Question

Enterprise Industries produces Fresh, a brand of liquid laundry detergent. In order to manage its inventory more effectively and make revenue projections, the company would like to better predict demand for Fresh. To develop a prediction model, the company has gathered data concerning demand for Fresh over the last 30 sales periods (each sales period is defined to be a four-week period). The demand data are presented in table concerning y (demand for Fresh liquid laundry detergent), x1x1 (the price of Fresh), x2x2 (the average industry price of competitors' similar detergents), and x3x3 (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 below. 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 Cwere 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 below 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 D_B 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. The table presents the Excel and Excel add-in (MegaStat) output of a regression analysis of the Fresh demand data by using the model

Historical Data Concerning Demand for Fresh Detergent
Sales Period	Price for Fresh, x1	Average Industry Price, x2	Advertising Expenditure for Fresh, x3	Demand for Fresh, y
1	3.89	3.86	5.56	7.36
2	3.74	4.05	6.72	8.52
3	3.75	4.35	7.21	9.20
4	3.78	3.78	5.52	7.55
5	3.61	3.85	7.03	9.33
6	3.66	3.86	6.54	8.25
7	3.65	3.78	6.71	8.77
8	3.81	3.81	5.23	7.86
9	3.80	3.66	5.27	7.14
10	3.80	4.04	6.06	8.06
11	3.90	4.11	6.53	7.80
12	3.96	4.05	6.25	8.13
13	3.75	4.17	7.02	9.15
14	3.71	4.27	6.96	8.89
15	3.74	4.16	6.85	8.98
16	3.80	4.17	6.87	8.86
17	3.70	4.22	7.10	9.26
18	3.82	4.35	7.04	9.03
19	3.79	4.14	6.88	8.78
20	3.85	3.76	6.58	7.90
21	3.81	3.77	6.23	7.65
22	3.73	3.69	6.09	7.28
23	3.73	3.92	6.51	8.04
24	3.51	3.66	7.03	8.57
25	3.60	4.15	6.89	8.76
26	3.64	4.25	6.89	9.26
27	3.70	3.61	6.52	8.22
28	3.70	3.74	5.76	7.60
29	3.81	3.80	5.81	7.90
30	3.74	4.25	6.86	9.23

Advertising Campaigns Used by Enter prise Industries
Sales Period	Advertising Campaign
1	B
2	B
3	B
4	A
5	C
6	A
7	C
8	C
9	B
10	C
11	A
12	C
13	C
14	A
15	B
16	B
17	B
18	A
19	B
20	B
21	C
22	A
23	A
24	A
25	A
26	B
27	C
28	B
29	C
30	C

Regression Statistics
Multiple R	.9698
R Square	.9405
Adjusted R Square	.9281
Standard Error	.1811
Observations	30

ANOVA	df	SS	MS	F	Significance F
Regression	5	12.4458	2.4892	75.8909	.0000
Residual	24	.7872	.0328
Total	29	13.2329

	Coefficients	Standard Error	t Stat	p-value	Lower 95%	Upper 95%
Intercept	8.7835	1.8061	4.863	.0001	5.0559	12.5111
Price X1	-2.6123	.4714	-5.542	.0000	-3.5851	-1.63947
Ind Price X2	1.5397	.2240	6.874	.0000	1.0774	2.0020
AdvExp X3	.5034	.0963	5.226	.0000	.3046	.7023
DB	-.2654	.0835	-3.178	.0041	-.4378	-.0930
DC	.2466	.0814	3.029	.0058	.0786	.4146

	Predicted values for: Demand using an Excel add-in (MegaStat)
	95% Confidence Interval			95% Prediction Interval
	Predicted	lower	upper	lower	upper	Leverage
	8.64161	8.51365	8.76957	8.24653	9.03669	.117

y = β₀ + β₁x1x1 + β₂ x2x2+ β₃x3x3 + β₄D_B + β₅D_C + ε

(a) In this model 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. 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. Also, find and report a 95 percent confidence interval for each of the above effects. (Round your answers to 4 decimal places.)

The point estimate of the effect on the mean of campaign B compared to campaign A is b4 = .

The 95% confidence interval = [, ].

The point estimate of the effect on mean of campaign C compared to campaign A is b5 = .

The 95% confidence interval = [, ].

(b) The prediction results at the bottom of the output correspond to a future period when Fresh’s price will be x1x1 = 3.70, the average price of similar detergents will be x2x2 = 3.90, Fresh’s advertising expenditure will be x3x3= 6.50, and advertising campaign C will be used. Show how yˆy^= 8.64161 is calculated. Then find, report, and interpret a 95 percent confidence interval for mean demand and a 95 percent prediction interval for an individual demand when x1x1 = 3.70, x2x2 = 3.90, x3x3 = 6.50, and campaign C is used. (Round your answers to 5 decimal places.)

yˆy^ =

Confidence interval = [, ]

Prediction interval = [, ]

(c) Consider the alternative model

y = β₀ + β₁x1x1 + β₂x2x2 + β₃x3x3 + β₄D_A + β₅D_C + ε

Here D_A equals 1 if advertising campaign A is used and equals 0 otherwise. Describe the effect represented by the regression parameter β₅.

β₅ = effect on mean of Campaign (Click to select)ABC compared to Campaign B.

(d) The Excel output of the least squares point estimates of the parameters of the model of part c is as follows.(Round your answer to 4 decimal places.)

	Coefficients	Standard Error	t Stat	p-value	Lower 95%	Upper 95%
Intercept	8.7835	1.8061	4.863	.0001	5.0559	12.5111
Price X1	-2.6123	.4714	-5.542	.0000	-3.5851	-1.6395
Ind Price X2	1.5397	.2240	6.874	.0000	1.0774	2.0020
AdvExp X3	.5034	.0963	5.226	.0000	.3046	.7023
DA	-.2654	.0835	-3.178	.0041	-.4378	-.0930
DC	.2466	.0814	3.029	.0058	.0786	.4146

Use the Excel output to test the significance of the effect represented by β₅ and find a 95 percent confidence interval for β₅. Interpret your results.

95 percent confidence interval for β5 [  ,  ]

β5 is significant at alpha = 0.1 and alpha = 0.05 because p-value = .

Thus there is strong evidence that β5 (Click to select)is greater thanis less thanis equal to 0.

Answer 1

(a)
From the output,
The point estimate of the effect on the mean of campaign B compared to campaign A is b4 = -.2654
The 95% confidence interval = [-.4378, -.0930].
The point estimate of the effect on mean of campaign C compared to campaign A is b5 = .2466
The 95% confidence interval = [.0786, .4146].

(b)
The estimated regression equation is,
yˆ= 8.7835 -2.6123x1 + 1.5397x2+ 0.5034x3 - 0.2654DB + 0.2466DC
For x1 = 3.70, x2 = 3.90, x3 = 6.50, DB = 0 and DC = 1
yˆ= 8.7835 -2.6123 * 3.70 + 1.5397 * 3.90 + 0.5034 * 6.50 - 0.2654 * 0 + 0.2466 * 1 = 8.6415
Confidence interval = [8.5137, 8.7696]
Prediction interval = [8.2465, 9.0367]

(c)
Since DC = 1, Campaign C is used, the answer is,
β5 = effect on mean of Campaign C compared to Campaign B.

(d)
95 percent confidence interval for β5 [0.0786 , 0.4146 ]
β5 is significant at alpha = 0.1 and alpha = 0.05 because p-value = 0.0058 (less than 0.1 and 0.05).
Since the confidence interval contains all positive values,
Thus there is strong evidence that β5 is greater than equal to 0.