# Question

The Tastee Bakery Company supplies a bakery product to many supermarkets in a metropolitan area. The company wishes to study the effect of the height of the shelf display employed by the supermarkets on monthly sales, y (measured in cases of 10 units each), for this product. Shelf display height has three levels— bottom (B), middle (M), and top (T). For each shelf display height, six supermarkets of equal sales potential are randomly selected, and each supermarket displays the product using its assigned shelf height for a month. At the end of the month, sales of the bakery product at the 18 participating stores are recorded, and the data in Table 14.18 are obtained. To compare the population mean sales amounts μB, μM, and m T that would be obtained by using the bottom, middle, and top display heights, we use the following dummy variable regression model: y = βB + βMDM + βTDT + ε, which we call Model 1. Here, DM equals 1 if a middle display height is used and 0 otherwise; DT equals 1 if a top display height is used and 0 otherwise.

a. Because the expression βB + βMDM + βTDT represents mean monthly sales for the bakery product, the definitions of the dummy variables imply, for example, that μT = βb + βM(0) + βT(1) = βB + βT. (1) In a similar fashion, show that m μB = βb and μM + βB + βM.

(2) By appropriately subtracting the expressions for μB, μM, and μT, show that μM – μB = βM, μT – μB = βT, and μM – μT = βM – βT.

b. Use the overall F statistic in Figure 14.36(a) to test H0: βM = βT, or, equivalently, H0: m μB = μM = μT. Interpret the practical meaning of the result of this test.

c. Consider the following two differences in means: μM – μB = βM and μT – μB = βT. Use information in Figure 14.36(a) to (1) find a point estimate of, (2) test the significance of, and (3) find a 95 percent confidence interval for each difference.

d. Consider the following alternative model: y = βT + βBDB + βMDM + ε, which we call Model 2. Here, DB equals 1 if a bottom display height is used and 0 otherwise. This model implies that μM – μT = βM. Use information in Figure 14.36(b) to (1) find a point estimate of, (2) test the significance of, and (3) find a 95 percent confidence interval for μM – μT = βM. Interpret your results.

e. Show by hand calculation that both Models 1 and 2 give the same point estimate y = 77.2 of mean monthly sales when using a middle display height.

f. Use information in Figure 14.36(c) to find (1) a 95 percent confidence interval for mean sales when using a middle display height, and (2) a 95 percent prediction interval for individual sales during a month at a supermarket that employs a middle display height.

a. Because the expression βB + βMDM + βTDT represents mean monthly sales for the bakery product, the definitions of the dummy variables imply, for example, that μT = βb + βM(0) + βT(1) = βB + βT. (1) In a similar fashion, show that m μB = βb and μM + βB + βM.

(2) By appropriately subtracting the expressions for μB, μM, and μT, show that μM – μB = βM, μT – μB = βT, and μM – μT = βM – βT.

b. Use the overall F statistic in Figure 14.36(a) to test H0: βM = βT, or, equivalently, H0: m μB = μM = μT. Interpret the practical meaning of the result of this test.

c. Consider the following two differences in means: μM – μB = βM and μT – μB = βT. Use information in Figure 14.36(a) to (1) find a point estimate of, (2) test the significance of, and (3) find a 95 percent confidence interval for each difference.

d. Consider the following alternative model: y = βT + βBDB + βMDM + ε, which we call Model 2. Here, DB equals 1 if a bottom display height is used and 0 otherwise. This model implies that μM – μT = βM. Use information in Figure 14.36(b) to (1) find a point estimate of, (2) test the significance of, and (3) find a 95 percent confidence interval for μM – μT = βM. Interpret your results.

e. Show by hand calculation that both Models 1 and 2 give the same point estimate y = 77.2 of mean monthly sales when using a middle display height.

f. Use information in Figure 14.36(c) to find (1) a 95 percent confidence interval for mean sales when using a middle display height, and (2) a 95 percent prediction interval for individual sales during a month at a supermarket that employs a middle display height.

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