Question

Short answer

The slope of the population regression line is not 0 and hence the variable, $x$is not useful for predicting the variable, $y$.

Interpretation: We can be $90\%$confident that the slope of the population regression line is somewhere between $-0.782$and $1.282$.

Step-by-step solution

Given information

The equation is $\hat{y}=1.75+0.25x$

We have to find that the predicting variable and the confidence level.

Determine the hypothesis 

STEP 1: The null and alternative hypotheses are

${\mathrm{H}}_0:{\beta }_1=0$( $x$is not useful for predicting $y$)

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

STEP 2: Decide on the significance level, $\alpha$.

We are to perform the hypothesis test at the $10\%$significance level, or $\alpha =0.10$.

Computation table

Determine the equation

$$\begin{gathered} S_{xy}=\sum x_i{y}_i-\left(\sum x_i\right)\left(\sum y_i\right)/n \\ =34-\left(12\right)\left(10\right)/4 \\ =34-120/4 \\ =34-30 \\ =4 \\ {S}_{xx}=\sum x_i^2-{\left(\sum x_i\right)}^2/n \\ =52-(12{)}^2/4 \\ =52-144/4 \\ =52-36 \\ =16 \end{gathered}$$

The total sum of square SST

$$\begin{gathered} S_{yy}=\sum y_i^2-{\left(\sum y_i\right)}^2/n \\ =30-(10{)}^2/4 \\ =30-100/4 \\ =30-25 \\ =5 \end{gathered}$$

Since the P-value $=0.5527>\alpha =0.10$. We do not reject our null hypothesis $H_0.$

At the $10\%$significance level, the data do not provide sufficient evidence to conclude that the slope of the population regression line is not 0 and hence the variable, $x$is not useful for predicting the variable, $y$.