Ex 11-2... question is attached...

Exercise 11-22 Break-even sales and sales mix for a service company Yellow Dove Airways provides air transportation services between Portland and Minneapolis. A single Portland to Minneapolis round-trip flight has the following operating statistics: Fuel and landing fees Flight crew salaries Airplane depreciation Variable cost per passenger- business class Variable cost per pasengerper passenger- economy class Round-trip ticket price- business class Round-trip ticket price- economy class $19,400 3,760 2,600 50 20 750 300 It is assumed that the fuel and landing fees, crew salaries, and airplane depreciation are fixed, regardless of the number of seats sold for the round-trip flight. a.) Compute the break-even number of seats sold on a single round-trip flight for the overall product. Assume that the overall product is 10% business class and 90% economy class tickets. b.) How many business class and economy seats would be sold at the break even point? PMBA 305 Spring '13 Quantitative Aspects of Decision Making Module #5: Multiple Regression Draft Due: April 6, 2012 Name Darrin Williams ID# 0337239 Professional Master of Business Administration Program The Ageno School of Business GOLDEN GATE UNIVERSITY [1] A consumer products company wants to measure the effectiveness of different types of advertising media in the promotion of its products. Specially, two types of advertising media are to be considered: radio and television advertising and newspaper advertising (including the cost of discount coupons). A sample of 22 cities with approximately equal population is selected for study during a test period of one month. Each city is allocated a specific expenditure level both for radio and television advertising and for newspaper advertising. The sales of the product (in thousands of dollars) and also the levels of media expenditure (in thousands of dollars) during the test month are recorded, with the data set ADVERTISE.XLS. Using EXCEL or PHStat2, answer the following: a) State the multiple regression equation. b) Interpret the meaning of the slopes b1 and b2 in this problem. c) Interpret the meaning of the regression coefficient b0 in this problem. d) Predict the average sales for a city in which radio and television advertising is $20,000 and newspaper advertising is $20,000. e) Set up a 95% confidence interval estimate for the average sales for cities in which radio and television advertising is $20,000 and newspaper advertising is $20,000. f) Set up a 95% confidence interval estimate for the sales for an individual city in which radio and television advertising is $20,000 and newspaper advertising is $20,000. g) Determine whether there is a significant relationship between sales and the two independent variables (radio advertising and newspaper advertising) at the 0.05 level of significance. h) Interpret the meaning of the pvalue. i) Determine the coefficient of multiple determination, r2. j) Determine the adjusted r2. k) Perform a residual analysis on your results and determine the adequacy of the fit of the model. l) Set up a 95% confidence interval estimate of the population slope between sales and radio and television advertising. m) At the 0.05 level of significance, determine whether each explanatory variable makes a significant contribution to the regression model. On the basis of these results, indicate the independent variables that should be included in this model. n) Compute the coefficients of partial determination, r2Y1.2.and r2Y2.1.and interpret their meaning. Solution: a) Regression Statistics Multiple R 0.8993 R Square 0.8087 Adjusted R Square 0.7886 Standard Error 158.9041 Observations 22 ANOVA df Regression SS MS F 2 2028032.6896 1014016.3448 40.1582 Residual 19 479759.9014 Total Significanc eF 0.0000 21 2507792.5909 Coefficients Standard Error Intercept 25250.5211 t Stat P-value Lower 95% Upper 95% 156.4304 126.7579 1.2341 0.2322 -108.8768 421.7377 Radio 13.0807 1.7594 7.4349 0.0000 9.3983 16.7631 Newspaper 16.7953 2.9634 5.6676 0.0000 10.5929 22.9977 The multiple regression equation is Y = 156.43 + 13.08 (Radio) + 16.80 (Newspaper) b) The regression coefficient b1 is 13.08. For every additional increase in radio, the sales increase by 13.08. What units are you using? The regression coefficient b2 is 16.80. For every additional increase in newspaper, the