Recall from Exercise 14.5 that Enterprise Industries has observed the historical data in Table 14.5 concerning y (demand for Fresh liquid laundry detergent), x1 (the price of Fresh), x2 (the average industry price of competitors’ similar detergents), and x3 (Enterprise Industries’ advertising expenditure for Fresh). To ultimately increase the demand for Fresh, Enterprise Industries’ marketing department is comparing the effectiveness of three different advertising campaigns. These campaigns are denoted as campaigns A, B, and C. Campaign A consists entirely of television commercials, campaign B consists of a balanced mixture of television and radio commercials, and campaign C consists of a balanced mixture of television, radio, newspaper, and magazine ads. To conduct the study, Enterprise Industries has randomly selected one advertising campaign to be used in each of the 30 sales periods in Table 14.5. Although logic would indicate that each of campaigns A, B, and C should be used in 10 of the 30 sales periods, Enterprise Industries has made previous commitments to the advertising media involved in the study. As a result, campaigns A, B, and C were randomly assigned to, respectively, 9, 11, and 10 sales periods. Furthermore, advertising was done in only the first three weeks of each sales period, so that the carryover effect of the campaign used in a sales period to the next sales period would be minimized. Table 14.12 lists the campaigns used in the sales periods. To compare the effectiveness of advertising campaigns A, B, and C, we define two dummy variables. Specifically, we define the dummy variable DB to equal 1 if campaign B is used in a sales period and 0 otherwise. Furthermore, we define the dummy variable DC to equal 1 if campaign C is used in a sales period and 0 otherwise. Figure 14.21 presents the Excel add-in (MegaStat) output of a regression analysis of the Fresh demand data by using the model
y = β0 + β1x1 + β2x2 + β3x3 + β4DB + β5DC + ε
a. Because this model does not use a dummy variable to represent advertising campaign A, the parameter β4 represents the effect on mean demand of advertising campaign B compared to advertising campaign A, and the parameter β5 represents the effect on mean demand of advertising campaign C compared to advertising campaign A. (1) Use the regression output to find and report a point estimate of each of the above effects and to test the significance of each of the above effects. (2) Find and report a 95 percent confidence interval for each of the above effects. (3) Interpret your results.
b. The prediction results at the bottom of the output correspond to a future period when Fresh’s price will be x1 = 3.70, the average price of similar detergents will be x2 = 3.90, Fresh’s advertising expenditure will be x3 6.50, and advertising campaign C will be used. (1) Show (within rounding) how 8.61621 is calculated. (2) Find, report, and interpret a 95 percent confidence interval for mean demand and a 95 percent prediction interval for an individual demand when x1 = 3.70, x2 = 3.90, x3 = 6.50, and campaign C is used.
c. Consider the alternative model
y = β0 + β1x1 + β2x2 + β3x3 + β4DA + β5DC = ε
Here DA equals 1 if advertising campaign A is used and equals 0 otherwise. Because this model does not use a dummy variable to represent advertising campaign B, the parameter β5 in this model represents the effect on mean demand of advertising campaign C compared to advertising campaign B. The Excel output of the least squares point estimates of the parameters of the alternative model are as follows.
Use the Excel output to (1) test the significance of the effect represented by β5 and (2) find a 95 percent confidence interval for b β5. (3) Interpret your results.