Statistics: Statistics: Multiple Linear Regression Please Solve problem. Page 1 of 4 Enterprise Industries produces Fresh,...

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Statistics: Statistics: Multiple Linear Regression Please Solve problem. Page 1 of 4 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 below concerning y (demand for Fresh liquid laundry detergent), x (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 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 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 og to equal 1 if campaign Bis used in a sales period and 0 otherwise. Furthermore, we define the dummy variable De to equal 1 if campaign C is used in a sales period and 0 otherwise. Table presents the JMP 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 Average Industry Advertising Expenditure Fresh, x1 1 3.85 Price, x2 for Fresh, x3 for Fresh, y Demand 3.80 5.50 7.38 2 3.75 4.00 6.75 8.51 3 3.70 4.30 7.25 9.52 4 3.70 3.70 5.50 7.50 3.60 3.85 7.00 9.33 6 3.60 3.80 6.50 8.28 7 3.60 3.75 6.75 8.75 8 3.80 3.85 5.25 7.87 9 3.80 3.65 5.25 7.10 10 3.85 4.00 6.00 8.00 11 3.90 4.10 6.50 7.89 12 3.90 4.00 6.25 8.15 13 3.70 4.10 7.00 9.10 14 3.75 4.20 6.90 8.86 15 3.75 4.10 6.80 8.90 16 3.80 4.10 6.80 8.87 17 3.70 4.20 7.10 9.26 18 3.80 4.30 7.00 9.00 19 3.70 4.10 6.80 8.75 20 3.80 3.75 6.50 7.95 21 3.80 3.75 6.25 7.65 22 3.75 3.65 6.00 7.27 23 3.70 3.90 6.50 8.00 24 3.55 3.65 7.00 8.50 25 3.60 4.10 6.80 8.75 26 3.65 4.25 6.80 9.21 27 3.70 3.65 6.50 8.27 28 3.75 3.75 5.75 7.67 29 3.80 3.85 5.80 7.93 30 3.70 4.25 6.80 9.26 Page 3 of 4 (a) In this model the parameter 84 represents the effect on mean demand of advertising campaign B compared to advertising campaign A, and the parameter es 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.) 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 the mean of campaign C compared to campaign A is b5 - The 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 z = 3.70, the average price of similar detergents will be 2 = 3.90, Fresh's advertising expenditure will be x3 = 6.50, and advertising campaign C will be used. Show how = 8.61621 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 z = 3.70, x2 = 3.90, x3 = 6.50, and campaign C is used. (Round your answers to 5 decimal places.) y-hat Confidence interval Prediction interval (c) Consider the alternative model y= 6 +61 +62 2 + 83 23 84DA +85C + Here D equals 1 if advertising campaign A is used and equals 0 otherwise. Describe the effect represented by the regression parameter 65. B5-effect on mean of Campaign compared to Campaign B. Price IndPrice PriceDif AdvExp Demand AdCamp DA DB DC X1 X2 X4 X3 Y 3.85 3.80 -0.05 5.50 7.38 3.75 4.00 0.25 6.75 8.51 3.70 4.30 0.60 7.25 9.52 3.70 3.70 0.00 5.50 7.50 3.60 3.85 0.25 7.00 9.33 3.60 3.80 0.20 6.50 8.28 3.60 3.75 0.15 6.75 8.75 3.80 3.85 0.05 5.25 7.87 3.80 3.65 -0.15 5.25 7.10 BBBACACOB 11 1000 0 0 1 0 0 0 0 1 1 0 1 0 3.85 4.00 0.15 6.00 8.00 C 0 0 1 3.90 4.10 0.20 6.50 7.89 A 1 0 0 3.90 4.00 0.10 6.25 8.15 3.70 4.10 0.40 7.00 9.10 00 0 1 0 0 1 3.75 4.20 0.45 6.90 8.86 A 3.75 4.10 0.35 6.80 8.90 B 1 3.80 4.10 0.30 6.80 8.87 B 1 3.70 4.20 0.50 7.10 9.26 B 0 1 3.80 4.30 0.50 7.00 9.00 A 1 0 3.70 4.10 0.40 6.80 8.75 B 3.80 3.75 -0.05 6.50 7.95 B 0 1 3.80 3.75 -0.05 6.25 7.65 C 0 0 3.75 3.65 -0.10 6.00 7.27 A 1 0 3.70 3.90 0.20 6.50 8.00 3.55 3.65 0.10 7.00 8.50 3.60 4.10 0.50 6.80 8.75 3.65 4.25 0.60 6.80 9.21 AAAB 1 1 1 0 0 0 1 0 3.70 3.65 -0.05 6.50 8.27 C 0 3.75 3.75 