1. Enterprise industries Produces Fresh, a brand of liquid laundary detergent. In order to more effectively manage its inventory and make revenue projections, the company would like to better predict demand of 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). y = the demand for the large size bottle of Fresh(in hundreds of thousands of bottles) in sales period. x1 = the price (in dollars) of Fresh as offered by Enterprise Industries in the sales. x2 = the average industry price(in dollars) of competitors’s similar detergents in the sales period. x3 = Enterprize Industries’ advertising expenditure (in hundreds of thousands of bottles) to promote Fresh in the sales period. x4 = x2 − x1 = the “price difference” in the sales period. Suppose that Enterprise Industries believes on theoritical grounds that the single indepen- dent variable x4 adequately describes the effects of x1 and x2 on y. (a) To ultimately increase the demand for Fresh, Enterprise Industries’ marketing depart- ment is comparing the effectiveness of three different advertising companions. The campions are denoted by A, B, C. For example. 1 if the advertising campion is A DA = 0 if the advertising campion is not A Similar definition is used in DB and DC columns. Create one column (with factors A, B and C) by merging the columns “DA”, “DB”, and “DC”. Assign the name of the newly created column as “D”, where D is a factor variable with three factors A, B and C. (b) Present scatter plots of y versus x4 and y versus x3. (c) Find regression model of y on x4. (d) Find quadratic model of y on x3. What is the rational of using quadratic model ? (e) Find combined model β0 + β1x4 + +β2x3 + β3x23. (f) Compare the effectiveness of advertising campions A, B, and C. Fit the following model y = β0 + β1x4 + +β2x3 + β3x23 + β4x4x3 + β5DB + β6DC + ε. (g) Is there a importance (using part f) of interaction between x4 and x3? (h) (use part f) If we set x4 = d and x3 = a find expression for mean response μ[d,a,A], μ[d,a,B] and μ[d,a,C]. Verify that μ[d,a,B] −μ[d,a,A] = β5; μ[d,a,C] −μ[d,a,A] = β6, μ[d,a,C] − μ[d,a,B] = β6 − β5. STAT 430/530 Take Home FInal Exam - Page 2 of 3 (i) Use the least square point estimates of the model parameters to find a point estimate of each of the three differences in means. Also, find a 95% confidence interval and test the significance of each of first two differences in means. (j) Find estimated demand fir fresh laundry detergent when x4 = 0.20 and x3 = 6.5 Identify and interpret 95% confidence interval for mean demand and 95% prediction interval for individual demand when x4 = 0.20 and x3 = 6.5, and campaign C is used. (k) Fit the following regression mode: y = β0 + β1x4 + +β2x3 + β3x23 + β4x4x3 + β5DB + β6DC + β7x3DB + β8x3DC + ε. When there are many independent variables in a model, we might not be able to trust the p-values to the p-values to tell us what is important. This is because of a condition called multicollinearity. Discuss about possible multicollinearity. (l) In the above model note, however, that the p-value for x3DC is the smallest of the p-values for the independenet variables DC, DB, X3DB, and x3DC. This might be as “some evidence” that “some interaction” exist between advertising expenditure and advertising campaign. Use advertising espenditure x3 and verify that μ[d,a,C] − μ[d,a,A] =β6+β8aandμ[d,a,C]−μ[d,a,B] =β6−β−5+β−8a−β7a. (m) Find the values of μ[d,a,C] − μ[d,a,A] and μ[d,a,C] − μ[d,a,B] when a = 6.2 and a = 6.6. Discuss why these results imply that the larger the advertising expenditure a is, the larger is the improvement in mean sales that is obtained by using advertising campaign C rather than advertising campaign A or B. (n) Display scatter plot of y versus x3. Plot the linear models (part k) for three adver- tising campaign with scatter plot in the same graph. Does the graph supports your conclusion in part m.