Chapter 8, Problem 37E

(a)

To determine

To describe the relationship.

Explanation of Solution

In the question 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. Thus, by looking at the scatterplot of the two we can say that the scatterplot is moderately strong and fairly straight and is positive in nature as the points is going in the upward direction. And possibly we can see that there are some outliers in the scatterplot.

(b)

To determine

To explain are there any students whose scores do not seem to fit the overall pattern.

Answer to Problem 37E

Yes, there are students whose scores do not seem to fit the overall pattern.

Explanation of Solution

In the question 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. Thus, there are students whose scores do not seem to fit the overall pattern. We can say that because in the scatterplot the student whose verbal SAT score is $500$ and the math SAT score is $800$ does not fit to the overall pattern and thus it is outlier of the scatterplot.

(c)

To determine

To interpret the statistic.

Explanation of Solution

In the question 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. It is given that,

  $r=0.685$

This interprets that the relationship between the math SAT score and the verbal SAT score is positive and fairly strong. And also,

  $\begin{array}{l}{R}^2=r^2\\ $ ={(0.685)}^2$=0.469\\ =46.9\%\end{array}$

Thus, we can say that $46.9\%$ of variation in math SAT scores can be explained by the verbal SAT scores.

(d)

To determine

To write the equation of the regression line.

Answer to Problem 37E

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

Explanation of Solution

In the question 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. Also, it is given that,

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

Thus, the equation of the regression line can be calculated as:

  $\begin{array}{l}\beta =r\times \frac{{\sigma }_1}{{\sigma }_2}\\ =0.685\times \frac{96.1}{99.5}\\ =0.662\\ \alpha ={\mu }_1-\beta \times {\mu }_2\\ =612.20-0.662\times 596.3\\ =217.7\end{array}$

Thus, the regression equation is as:

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

(e)

To determine

To interpret the slope of this line.

Explanation of Solution

In the question 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. Also, it is given that,

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

And the regression equation is:

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

Thus, we interpret that every point of verbal SAT scores adds $0.662$ points to the predicted average math SAT score.

(f)

To determine

To predict the math score of a student with a verbal score of $500$ .

Answer to Problem 37E

The math score of a student with a verbal score of $500$ is $548.7$ points.

Explanation of Solution

In the question 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. Also, it is given that,

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

And the regression equation is:

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

Thus, the math score of a student with a verbal score of $500$ can be calculated as:

  $\begin{array}{l}\text{<mover accent="true"><mo>M</mo><mo>^</mo></mover> ath}=217.7+0.662(\text{Verbal)}\\ =217.7+0.662\times 500\\ =548.7\end{array}$

(g)

To determine

To explain what would be that student’s math score residual if every year some student scores a perfect $1600$ .

Answer to Problem 37E

The student’s math score residual if every year some student scores a perfect $1600$ will be $53$ points.

Explanation of Solution

In the question 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. Also, it is given that,

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

And the regression equation is:

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

Thus, the math score of a student with a verbal score of $800$ can be calculated as:

  $\begin{array}{l}\text{<mover accent="true"><mo>M</mo><mo>^</mo></mover> ath}=217.7+0.662(\text{Verbal)}\\ =217.7+0.662\times 800\\ =747.30\end{array}$

Thus, the residual will be as:

  $\begin{array}{l}=800-747.30\\ =52.7\\ \approx 53\end{array}$