The magazine Consumer Reports publishes information on automobile gas mileage and variables that affect gas mileage. In one issue, data on gas mileage (in mpg) and engine displacement (in liters L) were published for \(121\) vehicles. Those data are stored on the Weiss Stats site.

a. obtain and interpret the standard error of the estimate.

b. obtain a residual plot and a normal probability plot of the residuals.

c. decide whether you can reasonably consider Assumptions \(1-3\) for regression inferences met by the two variables under considerations.

Given,

Find the standard error of the estimate by using MINITAB.

MINITAB procedure:

Step 1: Choose Stat > Regression > Regression.

Step 2: In Response, enter the column MPG.

Step 3: In Predictors, enter the columns DISP.

Step 4: Click OK.

MINITAB output:

Regression Analysis: MPG versus DISP

From the MINITAB output, the standard error of the estimate is \(2.248\)

Interpretation:

The predicted scores in the sample differs on average from the observed scores by \(2.248\).

Construct the residual plot by using MINITAB.

MINITAB procedure:

Step 1: Choose Stat > Regression > Regression.

Step 2: In Response, enter the column MPG

Step 3: In Predictors, enter the columns DISP.

Step 4: In Graphs, enter the columns DISP variables under Residuals versus the variables.

Step 5: Click OK.

MINITAB output:

Construct the normal probability plot of residuals by using MINITAB.

MINITAB procedure:

Step 1: Choose Stat > Regression > Regression.

Step 2: In Response, enter the column MPG

Step 3: In Predictors, enter the columns DISP.

Step 4: In Graphs, select Normal probability plot of residuals.

Step 5: Click OK.

MINITAB output:

Therefore, residual plot and Normal probability plot of residuals are obtained.

The assumption for regression inferences is given below:

Population regression line:

The conditional mean of the response variable \((Y)\) is \(\beta _{0}+\beta _{1}X\), for each value \(X\) of predictor variable.

Equal standard deviation:

The standard deviation for the response variable \((Y)\) is same for the standard deviation for the explanatory variable \((X)\). The standard deviation is denoted as \(\sigma\).

Normal populations:

The distribution of the response variable follows normal.

Independent observations:

The observations of the response variable are independent of each other.

Check whether the graph suggests violation of one or more of the assumptions for the regression inferences.

• From the residual plot, it is clear that the residuals are fall in the horizontal band.
• From the normal probability plot of residuals, it is clear that the residuals are in the linear pattern.

Hence, the assumption 1 for the regression inferences is violated for the variables Mpg and disp. Moreover, the interpretation for standard error obtained in part (a) does not validate.