sales increase by 16.80. What units are you using? c) The regression coefficient b0 is 156.43. When there is no change in radio and no change in newspaper, the sales is 156.43. What units are you using? Does this value make sense to you? d) Average sales = 156.43 + 13.08 (20) + 16.80 (20) = 753.95 What units are you using? e) For Average Predicted Y (YHat) Interval Half Width 130.9113 Confidence Interval Lower Limit 623.0383 Confidence Interval Upper Limit 884.8609 95% confidence interval estimate for the average sales for cities in which radio and television advertising is $20,000 and newspaper advertising is $20,000 is 623.04 to 884.86. f) For Individual Response Y Interval Half Width 357.4269 Prediction Interval Lower Limit 396.5227 Prediction Interval Upper Limit 1111.376 95% confidence interval estimate for the sales for an individual city in which radio and television advertising is $20,000 and newspaper advertising is $20,000 is 396.52 to 1111.38. What units are you using? g) Since pvalue of Fstatistics is less than 0.05, we reject the null hypothesis. There is significant relationship between sales and the two independent variables (radio advertising and newspaper advertising). h) If pvalue is less than significance level, null hypothesis is rejected or accepted otherwise. i) The coefficient of multiple determination, r2 is 0.8993. 89.93% of the total variation is explained by the regression model. j) The adjusted rsquare is 0.7886. k) Since it does not appear to follow particular pattern, the assumptions of linearity and equal variance are not violated. Are you sure about this? There is a pattern to me. Since it does not appear to follow particular pattern, the assumptions of linearity and equal variance are not violated. l) Coefficients Intercept Standard Error t Stat P-value Lower 95% Upper 95% 156.4304 126.7579 1.2341 0.2322 -108.8768 421.7377 Radio 13.0807 1.7594 7.4349 0.0000 9.3983 16.7631 Newspaper 16.7953 2.9634 5.6676 0.0000 10.5929 22.9977 95% confidence interval for the population slope between sales and radio is 9.3983 to 16.7631. 95% confidence interval for the population slope between sales and advertising is 10.5929 to 22.9977. m) Since the pvalue of both the variables is approximately zero and hence are significant to the model. n) Coefficient of partial determination cannot be calculated for this data according to phstat. [2] A collector of antique grandfather clocks believes that the price (in dollars) received for the clocks at an antique auction increases with the age of the clocks and with the number of bidders. Thus the model is hypothesized is where y = auction price, x1 = age of clock (years) and x2 = number of bidders. A sample of 32 auction prices of grandfather clocks, along with their ages and the number of bidders, is given below (clock_bidders.xls). Age (x1)Bidders (x2) Price (y) Age (x1) Bidders (x2) Price (y) 127 13 1235 170 14 2131 115 12 1080 182 8 1550 127 7 845 162 11 1884 150 9 1522 184 10 2041 156 6 1047 143 6 854 182 11 1979 159 9 1483 156 12 1822 108 14 1055 132 10 1253 175 8 1545 137 9 1297 108 6 729 113 9 946 179 9 1792 137 15 1713 111 15 1175 117 11 1024 187 8 1593 137 8 1147 111 7 785 153 6 1092 115 7 744 117 13 1152 194 5 1356 126 10 1336 168 7 1262 a) State the multiple regression equation. b) Interpret the meaning of the slopes b1 and b2 in the model. c) Interpret the meaning of the regression coefficient b0. d) Test H0: b = 0 against H1: b 2 2 0. Interpret your finding. e) Use a 95% confidence interval to estimate b . Interpret the pvalue corresponding to the estimate b . Does 2 2 the confidence interval support your interpretation in d)? f) Determine the coefficient of multiple determination r2 and interpret its meaning. g) Perform a residual analysis on your results and determine the adequacy of the fit of the model. h) Plot the residuals against the prices. Is there evidence of a pattern in the residuals? Explain. i) At a = 0.05, is there evidence of positive autocorrelation in the residuals? j) Suppose the collector, having observed many auctions, believes that the rate of increase of the auction price with age will be driven upward by a large number of bidders. In other words, the collector believes that the age of clock and the number of bidders should interact. Is there evidence to support his claim that the rate of change in the mean