Consider the QHIC data in Figure 13.21 (page 556). When we performed a regression analysis of these data by using the simple linear regression model, plots of the model's residuals versus x (home value) and Å· (predicted upkeep expenditure) both fanned out and had a "dip." or slightly curved appearance (see Figure 13.22, page 558). In order to remedy the indicated violations of the constant variance and correct functional form assumptions, we transformed the dependent variable by taking the square roots of the upkeep expenditures. An alternative approach consists of two steps. First, the slightly curved appearance of the residual plots implies that it is reasonable to add the squared term x~ to the simple linear regression model. This gives the quadratic regression model
y = β0 + β1x + β2x2 + ε
The MINITAB output below shows that the plot of this model's residuals versus x fans out, indicating a violation of the constant variance assumption.

To remedy this violation, we (in the second step) divide all terms in the quadratic model by x. This gives the transformed model

The MINITAB regression output and a residual plot versus x for this model are as follows:

a. Does the residual plot indicate the constant variance assumption holds for the transformed model?
b. Consider a home worth $220,000. We let µ0 represent the mean yearly upkeep expenditure for all homes worth $220,000, and we let y0 represent the yearly upkeep expenditure for an individual home worth $220,000. The bottom of the MINITAB output tells us that y0/220 = 5.635 is a point estimate of µ0/220 and a point prediction of y0/220. Multiply this result by 220 to obtain Å·. Multiply the ends of the confidence interval and prediction interval shown on the MINITAB output by 220. This will give a 95 percent confidence interval for µ0 and a 95 percent prediction interval for y0. Suppose that QHIC has decided to send a special, more expensive advertising brochure to any home whose value makes QHIC 95 percent confident that the mean upkeep expenditure for all homes having this value is at least $1.000. Will a home worth $220,000 be sent a special brochure?

Residuals Versus Value (response is Upkeep) 400 300 200 100 0 100 200 300 50 100 150 200 250 300 Value Predictor Noconstant 1/Value one Value Coef SE Coef Residuals Versus Value 83.20 -0.64 0.524 1.321 2.58 0.014 0.011224 0. 004627 2.43 0.020 -53.50 3.409 2 1 Predicted Values for New Observations Fit 5.635 (5.306, 5.964) (3.994, 7.276) 95% CI 95% PI 1 -53.50 --+ 3.409 + .01 1224(220) = 5.635 220 220 2 50 100 150 200 250 300 Value