The data table gives various characteristics of 318 types of cars sold in the United States during the 2011 model years. Use the combined mileage rating as the response and the horsepower of the engine (HP) and the weight of the car (given in thousands of pounds) as explanatory variables.
(a) Examine the calibration plot and the plot of the residuals e on the fitted values Yn for the multiple regression of the mileage rating on HP and weight. Do these plots reveal any problems in the fit of this model?
(b) Revise the variables in the model so all are on the scale defined by log10. Has this common transformation fixed the problems identified in part (a)? To answer this question, refit the model on the log scale and consider the calibration and residual plots for the revised model.
This multiple regression has all variables on a log scale as in Exercise 43, but using base 10 logs. To interpret this model, recall that the sum of logs is the log of the product, log10 x + log10 y = log10 (xy) and that b log10 x = log10 xb. Hence, an equation of the form
Log 10 y = b0 + b1 log10 x1 + b2 log10 x2
Is equivalent to the product
Y = 10b0 x1b1 x2b2
The slopes in the log-log regression are exponents in a model that estimates y as the product of the explanatory variables raised to different powers. These powers are the partial elasticities of the response with respect to the predictors. (See Chapter 20 for a discussion of elasticities.)
(c) Is the partial elasticity for weight equal to zero? Estimate the partial elasticity from the multiple regression of log10 MPG on log10 HP and log10 weight.
(d) Compare the partial elasticity for weight (the slope for log10 weight in the multiple regression) to the marginal elasticity of MPG with respect to weight (the slope for log10 weight in a simple regression of log10 MPG on log10 weight). Are these estimates very different? Use confidence intervals to measure the size of any differences.
(e) Does the path diagram for this model offer an explanation for the differences in the confidence intervals found in part (d)? Explain.
(f) Based on your analysis, describe the effect of weight on MPG. Does it have an effect? Do heavier cars get worse mileage on average?