Chapter 12.1, Problem 17E

(a)

To determine

To verify that the $99\%$ confidence interval of the slope of the population regression line is $(0.5785,1.001)$ .

Answer to Problem 17E

Yes, it is $(0.5785,1.001)$ .

Explanation of Solution

It is given in the question that,

  $\begin{array}{l}n=16\\ b=0.79021\\ {\text{SE}}_b=0.07104\end{array}$

The degrees of freedom is the sample size decreased by $2$ :

  $\begin{array}{l}df=n-2\\ =16-2\\ =14\end{array}$

The critical t -value can be found in table B in the row of $df=14$ and in the column of $c=99\%$ as:

  $t^{*}=2.977$

Thus, the confidence interval is as:

  $\begin{array}{l}b-t^{*}\times {\text{SE}}_b\\ =0.79021-2.977\times 0.07104\\ =0.5787\\ b+t^{*}\times {\text{SE}}_b\\ =0.79021+2.977\times 0.07104\\ =1.0017\end{array}$

We are $99\%$ confident that the slope of the true regression line is between $(0.5785,1.001)$ .

(b)

To determine

To explain why an appropriate pair of hypotheses for this test is $H_0:\beta =1$ versus $H_a:\beta \ne 1$ .

Explanation of Solution

Now, as the two variables both measures the tire wear in the same measurements units. If we want to know if there is a difference we assume that there is no difference and thus both need to increase by the same amounts resulting in the null hypotheses as:

  $H_0:\beta =1$

And the alternative hypothesis states the opposite of the null hypothesis as:

  $H_a:\beta \ne 1$ .

(c)

To determine

To explain what conclusion would you draw for this significance test based on your interval in part (a).

Explanation of Solution

We have define the hypotheses in the part (b) as:

  $\begin{array}{l}{H}_0:\beta =1\\ H_a:\beta \ne 1\end{array}$

And the confidence interval calculated in part (a) is as:

There is a $99\%$ confidence that the slope of the true regression line is between $(0.5785,1.001)$ .

Thus, the confidence interval contain one and thus it is likely to obtain $\beta =1$ , which means that there is not sufficient evidence to support the claim of a difference.