price of the clocks with age increases as the number of bidders increases? Should the interaction term (x1x x2) be included in the model? If so, what is the multiple regression equation? Solution: a) PhSTAT OUTPUT Regression Statistics Multiple R 0.9448 R Square 0.8927 Adjusted R Square 0.8853 Standard Error 133.1365 Observations 32 ANOVA df Regression SS 2 F 4277159.703 2138579.851 120.651 4 7 1 Residual 29 514034.5153 Total 31 Coefficient s MS Significanc e F 0.0000 17725.3281 4791194.218 8 Standard Error t Stat Pvalue Lower 95% Upper 95% 1336.7221 173.3561 7.7108 0.0000 1691.2751 982.1690 Age 12.7362 0.9024 14.1140 0.0000 10.8906 14.5818 Bidders 85.8151 8.7058 9.8573 0.0000 68.0099 103.6204 Intercept The multiple regression equation is Y = 1336.7221 + 12.7362 Age + 85.8151 Bidders b) The slope of age is 12.7362. For every additional increase in age, the price increases by 12.7362. The slope of bidders is 85.8151, for every additional increase in bidders, the price increases by 85.8151. c) The regression coefficient b0 is 1336.7221. This indicates that when age and bidders are zero, the price is 1336.7221. d) Since the pvalue of bidder is approximately zero, null hypothesis is rejected. There is sufficient evidence to conclude that the variable bidder is statistically significant. e) 95% confidence interval is 68.0099 to 103.6204. Since the interval does not contain zero, we are 95% confident that the population slope of bidder is not equal to zero. Using the pvalue, they are not correct about the claim. f) The coefficient of multiple determination is 0.8927. 89.27% of the total variation is explained by the regression model. g) Residual Plot for age Since it does not appear to follow particular pattern, the assumptions of linearity and equal variance are not violated. The normal probability plot is close to the straight line, therefore normality assumption is not violated. Since it does not appear to follow particular pattern, the assumptions of linearity and equal variance are not violated. The assumption of independence error is not violated. h) It does not appear to follow any pattern and hence there is a not major difference in the variability of the residuals. i) Durbin-Watson Calculations Sum of Squared Difference of Residuals 958303.3768 Sum of Squared Residuals 514034.5153 Durbin-Watson Statistic 1.86427827 Since DurbinWatson Statistic is greater than 1.57, therefore there is no evidence of positive autocorrelation among the residuals. j) Regression Statistics Multiple R 0.9769 R Square 0.9544 Adjusted R Square 0.9495 Standard Error 88.3674 Observations 32 ANOVA df Regression SS MS F 3 4572547.9872 1524182.6624 195.1880 Residual 28 218646.2316 Total 31 4791194.2188 7808.7940 Significanc eF 0.0000 Coefficients Standard Error Intercept Age Bidders Interaction term t Stat P-value Lower 95% Upper 95% 322.7544 293.3251 1.1003 0.2806 -278.0950 923.6037 0.8733 2.0197 0.4324 0.6688 -93.4099 29.7077 -3.1443 1.2979 0.2110 6.1504 -3.2638 5.0104 0.0039 -154.2633 -32.5565 0.0000 0.8656 1.7302 Since the pvalue of bidders and interaction term is almost zero, therefore they are statistically significant. The p value of age is greater than 0.05, and hence is not statistically significant. The variable age does not contribute significantly to the model. Therefore, we remove the variable age. Regression Statistics Multiple R 0.9768 R Square 0.9541 Adjusted R Square 0.9509 Standard Error 87.1199 Observations 32 ANOVA df Regression SS MS F 2 4571088.0103 2285544.0051 301.1309 Residual 29 220106.2085 Total Significanc eF 0.0000 31 4791194.2188 Coefficients Standard Error 7589.8693 t Stat Intercept 447.0965 57.0238 7.8405 Bidders -105.6313 9.0184 -11.7128 P-value Lower 95% 0.0000 Upper 95% 330.4698 563.7232 0.0000 -124.0760 -87.1865 Interaction term 1.3850 0.0617 22.4488 0.0000 1.2589 1.5112 Since the pvalue of bidder and interaction term is less than 0.05, both the variables are statistically significant. It does not appear to follow any particular pattern and hence the assumptions of linearity and equal variance are not violated. It does not appear to follow any particular pattern and hence the assumptions of linearity and equal variance are not violated. Since the data are not timedependent, the assumption of independence errors is not violated. We have two models Y = 1336.7221 + 12.7362 Age + 85.8151 Bidders