Chapter 27, Problem 22E

a)

To determine

To find the 90% confidence interval for the mean SAT-Math score for all students with an SAT-Verbal score of 500.

Answer to Problem 22E

We are 90% confidence that the mean SAT-Math score for all students with an SAT-Verbal score of 500 is between 534.0843 and 560.0987

Explanation of Solution

Given:

  

Formula:

Confidence interval for average of response variable is,

  $(\widehat{y}\mp t_c\times S{E}_{\widehat{\mu }})$

Standard error is,

  $S{E}_{\widehat{\mu }}=\sqrt{S{E}_b{}^2{(x-\overline{x})}^2+\frac{s{e}^2}{n}}$

Therefore, predicted value becomes,

  $\widehat{y}=209.554+0.675075\times 500=547.0915$

So, standard error is,

  $\begin{array}{l}S{E}_{\widehat{\mu }}=\sqrt{{0.0568}^2{(500-596.296)}^2+\frac{{71.75}^2}{162}}\\ S{E}_{\widehat{\mu }}=7.8546\end{array}$

The confidence level = 0.90

So, level of significance = $\alpha$ = 0.10

First need to find critical t value.

tc = 1.656 …Using excel formula, =TINV(0.10,160)

Using formula, 90% confidence interval is,

  $\begin{array}{l}(547.0915-1.656\times 7.8546,547.0915+1.656\times 7.8546)\\ (534.0843,560.0987)\end{array}$

Hence, we are 90% confidence that the mean SAT-Math score for all students with an SAT-Verbal score of 500 is between 534.0843 and 560.0987

b)

To determine

To find the 90% prediction interval for the Math score of the senior class president if she scored 710 on the verbal section.

Answer to Problem 22E

We are 90% confidence that the mean SAT-Math score for all students with an SAT-Verbal score of 710 is between 569.1942 and 808.5203.

Explanation of Solution

Given:

  

Formula:

Confidence interval for average of response variable is,

  $(\widehat{y}\mp t_c\times S{E}_{\widehat{\mu }})$

Standard error is,

  $S{E}_{\widehat{\mu }}=\sqrt{S{E}_b{}^2{(x-\overline{x})}^2+\frac{s{e}^2}{n}+s{e}^2}$

Therefore, predicted value becomes,

  $\widehat{y}=209.554+0.675075\times 710=688.85725$

So, standard error is,

  $\begin{array}{l}S{E}_{\widehat{\mu }}=\sqrt{{0.0568}^2{(710-596.296)}^2+\frac{{71.75}^2}{162}+{71.75}^2}\\ S{E}_{\widehat{\mu }}=72.2603\end{array}$

The confidence level = 0.90

So, level of significance = $\alpha$ = 0.10

First need to find critical t value.

tc = 1.656 …Using excel formula, =TINV(0.10,160)

Using formula, 90% confidence interval is,

  $\begin{array}{l}(688.85725-1.656\times 72.2603,688.85725+1.656\times 72.2603)\\ (569.1942,808.5203)\end{array}$

Hence, we are 90% confidence that the mean SAT-Math score for all students with an SAT-Verbal score of 710 is between 569.1942 and 808.5203.