0.00 5.75 7.67 B 1 0 3.80 3.85 0.05 5.80 7.93 0 1 3.70 4.25 0.55 6.80 9.26 0 1 Page 4 of 4 Statistics: Statistics: Multiple Linear Regression Please Solve problem. Page 1 of 4 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 below concerning y (demand for Fresh liquid laundry detergent), x (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 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 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 og to equal 1 if campaign Bis used in a sales period and 0 otherwise. Furthermore, we define the dummy variable De to equal 1 if campaign C is used in a sales period and 0 otherwise. Table presents the JMP 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 Average Industry Advertising Expenditure Fresh, x1 1 3.85 Price, x2 for Fresh, x3 for Fresh, y Demand 3.80 5.50 7.38 2 3.75 4.00 6.75 8.51 3 3.70 4.30 7.25 9.52 4 3.70 3.70 5.50 7.50 3.60 3.85 7.00 9.33 6 3.60 3.80 6.50 8.28 7 3.60 3.75 6.75 8.75 8 3.80 3.85 5.25 7.87 9 3.80 3.65 5.25 7.10 10 3.85 4.00 6.00 8.00 11 3.90 4.10 6.50 7.89 12 3.90 4.00 6.25 8.15 13 3.70 4.10 7.00 9.10 14 3.75 4.20 6.90 8.86 15 3.75 4.10 6.80 8.90 16 3.80 4.10 6.80 8.87 17 3.70 4.20 7.10 9.26 18 3.80 4.30 7.00 9.00 19 3.70 4.10 6.80 8.75 20 3.80 3.75 6.50 7.95 21 3.80 3.75 6.25 7.65 22 3.75 3.65 6.00 7.27 23 3.70 3.90 6.50 8.00 24 3.55 3.65 7.00 8.50 25 3.60 4.10 6.80 8.75 26 3.65 4.25 6.80 9.21 27 3.70 3.65 6.50 8.27 28 3.75 3.75 5.75 7.67 29 3.80 3.85 5.80 7.93 30 3.70 4.25 6.80 9.26 Page 3 of 4 (a) In this model the parameter 84 represents the effect on mean demand of advertising campaign B compared to advertising campaign A, and the parameter es 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.) 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 the mean of campaign C compared to campaign A is b5 - The 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 z = 3.70, the average price of similar detergents will be 2 = 3.90, Fresh's advertising expenditure will be x3 = 6.50, and advertising campaign C will be used. Show how = 8.61621 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 z = 3.70, x2 = 3.90, x3 = 6.50, and campaign C is used. (Round your answers to 5 decimal places.) y-hat Confidence interval Prediction interval (c) Consider the alternative model y= 6 +61 +62 2 + 83 23 84DA +85C + Here D equals 1 if advertising campaign A is used and equals 0 otherwise. Describe the effect represented by the regression parameter 65. B5-effect on mean of Campaign compared to Campaign B. Price IndPrice PriceDif AdvExp Demand AdCamp DA DB DC X1 X2 X4 X3 Y 3.85 3.80 -0.05 5.50 7.38 3.75 4.00 0.25 6.75 8.51 3.70 4.30 0.60 7.25 9.52 3.70 3.70 0.00 5.50 7.50 3.60 3.85 0.25 7.00 9.33 3.60 3.80 0.20 6.50 8.28 3.60 3.75 0.15 6.75 8.75 3.80 3.85 0.05 5.25 7.87 3.80 3.65 -0.15 5.25 7.10 BBBACACOB 11 1000 0 0 1 0 0 0 0 1 1 0 1 0 3.85 4.00 0.15 6.00 8.00 C 0 0 1 3.90 4.10 0.20 6.50 7.89 A 1 0 0 3.90 4.00 0.10 6.25 8.15 3.70 4.10 0.40 7.00 9.10 00 0 1 0 0 1 3.75 4.20 0.45 6.90 8.86 A 3.75 4.10 0.35 6.80 8.90 B 1 3.80 4.10 0.30 6.80 8.87 B 1 3.70 4.20 0.50 7.10 9.26 B 0 1 3.80 4.30 0.50 7.00 9.00 A 1 0 3.70 4.10 0.40 6.80 8.75 B 3.80 3.75 -0.05 6.50 7.95 B 0 1 3.80 3.75 -0.05 6.25 7.65 C 0 0 3.75 3.65 -0.10 6.00 7.27 A 1 0 3.70 3.90 0.20 6.50 8.00 3.55 3.65 0.10 7.00 8.50 3.60 4.10 0.50 6.80 8.75 3.65 4.25 0.60 6.80 9.21 AAAB 1 1 1 0 0 0 1 0 3.70 3.65 -0.05 6.50 8.27 C 0 3.75 3.75 0.00 5.75 7.67 B 1 0 3.80 3.85 0.05 5.80 7.93 0 1 3.70 4.25 0.55 6.80 9.26 0 1 Page 4 of 4