Y = 447.0965 - 105.6313 Bidders + 1.3850 Interaction term Hence, the interaction term should be included and remove the variable age from the data. PMBA 305 Spring '13 Quantitative Aspects of Decision Making Module #5: Multiple Regression Draft Due: April 6, 2012 Name ID# Professional Master of Business Administration Program The Ageno School of Business GOLDEN GATE UNIVERSITY PMBA 305 Spring '13 Module #5 [1] A consumer products company wants to measure the effectiveness of different types of advertising media in the promotion of its products. Specially, two types of advertising media are to be considered: radio and television advertising and newspaper advertising (including the cost of discount coupons). A sample of 22 cities with approximately equal population is selected for study during a test period of one month. Each city is allocated a specific expenditure level both for radio and television advertising and for newspaper advertising. The sales of the product (in thousands of dollars) and also the levels of media expenditure (in thousands of dollars) during the test month are recorded, with the data set ADVERTISE.XLS. Using EXCEL or PHStat2, answer the following: a) State the multiple regression equation. b) Interpret the meaning of the slopes b1 and b2 in this problem. c) Interpret the meaning of the regression coefficient b0 in this problem. d) Predict the average sales for a city in which radio and television advertising is $20,000 and newspaper advertising is $20,000. e) Set up a 95% confidence interval estimate for the average sales for cities in which radio and television advertising is $20,000 and newspaper advertising is $20,000. f) Set up a 95% confidence interval estimate for the sales for an individual city in which radio and television advertising is $20,000 and newspaper advertising is $20,000. g) Determine whether there is a significant relationship between sales and the two independent variables (radio advertising and newspaper advertising) at the 0.05 level of significance. h) Interpret the meaning of the pvalue. i) Determine the coefficient of multiple determination, r2. j) Determine the adjusted r2. k) Perform a residual analysis on your results and determine the adequacy of the fit of the model. l) Set up a 95% confidence interval estimate of the population slope between sales and radio and television advertising. m) At the 0.05 level of significance, determine whether each explanatory variable makes a significant contribution to the regression model. On the basis of these results, indicate the independent variables that should be included in this model. n) Compute the coefficients of partial determination, r2Y1.2.and r2Y2.1.and interpret their meaning. Solution: a) Regression Statistics Multiple R 0.8993 R Square 0.8087 Adjusted R Square 0.7886 Standard Error 158.9041 Observations 22 ANOVA df Regression 2 SS 2028032.689 6 MS 1014016.34 48 F 40.158 2 Significanc eF 0.0000 PMBA 305 Spring '13 Module #5 Residual 19 Total 21 479759.9014 2507792.590 9 Coefficien ts 156.4304 13.0807 16.7953 Standard Error 126.7579 1.7594 2.9634 Intercept Radio Newspaper 25250.5211 t Stat 1.2341 7.4349 5.6676 Pvalue 0.2322 0.0000 0.0000 Lower 95% -108.8768 9.3983 10.5929 Upper 95% 421.7377 16.7631 22.9977 The multiple regression equation is Y = 156.43 + 13.08 (Radio) + 16.80 (Newspaper) b) The regression coefficient b1 is 13.08 (in thousands of dollars). For every additional increase in radio (what is increased?), the sales increase by 13.08 (in thousands of dollars). The regression coefficient b2 is 16.80 (in thousands of dollars). For every additional increase in newspaper (what is increased?), the sales increase by 16.80 (in thousands of dollars). c) The regression coefficient b0 is 156.43 ((in thousands of dollars). When there is no change in radio and no change in newspaper, the sales is 156.43. What kind of change are you talking about here? Will this make sense to you? d) Average sales = 156.43 + 13.08 (20) + 16.80 (20) = 753.95 (in thousands of dollars) e) For Average Predicted Y (YHat) Interval Half Width 130.9113 Confidence Interval Lower Limit 623.0383 Confidence Interval Upper Limit 884.8609 95% confidence interval estimate for the average sales for cities in which radio and television advertising is $20,000 and newspaper advertising is $20,000 is 623.04 to 884.86. f) For Individual