ProblemSet_3_08B solutions

ProblemSet_3_08B solutions - ESM 206 Problem set 3...

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ESM 206 Problem set 3 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, all else being equal, increase mileage by 0.73 MPG. 2) The equation is 0 1 i i i H W β ε = + + . Variables Parameters Residual The estimate of β 0 is 51.58, the estimate of β 1 is -0.0073, and the estimate of the residual variance is 9.96. The 95% confidence interval for β 0 is 48.10 to 55.05, and for β 1 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. 25 30 35 40 45 50 HighwayMPG Figure 1: Highway fuel efficiency (in miles per gallon) as a function of vehicle weight. 4) See Figure 1 for the plot. Modelling fuel consumption improves the fit slightly: the R 2 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
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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. Table 1: Type-specific intercepts (for the constant slope model) and slopes (for the constant intercept model) for the two models discussed in problem 5. Type Intercept Slope Compact 0.012648 7.266e-06 Large 0.010747 6.742e-06 Midsize 0.013061 7.389e-06 Small 0.012050 7.053e-06 Sporty 0.014257 7.802e-06 Van 0.018143 8.689e-06 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 HiwayGPM 1500 2000 2500 3000 3500 4000 Weight Compact Large Midsize Small Sporty Van Figure 2: Highway fuel consumption as a function of weight, for the model with identical slopes but different intercepts (this plot cannot be easily made in RCmdr; you
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ProblemSet_3_08B solutions - ESM 206 Problem set 3...

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