Cars (introduced in Chapter 19) The cases that make up this dataset are types of cars. For each of 318 types of cars sold in the United States during the 2011 model year, we have the combined mileage and the horsepower of the engine (HP). In previous exercises, we found that a model for the association of mileage and horsepower required taking logs of both variables. (We used base 10 logs.) The column Vehicle Type denotes whether the vehicle is a car, truck, or SUV. (This is a bit arbitrary; we classified station wagons and vans together with SUVs.)
(a) Plot the log10 of mileage on the log10 of horse- power for vehicles of all three types in one scatterplot. Use color-coding or distinct symbols to distinguish the groups. Does it appear that the relationship is different among the groups, or can you capture the association with a single simple regression?
(b) Add two dummy variables (one for trucks and one for SUVs) and their interactions with log10 of HP to the model. Does the fit of this model meet the conditions for the MRM? Comment on the consequences of any problem that you identify.
(c) Assume that the model meets the conditions for the MRM. Use the incremental F-test to assess the size of the change in R2 produced when the dummy variables and interactions are added to the model. (See the discussion of this test in Exercise 45.) Does the test agree with your visual impression? (The value of kfull for the model with dummy variables and interactions is 5, with 4 slopes added. You will need to fit the simple regression of log10 of mileage on log10 of horsepower to get its R2 statistic.)
(d) Compare the conclusion of the incremental F-test to those of the tests of the coefficients of the dummy variable and interaction separately. Do these agree? Explain the similarity or difference.
(e) What does the addition of dummy variables and interactions to the regression tell you about these vehicles? A plot might help of the several fits might help you.