In this Exercise 14.54, we repeat the information from Exercise 14.18.

a. Decide, at the $10\%$significance level, whether the data provide sufficient evidence to conclude that $x$is useful for predicting $y$:

b. Find a $90\%$confidence interval for the slope of the population regression line.

role="math" localid="1652363608630" $\begin{array}{|lllll|}\hline x& 1& 3& 4& 4\\ y& 8& 0& 3& 1\\ \hline\end{array}$ $\hat{y}=9-2x$

$\begin{array}{|lllll|}\hline x& 1& 3& 4& 4\\ y& 8& 0& 3& 1\\ \hline\end{array}$

role="math" localid="1652363652491" $\hat{y}=9-2x$

Decide the Null and Alternate hypothesis

$H_0:{\beta }_1=0$ ($x$is not useful for predicting $y$)

$H_{\alpha }:{\beta }_1\ne 0$ ($x$is useful for predicting $y$).

If $p$-value $\le \alpha (=0.10)$, then reject the null hypothesis $\left(H_0\right)$

Using Excel, find the test statistic and p-value.

Step 1: Choose Data >Data Analysis >Regression from the drop-down menu.

Step 2: In the Input Y Range box, type $y$.

Step 3:In the Input X Range box, type $x$.

Step 4: Select Confidence Level $90$from the drop-down menu.

Step 5: Click the OK button.

OUTPUT

The test statistic's value is $-1.8516$, and the $p$-value is$0.2052$, according to the Excel output.

Use the $\alpha =0.10$significance level.

The $p$-value is higher than the level of significance in this case.

To put it another way, $p$-value $(=0.2052)>\alpha (=0.10)$.

As a result of the rejection rule, it may be argued that at $\alpha =0.10$, there is no evidence to reject the null hypothesis ($H_0$).

$\begin{array}{|lllll|}\hline x& 1& 3& 4& 4\\ y& 8& 0& 3& 1\\ \hline\end{array}$

$\hat{y}=9-2x$

Using Excel, calculate the $90\%$confidence interval for the slope of the regression line.

From the Excel output in part (a), We get as$(-5.1539,1.1539)$.