Following are the age and price data for corvettes, use $\alpha =0.10$

presuming that the assumption for regression inference are met, decide at the specified significance level whether the data provide sufficient evidence to conclude that the predictor variable is useful for providing the response variable.

Given table is

and,$\alpha =0.10$. we have to find out whether the data provide sufficient evidence to conclude that the predictor variable is useful for providing the response variable.

The null and alternative hypothesis are as follows:

$H_0:{\beta }_1=0$

$H_{\alpha }:{\beta }_1\ne 0$

The formula of test statistic is$t=\frac{b_1}{s_e/\sqrt{S_{xx}}}$

We extract the regression output using Minitab software and instructions.

a. Separate the DISP and MPG values into columns designated DISP and MPG.

c. choose Graph? Scatter plot...

c. Click OK after selecting Simple scatter plot.

d. In the $Y$variables text box, type MPG.

e. In the$X$ variables text box, type DISP.

f. Select OK.

From the minitab regression output we found that

$b_1=-27.903$

$s_e=14.2465$

$n=10$

To find $s_{xx}$following calculation is needed.

$$\begin{gathered} S_{xx}=\sum x_i^2-\frac{{\left(\sum x_i\right)}^2}{n} \\ =199-\frac{(41{)}^2}{10} \\ =30.9 \end{gathered}$$

Test statistic is

localid="1651914221591" $$\begin{gathered} t=\frac{b_1}{s_e/\sqrt{S_{xx}}} \\ =\frac{-27.903}{14.2465/\sqrt{30.9}} \\ =-10.89 \end{gathered}$$

The P-value obtained with software is$0.000<\alpha =0.10$. Our null hypothesis $H_0$is rejected.

The data give adequate evidence to show that age$\left(x\right)$is a useful predictor of Corvette pricing$\left(y\right)$at a$10\%$significance level.