Response Y Interval Half Width 357.4269 Prediction Interval Lower Limit 396.5227 Prediction Interval Upper Limit 1111.376 95% confidence interval estimate for the sales for an individual city in which radio and television advertising is $20,000 and newspaper advertising is $20,000 is 396.52 to 1111.38. g) Since pvalue of Fstatistics is less than 0.05, we reject the null hypothesis. There is significant relationship between sales and the two independent variables (radio advertising and newspaper advertising). What are you testing here? What is your null hypothesis? h) If pvalue is less than significance level, null hypothesis is rejected or accepted otherwise. PMBA 305 Spring '13 Module #5 This is not the meaning of the pvalue. This is how to use the pvalue for a hypothesis test. i) The coefficient of multiple determination, r 2 is 0.8993. 89.93% of the total variation is explained by the regression model. j) The adjusted rsquare is 0.7886. k) Since it does not appear to follow particular pattern, the assumptions of linearity and equal variance are not violated. You don't think that there is a pattern. PMBA 305 Spring '13 Module #5 Since it appears to follow particular pattern, the assumptions of linearity and equal variance are violated. You need more plots for the residual analysis. See 14.3. l) Intercept Radio Newspap er Coefficients 156.4304 13.0807 Standard Error 126.7579 1.7594 16.7953 2.9634 t Stat P-value 1.2341 0.2322 7.4349 0.0000 5.6676 0.0000 Lower 95% -108.8768 9.3983 Upper 95% 421.7377 16.7631 10.5929 22.9977 95% confidence interval for the population slope between sales and radio is 9.3983 to 16.7631. 95% confidence interval for the population slope between sales and advertising is 10.5929 to 22.9977. m) Since the pvalue of both the variables is approximately zero and hence are significant to the model. What are you testing here? n) Coefficient of partial determination cannot be calculated for this data according to phstat. What version of the PHStat are you using? I can't calculate them without any problem. PMBA 305 Spring '13 Module #5 [2] A collector of antique grandfather clocks believes that the price (in dollars) received for the clocks at an antique auction increases with the age of the clocks and with the number of bidders. Thus the model is hypothesized is Y 0 1 x1 2 x 2 where y = auction price, x1 = age of clock (years) and x2 = number of bidders. A sample of 32 auction prices of grandfather clocks, along with their ages and the number of bidders, is given below (clock_bidders.xls). Age (x1) Bidders (x2) Price (y) Age (x1) Bidders (x2) Price (y) 127 115 127 150 156 182 156 132 137 113 137 117 137 153 117 126 13 12 7 9 6 11 12 10 9 9 15 11 8 6 13 10 1235 1080 845 1522 1047 1979 1822 1253 1297 946 1713 1024 1147 1092 1152 1336 170 182 162 184 143 159 108 175 108 179 111 187 111 115 194 168 14 8 11 10 6 9 14 8 6 9 15 8 7 7 5 7 2131 1550 1884 2041 854 1483 1055 1545 729 1792 1175 1593 785 744 1356 1262 a) State the multiple regression equation. b) Interpret the meaning of the slopes b1 and b2 in the model. c) Interpret the meaning of the regression coefficient b0. d) Test H0: b = 0 against H1: b 0. Interpret your finding. 2 2 e) Use a 95% confidence interval to estimate b . Interpret the pvalue corresponding to the estimate b . 2 2 Does the confidence interval support your interpretation in d)? f) Determine the coefficient of multiple determination r2 and interpret its meaning. g) Perform a residual analysis on your results and determine the adequacy of the fit of the model. h) Plot the residuals against the prices. Is there evidence of a pattern in the residuals? Explain. i) At a = 0.05, is there evidence of positive autocorrelation in the residuals? j) Suppose the collector, having observed many auctions, believes that the rate of increase of the auction price with age will be driven upward by a large number of bidders. In other words, the collector believes that the age of clock and the number of bidders should interact. Is there evidence to support his claim that the rate of change in the mean price of the clocks with age increases as the number of bidders increases? Should the interaction term (x1x x2) be included in the model? If so, what is the multiple regression equation? Solution: a) PhSTAT OUTPUT PMBA 305 Spring '13 Module #5 Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.9448 