Chapter 12, Problem R12.4RE

To determine

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

Answer to Problem R12.4RE

We are $95\%$ confident that the slope of the true regression line is between $-1.3168396\text{ and }-0.5253604$ .

Explanation of Solution

A study was conducted to know that do taller students require fewer steps to walk a fixed distance. The scatterplot given in the question shows the relationship between height and the number of steps require to walk the school hallway. And it has been assumed that the conditions for inferences are met.Thus, it is given that,

  $\begin{array}{l}n=36\\ \alpha =0.05\end{array}$

The slope $b_1$ is given in the row “Height” and in the column “Coef” of the given computer output as:

  $b_1=-0.9211$

The standard error of the slope $S{E}_{b_1}$ is given in the row “Height” and in the column “SE Coef” of the given computer output as:

  $S{E}_{b_1}=0.1938$

Now, we need to find the degrees of freedom:

  $\begin{array}{l}df=n-2\\ =36-2\\ =34\end{array}$

Thus, the t-value can be found in the student’s T-distribution table, then,

  $t^{*}=2.042$

Thus, the confidence interval is:

  $\begin{array}{l}b-t^{*}\times S{E}_b\\ =-0.9211-2.042\times 0.1938\\ =-1.3168396\\ b+t^{*}\times S{E}_b\\ =-0.9211+2.042\times 0.1938\\ =-0.5253604\end{array}$

Thus, we are $95\%$ confident that the slope of the true regression line is between $-1.3168396\text{ and }-0.5253604$ .