Question

Q7. 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).

Summary of Fit
RSquare	0.940514
RSquare Adj	0.928121
Root Mean Square Error	0.18111
Mean of Response	8.382667
Observations (or Sum Wgts)	30

Analysis of Variance
Source	DF	Sum of Squares	Mean Square	F Ratio
Model	5	12.445759	2.48915	75.8909
Error	24	0.787178	0.03280	Prob > F
C. Total	29	13.232937		<.0001*

Term		Estimate	Std Error	t Ratio	Prob>|t|	Lower 95%	Upper 95%
Intercept		8.783477	1.806097	4.86	<0.0001*	5.0558749	12.511079
Price(X1)		−2.612300	0.471357	−5.54	<0.0001*	−3.58513	−1.639466
IndPrice(X2)		1.5396615	0.223996	6.87	<0.0001*	1.0773571	2.0019659
AdvExp(X3)		0.5034394	0.096329	5.23	<0.0001*	0.304627	0.702252
DB		0.265420	0.083525	3.18	0.0040530*	0.4378066	0.0930342
DC		0.2466072	0.081402	3.03	0.0057851*	0.0786017	0.4146128

	Predicted Demand	Lower 95% Mean Demand	Upper 95% Mean Demand	Lower 95% Indiv Demand	Upper 95% Indiv Demand
31	8.599070456	8.477963157	8.720177756	8.206157656	8.991983257

y = β₀ + β₁x1+ β₂x2 + β₃x3+ 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. Interpret your results. (Round your answers to 4 decimal places.) Point estimate of the effect on the mean of campaign B compared to campaign A is b4 = ____ The 95% confidence interval = [ ___ , ___ ]   Point estimate of the effect on the mean of campaign C compared to campaign A is b5 = ____   he 95% confidence interval = [ ___ , ___ ] Campaign ____ is probably most effective even though intervals overlap

(b) The prediction results at the bottom of the output correspond to a future period when Fresh’s price will be x1

= 3.73, the average price of similar detergents will be x2 = 3.92, Fresh’s advertising expenditure will be x3 = 6.51, and advertising campaign C will be used. Show how yˆ = 8.59907 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 x1 = 3.73, x2 = 3.92, x3= 6.51, and campaign C is used. (Round your answers to 5 decimal places.)   y-hat _____ Confidence interval [ __ , __ ]    Prediction interval [ __ , __ ]

(c) Consider the alternative model      y = β₀ + β₁x1+ β₂x2 + β₃x3+ 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.

Answer 1

(ai) The point estimate of the effect on the mean of campaign B compared to campaign A is

b4 = 0.2695
p-value = 0.0007
Since the p-value = 0.0007 < alpha = .05, reject Ho
Based on the sample data, the demand for Fresh using advertising campaign B is significantly greater that the demand for Fresh using advertising campaign A.

(aii) The 95% confidence interval for beta4

(0.1262, 0.4128)
We are 95% confident that the average demand for Fresh using advertising campaign B is between 0.1262 and 0.4128 HIGHER (since the CI is entirely positive), on average, than using using advertising campaign A.

(aiii) The point estimate of the effect on mean of campaign C compared to campaign A is

b5 = 0.4396
p-value = 1.8506E-06 = 0.0000018506 --> p-value = 0.0000
Since the p-value = 0.0000 < alpha = .05, reject Ho
Based on the sample data, the demand for Fresh using advertising campaign C is significantly greater that the demand for Fresh using advertising campaign A.

(aiv) The 95% confidence interval for beta5

(0.2944, 0.5847)
We are 95% confident that the average demand for Fresh using advertising campaign C is between 0.2944 and 0.5847 HIGHER (since the CI is entirely positive), on average, than using using advertising campaign A.

(av) Campaign (Click to select)ABC is probably most effective even though intervals overlap.

Campaign C is is probably most effective, because b5 = 0.4396 > b4 = 0.2695.

(b)

CI: (8.51380, 8.71862)
We are 95% confident the average demand over ALL 4-week sales periods when the price for Fresh is 3.70, when the average industry price of similar detergents is 3.90 and and when 6.5 is spent on Advertisting using campaign C is somewhere between 8.51380 and 8.71862.

PI: (8.28958, 8.94285)
We are 95% confident the average demand over a (ONE) individual 4-week sales period when the price for Fresh is 3.70, when the average industry price of similar detergents is 3.90 and and when 6.5 is spent on Advertisting using campaign C is somewhere between 8.51380 and 8.71862.

(ci)

The regression equation is: yhat = 8.9849 + -2.7680x1 + 1.6667x2 + 0.4927x3 - 0.2695x4 + 0.1701x5

(cii)

b5 = 0.1701 <-- The effect of campaign C compared to campaign B

Note: Whatever outcome is left out of the equation (in this case campaign B), the other dummy variable coefficient are relative to that outcome.

(ciii)

95% CI for beta5: (0.0320, 0.3081)

(civ)
p-value = 0.0179
Since the p-value = 0.0179 < alpha = .10, reject Ho --> significant at alpha = .10
Since the p-value = 0.0179 < alpha = .05, reject Ho --> significant at alpha = .05
There is strong evidence that beta5 _is_ greater than 0.