0.8927 0.8853 133.1365 32 ANOVA df Regression Residual Total Intercept Age Bidders SS 2 29 31 4277159.7034 514034.5153 4791194.2188 Coefficient s 1336.7221 12.7362 85.8151 Standard Error 173.3561 0.9024 8.7058 MS 2138579.851 7 17725.3281 F 120.651 1 t Stat 7.7108 14.1140 9.8573 Significance F 0.0000 Pvalue 0.0000 0.0000 0.0000 Lower 95% 1691.2751 10.8906 68.0099 Upper 95% 982.1690 14.5818 103.6204 The multiple regression equation is Y = 1336.7221 + 12.7362 Age + 85.8151 Bidders b) The slope of age is 12.7362. For every additional increase in age, the price increases by 12.7362. The slope of bidders is 85.8151, for every additional increase in bidders, the price increases by 85.8151. c) The regression coefficient b0 is 1336.7221. This indicates that when age and bidders are zero, the price is 1336.7221. d) Since the pvalue of bidder is approximately zero, null hypothesis is rejected. There is sufficient evidence to conclude that the variable bidder is statistically significant. e) 95% confidence interval is 68.0099 to 103.6204. Since the interval does not contain zero, we are 95% confident that the population slope of bidder is not equal to zero. Using the pvalue, they are not correct about the claim. f) The coefficient of multiple determination is 0.8927. 89.27% of the total variation is explained by the regression model. g) Residual Plot for age PMBA 305 Spring '13 Module #5 Since it does not appear to follow particular pattern, the assumptions of linearity and equal variance are not violated. The normal probability plot is close to the straight line, therefore normality assumption is not violated. PMBA 305 Spring '13 Module #5 Since it does not appear to follow particular pattern, the assumptions of linearity and equal variance are not violated. The assumption of independence error is not violated. h) PMBA 305 Spring '13 Module #5 It does not appear to follow any pattern and hence there is a not major difference in the variability of the residuals. i) Durbin-Watson Calculations Sum of Squared Difference of Residuals Sum of Squared Residuals Durbin-Watson Statistic 958303.3768 514034.5153 1.86427827 Since DurbinWatson Statistic is greater than 1.57, therefore there is no evidence of positive autocorrelation among the residuals. j) Regression Statistics Multiple R 0.9769 R Square 0.9544 PMBA 305 Spring '13 Adjusted R Square Standard Error Observations Module #5 0.9495 88.3674 32 ANOVA df Regression Residual 3 28 Total 31 SS 4572547.987 2 218646.2316 4791194.218 8 Coefficien ts 322.7544 0.8733 -93.4099 1.2979 Standard Error 293.3251 2.0197 29.7077 0.2110 Intercept Age Bidders Interaction term MS 1524182.66 24 7808.7940 F 195.188 0 t Stat 1.1003 0.4324 -3.1443 6.1504 P-value 0.2806 0.6688 0.0039 0.0000 Significanc eF 0.0000 Lower 95% -278.0950 -3.2638 -154.2633 0.8656 Upper 95% 923.6037 5.0104 -32.5565 1.7302 Since the pvalue of bidders and interaction term is almost zero, therefore they are statistically significant. The pvalue of age is greater than 0.05, and hence is not statistically significant. The variable age does not contribute significantly to the model. Therefore, we remove the variable age. Regression Statistics Multiple R 0.9768 R Square 0.9541 Adjusted R Square 0.9509 Standard Error 87.1199 Observations 32 ANOVA df Regression Residual 2 29 Total 31 SS 4571088.010 3 220106.2085 4791194.218 8 Coefficien ts 447.0965 -105.6313 1.3850 Standard Error 57.0238 9.0184 0.0617 Intercept Bidders Interaction term MS 2285544.00 51 7589.8693 F 301.130 9 t Stat 7.8405 -11.7128 22.4488 P-value 0.0000 0.0000 0.0000 Significanc eF 0.0000 Lower 95% 330.4698 -124.0760 1.2589 Upper 95% 563.7232 -87.1865 1.5112 PMBA 305 Spring '13 Module #5 Since the pvalue of bidder and interaction term is less than 0.05, both the variables are statistically significant. It does not appear to follow any particular pattern and hence the assumptions of linearity and equal variance are not violated. PMBA 305 Spring '13 Module #5 It does not appear to follow any particular pattern and hence the assumptions of linearity and equal variance are not violated. PMBA 305 Spring '13 Since the data are not timedependent, the assumption of independence errors is not violated. Module #5 PMBA 305 Spring '13 We have two models Y = 1336.7221 + 12.7362 Age + 85.8151 Bidders Y = 447.0965 - 105.6313 Bidders + 1.3850 Interaction term Hence, the