Question

R12.4 Long legs Construct and interpret a 95% confidence interval for the slope of the population regression line. Assume that the conditions for inference are met. Explain how the interval provides more information than the test in R12.3.

Short answer

A $95\%$ confidence level that the true regression line's slope is between $-1.3168396$ and $-0.5253604$.

Step-by-step solution

Given information

To construct and interpret a $95\%$ confidence interval for the slope of the population regression line

Explanation

A analysis was conducted to see if taller students needed fewer steps to go a certain distance. The box plot in the question indicates the link between height and the number of steps needed to walk down the school corridor. And it's been assumed that the inference requirements are met.
As a result, it is assumed that
$n=36$
$\alpha =0.05$
The slope $b_1$ is calculated as follows in the computer output's row "Height" and column "Coef":
$b_1=-0.9211$
In the row "Height" and the column "SE Coef" of the given computer output, the standard error of the slope $S{E}_{b_1}$is provided as:
$S{E}_{b_1}=0.1938$

Explanation

To find the degrees of freedom, as follows:
$df=n-2$
$=36-2$
$=34$
The t-value can then be discovered in the T-distribution table of the student.
$t^{*}=2.042$
As a result, the confidence interval is as follows:
$b-t^{*}\times S{E}_b$
$=-0.9211-2.042\times 0.1938$
$=-1.3168396$
$b+t^{*}\times S{E}_b$
$=-0.9211+2.042\times 0.1938$
$=-0.5253604$
As a result, a $95\%$ confidence level that the true regression line's slope is between $-1.3168396$ and $-0.5253604$.