Question

In Exercises $14.48-14.57$, we repeat the information from Exercises $14.12-14.21$.

a. Deride, at the lore significance level, whether the data provide sufficient evidence to conclude that $x$is useful for predicting $y$.

b. Find a localid="1653240248143" $90\%$confidence interval for the slope of the population regression line.

$y=1+2x$

Short answer

(a)The data does not support the conclusion that $x$can predict$y$.

(b)The $90\%$confidence interval for the slope of the regression line is $(-8.9357,12.9357)$, based on the Excel result in component (a).

Step-by-step solution

Given Information (part a)

The following are the proper hypotheses:

Hypothesis of nullity:

$H_0:{\beta }_1=0$

That is, the variable $x$has no predictive value for $y$.

Another possibility:

$H_a:{\beta }_1\ne 0$

That is, the variable $x$ can be used to forecast $y$.

Rejection Rule:

Reject the null hypothesis $\left(H_0\right)$if $p$-value $\le \alpha (=0.10)$.

Explanation (part a)

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

Procedure in Excel:

Step 1: Select Data > Data Analysis> Regression from the menu bar.

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

Step 3: Enter the column $x$in the Input $X$
Range.

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

Step 5: Click the OK button.

Output from Excel:

The test statistic value is $1.155$, and the $p$value is $0.4544$, according to the Excel output.

Step 3:Conclution (part a)

Use $\alpha =0.10$as your significance threshold.

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

That is, $p$-value$(=0.4544)>\alpha (=0.10)$

According to the rejection rule, there is insufficient evidence to reject the null hypothesis $\left(H_0\right)$ at $\alpha =0.10$.

As a result, the data is insufficient to establish that $x$ is beneficial for forecasting $y$.

Step 4:Given Information (part b)

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

Step 5:Explanation (part b)

The slope of the regression line has a $90\%$ confidence interval of $(-8.9357,12.9357)$, according to the Excel output in part (a).