ESM 206 Problem set 2
Solutions
Part A:
1) A regression of Highway MPG on weight in pounds has an estimated slope of -
0.0073. Thus a 100-pound reduction in weight should, on average, increase
mileage by 0.73 MPG.
2) The equation is
0 1i i i
H W
.
Variables Parameters Residual
The estimate of b0 is 51.58, the estimate of b1 is -0.0073, and the estimate of the
residual variance is 9.96. The 95% confidence interval for b0 is 48.10 to 55.05,
and for b1 is -0.0084 to -0.0062.
3) The interaction term shows how engine size affects the relationship between
weight and mileage. Since the parameter estimate is positive, increasing engine
size seems to decrease the negative effect of weight on mileage. Another way of
looking at it is that increasing engine size improves fuel economy after accounting
for car weight, and that this effect gets stronger the heavier the car is.
4) The fit improves slightly. The R2 goes from 0.65 to 0.68, and the F ratio for the
entire model goes from 171 to 194. Some slight curvature in the relationship is
eliminated, and the unusually large residuals at low weight are brought under
control. This makes sense, for I would expect that an increase in weight should
produce a proportional increase in fuel consumption, which is the inverse of
mileage. (Note that in metric countries, fuel efficiency is generally measured in
liters per 100 kilometers)
5) I first ran a model with weight, type, and the interaction between them. The P
values for the latter two were very large, so I removed the interaction (which had
the larger P). Then both weight and type were strongly significant (P < 0.0001;
figure 1). Thus, the different types have different inherent fuel efficiencies
(intercepts) but the rate at which fuel consumption increases with weight is the
same for all of them. This may have a lot to do with aerodynamics, for the vans
had the highest fuel consumption, given their weight.
Alternatively, the model with weight and the interaction describes the data
almost as well (F = 48.9, vs. 49.3 for the previous model). Vans again stand out,
having the steepest slope (figure 2). The reason both models do nearly equally
well is that the dominant effect is van’s increased consumption for their weight,
and that all the vans are heavy: there are no data on light vans that would tell us
whether the lines should be parallel.
The type-specific coefficients for both models are in table 1.
