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 will be randomly selected, and each supermarket will display 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 will be recorded. When the experiment is carried out. the data in Table 14.11 are obtained. Here we assume that the set of sales amounts for each display height is a sample that has been randomly selected from the population of all sales amounts that could be obtained (at supermarkets of the given sales potential) when using that display height. To compare the population mean sales amounts µB, µM, and µT that would be obtained by using the bottom, middle, and top display heights, we use the following dummy variable regression model:
y = βB + BMDM + BTDT + ε
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. Figure 14.20 on the next page presents the MINITAB output of a regression analysis of the bakery sales study data using this model.1
FIGURE 14.20
MINITAB Output of a Dummy Variable Regression Analysis of the Bakery Sales Data in Table 14.11

a. By using the definitions of the dummy variables, show that
µB - βB µm = βB + βM µT = βB + βT
b. Use the overall P statistic to test H0: βM = βT = 0, or, equivalently, H0: µB = µM = µT. Interpret the practical meaning of the result of this test.
c. Show that your results in part a. imply that
µM - µB - βM µT - µB = βT µM - µT = βM - βT
Then use the least squares point estimates of the model parameters to find a point estimate of each of the three differences in means. Also, find a 95 percent confidence interval for and test the significance of each of the first two differences in means. Interpret your results.
d. Find a point estimate of mean sales when using a middle display height, a 95 percent confidence interval for mean sales when using a middle display height, and a 95 percent prediction interval for sales at an individual supermarket that employs a middle display height (see the bottom of the MINITAB output in Figure 14.20).
e. Consider the following alternative model
y = βT + βBDB + βMDM + ε
Here Dβ equals 1 if a bottom display height is used and 0 otherwise. The MINITAB output of the least squares point estimates of the parameters of this model is as follows:

Since βM expresses the effect of the middle display height with respect to the effect of the top display height, βM equals µM - µT. Use the MINITAB output to calculate a 95 percent confidence interval for and test the significance of µM - µT Interpret your results.

Transcribed Image Text:

The regression equation is Bakery Sales # 55.8 + 21.4 DXriddle- 4.30 DTop Predictor Coef SE Coef Constant 55.800 1.013 55.07 0.000 DMİddIt 21.400 1.433 14.930.000 DTop -4.300 1.433 3.00 0.009 S 2.48193 R-Sq 96.1% R-Sq (adj) -95.6% " - Analysis of Variance Source Regression Residual Error 15 92.4 6.2 Total DF MS 2 2273.9 1136.9 184.57 0.000 17 2366.3 Values of Predictors for New Obs New Ob DMİdd1@ DTop Predicted Values for New Obaervations New Oba Fit SE Fit 95 CI 95% PI 1 77.200 1.013 75.040, 79.360) (71.486 82.914 Predictor Constant DBottom DMiddle Coef 51.500 4.300 25.700 SE Coef 1.013 50.83 0.000 1.433 1.433 17.94 0.000 3.00 0.009