Question

Gas Guzzelrs. Use the data on the WeissStats site for gas mileage and engine displacement for $121$vehicles referred to in Exercise 14.41

a. Decide whether you can reasonably apply the regression $t$-test. If so, then also do part (b).

b. Decide, at the $5\%$significance level, whether the data provide sufficient evidence to conclude that the predictor variable is useful for predicting the response variable.

Short answer

a). As a result, the linear model is ineffective. As a result, for the variables Mpg and disp, regression inferences of assumption $1$is broken.

b). As a result, the regression $t$-test is not appropriate for the data at hand.

Step-by-step solution

Construction of residual plot using MINITAB (Part a)

Step 1: From the drop-down menu, select Stat $>$Regression $>$Regression.

Step 2: In the Response column, type MPG.

Step 3: In Predictors, type Disp into the columns.

Step 4: In Graphs, under Residuals vs the variables, type the columns Disp.

Step 5: Click the OK button.

MINITAB Output:

Construction of normal probability of residuals using MINITAB

Step 1: From the drop-down menu, select Stat $>$Regression $>$Regression.

Step 2: In the Response column, Enter MPG.

Step 3: In Predictors, Enter Disp into the columns.

Step 4: In Graphs, Enter normal probability plot of residuals.

Step 5: Click the OK button.

MINITAB Output:

Regression inferences assumptions

The following is the regression inferences assumptions:

Line of population regression:

• For each value $X$of the predicator variable, the response variable conditional mean $Y$is ${\beta }_0+{\beta }_{1}X$.

Standard deviations are equal:

• The response variable's $\left(Y\right)$standard deviation is the same as the explanatory variable's $\left(X\right)$standard deviation. $\sigma$is standard deviation

Typical populations include:

• The response variable follows a normal distribution.

Observations made independently:

• The response variable observations are unrelated to one another.

To examine whether the graph shows a violation of one or more of the regression inference assumptions.

To examine whether the graph shows a violation of one or more of the regression inference assumptions.

- There is a concave upward curve in the residual plot vs engine displacement.

- The presence of outliers in the data is evident from the normal probability plot of residuals and the residual plot. As a result, the linear model is ineffective.

As a result, for the variables Mpg and disp, regression inferences of assumption $1$ is broken.

Explanation for Part (b)

• Part (a) clearly shows that the regression inference assumptions have been broken.
• As a result, it is impossible to determine whether the data are sufficient to establish that the predictor variable is effective for predicting the responder variable.
• That is, the regression $t$- test is not appropriate for the data at hand.