Chapter 27, Problem 31E

a)

To determine

To find the 90% confidence interval for the slope of the regression line.

Answer to Problem 31E

We are 90% confident that the slope of the population of regression line is between 3.29 and 10.01.

Explanation of Solution

Given:

  

Formula:

Confidence interval for slope is,

  $(Coefficient(b)\mp t_c\times S{E}_b)$

The confidence level = 0.90

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

First need to find critical t value.

tc = 1.761 …Using excel formula, =TINV(0.1,14)

Using formula, 90% confidence interval for the slope of the regression line.

  $\begin{array}{l}(6.650-1.761\times 1.910,6.650+1.761\times 1.910)\\ (3.29,10.01)\end{array}$

Therefore, we are 90% confident that the slope of the population of regression line is between 3.29 and 10.01.

b)

To determine

To find the 90% confidence interval for the mean ozone level.

Answer to Problem 31E

We are 90% confidence that the mean ozone level for cities with 0.6 million people is between 18.47 ppm and 27.29 ppm.

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,

  $\widehat{y}=18.892+6.650\times 0.6=22.882$

So, standard error is,

  $\begin{array}{l}S{E}_{\widehat{\mu }}=\sqrt{{1.910}^2{(0.6-1.7)}^2+\frac{{5.454}^2}{16}}\\ S{E}_{\widehat{\mu }}=2.5047\end{array}$

The confidence level = 0.90

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

First need to find critical t value.

tc = 1.761 …Using excel formula, =TINV(0.10,14)

Using formula, 90% confidence interval is,

  $\begin{array}{l}(22.882-1.761\times 2.5047,22.882+1.761\times 2.5047)\\ (18.47,27.29)\end{array}$

Hence, we are 90% confidence that the mean ozone level for cities with 0.6 million people is between 18.47 ppm and 27.29 ppm.