A short-term study was conducted to investigate the effect of mean monthly daily temperature x1 and cost per kilowatt-hour x2 on the mean daily consumption of electricity (in kilowatt-hours, kWh) per household. The investigators expected the demand for electricity to rise in cold weather (due to heating), fall when the weather was moderate, and rise again when the temperature rose and there was need for air-conditioning. They expected demand to decrease as the cost per kilowatt-hour increased, reflecting greater attention to conservation. Data were available for 2 years, a period in which the cost per kilowatt-hour x2 increased because of the increasing cost of fuel. The company fitted the model
E(y) = β0 + β1x1 + β2x21 + β3x2 + β4x1x2 + β5x21x2
to the data shown in the table. The Excel printout for this multiple regression problem is also provided.

Excel output for Exercise 13.28

a. Do the data provide sufficient evidence to indicate that the model contributes information for the prediction of mean daily kilowatt-hour consumption per household? Test at the 5% level of significance.
b. Graph the curve depicting Å· as a function of temperature x1 when the cost per kilowatt hour is x2 = 8¢. Construct a similar graph for the case when x2 = 10¢ per kilowatt-hour. Are the consumption curves different?
c. If cost per kilowatt-hour is unimportant in predicting use, then you do not need the terms involving x2 in the model. Therefore, the null hypothesis H0: x2 does not contribute information for the prediction of y is equivalent to the null hypothesis H0: β3 = β4 = β5 = 0 (if β3 = β4 = β5 = 0, the terms involving x2 disappear from the model). The MINITAB printout, obtained by fitting the reduced model
E(y) = β0 = β1x1 = β2x21
to the data, is shown here. Use the methods of Section 13.5 to determine whether price per kilowatt hour x2 contributes significant information for the prediction of y.
Excel output for Exercise 13.28

d. Compare the values of R2(adj) for the two models fit in this exercise. Which of the two models would you recommend?

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Price per|Daily Temperature kWh, x2and Consumption Mean Daily Consumption (kWh) per Household 31 34 39 4247 56 Mean daily Mean daily Mean daily Mean daily 8p temperature F, X6 66 68 71 75 78 55 49 46 47 40 43 41 46 44 51 62 73 32 36 3942 48 56 62 66 68 72 75 79 50 44 42 42 38 40 39 44 40 44 50 55 consumption, y 10p temperature, x consumption, y SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square 0.870 Standard Erro 2.908 Observations 0.948 0.898 24 ANOVA MS F Significance F 0.000 Regresslon Residual Total 1346.448 269.290 31.852 18 23 152.177 8.454 1498.625 8 Coefficients Standard Eror t Stat Pvalue 83.064 3.920 0.001 3.239 -3.515 0.002 0.029 3.854 0.001 9.224 -2.352 0.030 0.359 2.433 0.026 0.003 -2.7230.014 325.606 -11.383 0.113 21.699 0.873 -0.009 x1-sq xisqx2 SUMMARY OUTPUT Regression Statistics Multple R R Square Adjusted R Square Standard Emor Observations 0.8304 0.6896 0.6601 4.7063 24 ANOVA df MS F Significance F 0.000 Regression Residual Total 1033.490 516.745 23.330 21 465.135 22.149 1498.625 Coefficients Standard Error 14.876 23 Intercept x1 x1-s0 130.009 -3.502 0.033 t Stat P-value 8.740 0.000 0.579 -.049 0.000 6.349 0.000 0.005