interaction term should be included and remove the variable age from the data. Module #5 PMBA 305 Spring '13 Quantitative Aspects of Decision Making Module #5: Multiple Regression Draft Due: April 6, 2012 Name ID# Professional Master of Business Administration Program The Ageno School of Business GOLDEN GATE UNIVERSITY PMBA 305 Spring '13 Module #5 [1] A consumer products company wants to measure the effectiveness of different types of advertising media in the promotion of its products. Specially, two types of advertising media are to be considered: radio and television advertising and newspaper advertising (including the cost of discount coupons). A sample of 22 cities with approximately equal population is selected for study during a test period of one month. Each city is allocated a specific expenditure level both for radio and television advertising and for newspaper advertising. The sales of the product (in thousands of dollars) and also the levels of media expenditure (in thousands of dollars) during the test month are recorded, with the data set ADVERTISE.XLS. Using EXCEL or PHStat2, answer the following: a) State the multiple regression equation. b) Interpret the meaning of the slopes b1 and b2 in this problem. c) Interpret the meaning of the regression coefficient b0 in this problem. d) Predict the average sales for a city in which radio and television advertising is $20,000 and newspaper advertising is $20,000. e) Set up a 95% confidence interval estimate for the average sales for cities in which radio and television advertising is $20,000 and newspaper advertising is $20,000. f) Set up a 95% confidence interval estimate for the sales for an individual city in which radio and television advertising is $20,000 and newspaper advertising is $20,000. g) Determine whether there is a significant relationship between sales and the two independent variables (radio advertising and newspaper advertising) at the 0.05 level of significance. h) Interpret the meaning of the pvalue. i) Determine the coefficient of multiple determination, r2. j) Determine the adjusted r2. k) Perform a residual analysis on your results and determine the adequacy of the fit of the model. l) Set up a 95% confidence interval estimate of the population slope between sales and radio and television advertising. m) At the 0.05 level of significance, determine whether each explanatory variable makes a significant contribution to the regression model. On the basis of these results, indicate the independent variables that should be included in this model. n) Compute the coefficients of partial determination, r2Y1.2.and r2Y2.1.and interpret their meaning. Solution: a) Regression Statistics Multiple R 0.8993 R Square 0.8087 Adjusted R Square 0.7886 Standard Error 158.9041 Observations 22 ANOVA df Regression 2 SS 2028032.689 6 MS 1014016.34 48 F 40.158 2 Significanc eF 0.0000 PMBA 305 Spring '13 Module #5 Residual 19 Total 21 479759.9014 2507792.590 9 Coefficien ts 156.4304 13.0807 16.7953 Standard Error 126.7579 1.7594 2.9634 Intercept Radio Newspaper 25250.5211 t Stat 1.2341 7.4349 5.6676 Pvalue 0.2322 0.0000 0.0000 Lower 95% -108.8768 9.3983 10.5929 Upper 95% 421.7377 16.7631 22.9977 The multiple regression equation is Y = 156.43 + 13.08 (Radio) + 16.80 (Newspaper) b) The regression coefficient b1 is 13.08 (in thousands of dollars). For every one unit increase in radio, the sales increase by 13.08 units. The regression coefficient b2 is 16.80 (in thousands of dollars). For every one unit increase in newspaper, the sales increase by 16.80 units. c) The regression coefficient b0 is 156.43 ((in thousands of dollars). When there is no change in the units of radio and newspaper, the sales is 156.43 units d) Average sales = 156.43 + 13.08 (20) + 16.80 (20) = 753.95 (in thousands of dollars) e) For Average Predicted Y (YHat) Interval Half Width 130.9113 Confidence Interval Lower Limit 623.0383 Confidence Interval Upper Limit 884.8609 95% confidence interval estimate for the average sales for cities in which radio and television advertising is $20,000 and newspaper advertising is $20,000 is 623.04 to 884.86. f) For Individual Response Y Interval Half Width 357.4269 Prediction Interval Lower Limit 396.5227 Prediction Interval Upper Limit 1111.376 95% confidence interval estimate for the sales for an individual city in which radio and television advertising is $20,000 and newspaper advertising is $20,000 is 396.52 to 1111.38. g) Since pvalue of Fstatistics is less than 0.05, we reject the null hypothesis. There is significant relationship between sales and the two independent variables (radio advertising and newspaper advertising). Null Hypothesis (Ho): There is no significant relationship between sales and independent variables. PMBA 305 Spring '13 h) i) j) Module #5 The pvalue for the F statistic is the smallest value of alpha at which we could reject the null hypothesis in the F test. The coefficient of multiple determination, r 2 is 0.8993. 