Question

An investigation of the relationship between traf. fic flow \(x\) (thousands of cars per \(24 \mathrm{hr}\) ) and lead content \(y\) of bark on trees near the highway (mg/g dry weight) yielded the accompanying data. A simple linear regression model was fit, and the resulting estimated regression line was \(\hat{y}=28.7+33.3 x .\) Both residuals and standardized residuals are also given. a. Plot the \((x\), residual \()\) pairs. Does the resulting plot suggest that a simple linear regression model is an appropriate choice? Explain your reasoning. b. Construct a standardized residual plot. Does the plot differ significantly in general appearance from the plot in Part (a)?

Short answer

A simple linear regression model might be appropriate if the residuals scatter randomly around the horizontal axis in the residual plot. Standardized residuals are used to check whether the variances of the raw residuals stay constant or not. The comparison of the two plots helps to verify this.

Step-by-step solution

Understand residuals

The residuals in a regression model are the difference between the observed value of the target variable (y) and the predicted value ( \( \hat{y} \) ). They are used to understand the discrepancy between the model prediction and the actual result.

Plot (x, residual) pairs

To plot the residuals, create a scatter plot where the x-axis represents the traffic flow \( x \) and the y-axis represents the residuals. Usually, if the regression model is a good fit, residuals should randomly scatter around the horizontal axis.

Interpreting the residual plot

If the points in the plot are randomly dispersed around the horizontal axis, a linear regression model is appropriate for the data. Otherwise, if there are clear patterns (like curvilinear patterns), then the linear model might not be an appropriate choice.

Understand standardized residuals

A standardized residual plot has the same concept as the residual plot. However, instead of using raw residuals, it uses the standardized version of the residuals, which take into consideration the variability of the residuals.

Construct a standardized residual plot

In this step, create a scatter plot where the x-axis represents the traffic flow (\( x \)) and the y-axis represents the standardized residuals. This plot will be used to compare with the plot in Part (a).

Comparing the residual plots

Compare the two plots. If the plots do not show any significant differences, this would typically mean that the standardization process did not reveal any new information about the residuals and that the model's residuals are homoscedastic (constant variance).