Question

(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.

Short answer

(a) The data does not provide sufficient evidence to conclude that $x$is useful for predicting $y.$

(b) The $90\%$confidence interval for the slope of the regression line is$(-8.9357,12.9357)$

Step-by-step solution

Given information (a) 

The equation is$\hat{y}=1+2x$.
We have to find that the sufficient evidence.

Determine the hypothesis and the rejection rule

The appropriate hypotheses are given below:

Null hypothesis:

$H_0:{\beta }_1=0$

That is, the variable $x$is not useful for predicting $y$.

Alternative hypothesis:

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

That is, the variable $x$is useful for predicting $y.$

Rejection Rule:

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

Find the test statistic and p-value by using Excel.

Excel procedure:

Step 1: Choose Data > Data Analysis > Regression.

Step 2: In the Input Y Range, enter the column $y$.

Step 3: In the Input X Range, enter the column $x$.

Step 4: Choose your confidence level, enter$90$

Step 5: Click "OK."

Excel output:

From the Excel output, the value of the test statistic is $1.155$and the $p$-value is$0.4544.$

Determine the sufficient evidence 

Conclusion:

Use the significance level, $\alpha =0.10$.

Here, $p$-value is greater than the level of significance.

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

Therefore, by the rejection rule, it can be concluded that there is no evidence to reject the null hypothesis $\left(H_0\right)$at $\alpha =0.10$.

Thus, the data does not provide sufficient evidence to conclude that $x$is useful for predicting$y$

(b) Determine the confidence interval 

Compute the $90\%$confidence interval for the slope of the regression line by using Excel.

From the Excel output in part (a), the $90\%$confidence interval for the slope of the regression line is $(-8.9357,12.9357)$.