta \times {\mu }_2\\ =596.3-0.71\times 612.20\\ =162.1\end{array}$

Thus, the regression equation will be as:

  $\begin{array}{l}\text{<mover accent="true"><mo>V</mo><mo>^</mo></mover> erbal}=\alpha +\beta (\text{Math)}\\ =162.1+0.71(\text{Math)}\end{array}$

(c)

To determine

To explain what would a positive residual mean in this context.

Explanation of Solution

From the previous exercise $37$ , we have the following information as:

The SAT scores of the students are examined and the scatterplot is given for the Math SAT versus the Verbal SAT scores. The researchers wants to study that is the Verbal SAT scores helps in the scores of the Math SAT scores and vice versa. Now, we are supposing that we use math score to estimate verbal scores. Also, it is given that,

  $\begin{array}{l}r=0.685\\ {\mu }_1=596.3\\ {\mu }_2=612.20\\ {\sigma }_1=99.5\\ {\sigma }_2=96.1\end{array}$

And the regression equation is:

  $\text{<mover accent="true"><mo>V</mo><mo>^</mo></mover> erbal}=162.1+0.71(\text{Math)}$

Thus, a positive residual means in this context that the observed verbal score is higher than the predicted verbal score from the math score. As we know that the residual is calculated as the actual score minus the predicted score from the regression line.

(d)

To determine

To predict a person verbal score if her math score was $500$ .

Answer to Problem 39E

The person’s verbal score will be $516.7$ if her math score was $500$ .

Explanation of Solution

From the previous exercise $37$ , we have the following information as:

The SAT scores of the students are examined and the scatterplot is given for the Math SAT versus the Verbal SAT scores. The researchers wants to study that is the Verbal SAT scores helps in the scores of the Math SAT scores and vice versa. Now, we are supposing that we use math score to estimate verbal scores. Also, it is given that,

  $\begin{array}{l}r=0.685\\ {\mu }_1=596.3\\ {\mu }_2=612.20\\ {\sigma }_1=99.5\\ {\sigma }_2=96.1\end{array}$

And the regression equation is:

  $\text{<mover accent="true"><mo>V</mo><mo>^</mo></mover> erbal}=162.1+0.71(\text{Math)}$

Thus, a person’s predicted verbal score can be calculated as, if her math score was $500$ :

  $\begin{array}{l}\text{<mover accent="true"><mo>V</mo><mo>^</mo></mover> erbal}=162.1+0.71(\text{Math)}\\ =162.1+0.71\times 500\\ =516.7\end{array}$

Therefore we have that the person’s verbal score will be $516.7$ if her math score was $500$ .

(e)

To determine

To predict her math score using that predicted verbal score in part (d) and the equation you created in exercise $37$ .

Answer to Problem 39E

The predicted math score will be $559.76$ .

Explanation of Solution

From the previous exercise $37$ , we have the following information as:

The SAT scores of the students are examined and the scatterplot is given for the Math SAT versus the Verbal SAT scores. The researchers wants to study that is the Verbal SAT scores helps in the scores of the Math SAT scores and vice versa. Now, we are supposing that we use math score to estimate verbal scores. Now, we have to predict the math score using the predicted verbal score in part (d) and the equation you created in exercise $37$ . Thus, from the previous exercise the regression line will be as:

  $\text{<mover accent="true"><mo>M</mo><mo>^</mo></mover> ath}=217.7+0.662(\text{Verbal)}$

Thus, the predicted math score will be as:

  $\begin{array}{l}\text{<mover accent="true"><mo>M</mo><mo>^</mo></mover> ath}=217.7+0.662(\text{Verbal)}\\ =217.7+0.662\times 516.7\\ =559.76\end{array}$

(f)

To determine

To explain why doesn’t the result in part (e) comes out to $500$ .

Explanation of Solution

From the previous exercise $37$ , we have the following information as:

The SAT scores of the students are examined and the scatterplot is given for the Math SAT versus the Verbal SAT scores. The researchers wants to study that is the Verbal SAT scores helps in the scores of the Math SAT scores and vice versa. Now, we are supposing that we use math score to estimate verbal scores. Also, it is given that,

  $\begin{array}{l}r=0.685\\ {\mu }_1=596.3\\ {\mu }_2=612.20\\ {\sigma }_1=99.5\\ {\sigma }_2=96.1\end{array}$

And the regression equation is:

  $\text{<mover accent="true"><mo>V</mo><mo>^</mo></mover> erbal}=162.1+0.71(\text{Math)}$

Thus, the result in part (e) does not come out to $500$ because someone whose math score is below average is predicted to have a verbal score below average but not as far. So, if we use the verbal score to predict math they will be even closer to the mean in predicted math score than their observed math score. If we kept cycling back and forth eventually we would predict the mean of each and stay there.