7. (Continues Exercise 7 in Section 8.3.) To try to improve the prediction of FEV1, additional independent variables are included in the model. These new variables are Weight (in kg), the product (interaction) of Height and Weight, and the ambient temperature (in ◦C). The following MINITAB output presents results of fitting the model FEV1 = β0 + β1 Last FEV1 + β2 Gender + β3 Height + β4 Weight + β5 Height · Weight + β6 Temperature + β7 Pressure + ε The regression equation is FEV1 = —0.257 + 0.778 Last FEV — 0.105 Gender + 1.213 Height — 0.00624 Weight + 0.00386 Height*Weight — 0.00740 Temp — 0.00148 Pressure Predictor Coef StDev T P Constant —0.2565 0.7602 —0.34 0.736 Last FEV 0.77818 0.05270 14.77 0.000 Gender —0.10479 0.03647 —2.87 0.005 Height 1.2128 0.4270 2.84 0.005 Weight —0.0062446 0.01351 —0.46 0.645 Height*Weight 0.0038642 0.008414 0.46 0.647 Temp —0.007404 0.009313 —0.79 0.428 Pressure —0.0014773 0.0005170 —2.86 0.005 S = 0.22189 R-Sq = 93.5% R-Sq(adj) = 93.2% Analysis of Variance Source DF SS MS F P Regression 7 111.35 15.907 323.06 0.000 Residual Error 157 7.7302 0.049237 Total 164 119.08

a. The following MINITAB output, reproduced from Exercise 7 in Section 8.3, is for a reduced model in which Weight, Height · Weight, and Temp have been dropped. Compute the F statistic for testing the plausibility of the reduced model. The regression equation is FEV1 = —0.219 + 0.779 Last FEV — 0.108 Gender + 1.354 Height — 0.00134 Pressure Predictor Coef StDev T P Constant —0.21947 0.4503 —0.49 0.627 Last FEV 0.779 0.04909 15.87 0.000 Gender —0.10827 0.0352 —3.08 0.002 Height 1.3536 0.2880 4.70 0.000 Pressure —0.0013431 0.0004722 —2.84 0.005 S = 0.22039 R-Sq = 93.5% R-Sq(adj) = 93.3% Analysis of Variance Source DF SS MS F P Regression 4 111.31 27.826 572.89 0.000 Residual Error 160 7.7716 0.048572 Total 164 119.08

b. How many degrees of freedom does the F statistic have?

c. Find the P-value for the F statistic. Is the reduced model plausible?

d. Someone claims that since each of the variables being dropped had large P-values, the reduced model must be plausible, and it was not necessary to perform an F test. Is this correct? Explain why or why not.

e. The total sum of squares is the same in both models, even though the independent variables are different. Is there a mistake? Explain.