89.93% of the total variation is explained by the regression model. The adjusted rsquare is 0.7886. k) Since it does not appear to follow particular pattern, the assumptions of linearity and equal variance are not violated. PMBA 305 Spring '13 Module #5 Since it appears to follow particular pattern, the assumptions of linearity and equal variance are violated. l) Intercept Radio Newspap er Coefficients 156.4304 13.0807 Standard Error 126.7579 1.7594 16.7953 2.9634 t Stat P-value 1.2341 0.2322 7.4349 0.0000 5.6676 0.0000 Lower 95% -108.8768 9.3983 Upper 95% 421.7377 16.7631 10.5929 22.9977 95% confidence interval for the population slope between sales and radio is 9.3983 to 16.7631. 95% confidence interval for the population slope between sales and advertising is 10.5929 to 22.9977. m) Since the pvalue of both the variables is approximately zero and hence are significant to the model. The significance of the independent variables are tested here. What are you testing here? n) Coefficient of partial determination cannot be calculated for this data according to phstat. What version of the PHStat are you using? I can't calculate them without any problem. Phstat2 PMBA 305 Spring '13 Module #5 [2] A collector of antique grandfather clocks believes that the price (in dollars) received for the clocks at an antique auction increases with the age of the clocks and with the number of bidders. Thus the model is hypothesized is Y 0 1 x1 2 x 2 where y = auction price, x1 = age of clock (years) and x2 = number of bidders. A sample of 32 auction prices of grandfather clocks, along with their ages and the number of bidders, is given below (clock_bidders.xls). Age (x1) Bidders (x2) Price (y) Age (x1) Bidders (x2) Price (y) 127 115 127 150 156 182 156 132 137 113 137 117 137 153 117 126 13 12 7 9 6 11 12 10 9 9 15 11 8 6 13 10 1235 1080 845 1522 1047 1979 1822 1253 1297 946 1713 1024 1147 1092 1152 1336 170 182 162 184 143 159 108 175 108 179 111 187 111 115 194 168 14 8 11 10 6 9 14 8 6 9 15 8 7 7 5 7 2131 1550 1884 2041 854 1483 1055 1545 729 1792 1175 1593 785 744 1356 1262 a) State the multiple regression equation. b) Interpret the meaning of the slopes b1 and b2 in the model. c) Interpret the meaning of the regression coefficient b0. d) Test H0: b = 0 against H1: b 0. Interpret your finding. 2 2 e) Use a 95% confidence interval to estimate b . Interpret the pvalue corresponding to the estimate b . 2 2 Does the confidence interval support your interpretation in d)? f) Determine the coefficient of multiple determination r2 and interpret its meaning. g) Perform a residual analysis on your results and determine the adequacy of the fit of the model. h) Plot the residuals against the prices. Is there evidence of a pattern in the residuals? Explain. i) At a = 0.05, is there evidence of positive autocorrelation in the residuals? j) Suppose the collector, having observed many auctions, believes that the rate of increase of the auction price with age will be driven upward by a large number of bidders. In other words, the collector believes that the age of clock and the number of bidders should interact. Is there evidence to support his claim that the rate of change in the mean price of the clocks with age increases as the number of bidders increases? Should the interaction term (x1x x2) be included in the model? If so, what is the multiple regression equation? Solution: a) PhSTAT OUTPUT PMBA 305 Spring '13 Module #5 Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.9448 0.8927 0.8853 133.1365 32 ANOVA df Regression Residual Total Intercept Age Bidders SS 2 29 31 4277159.7034 514034.5153 4791194.2188 Coefficient s 1336.7221 12.7362 85.8151 Standard Error 173.3561 0.9024 8.7058 MS 2138579.851 7 17725.3281 F 120.651 1 t Stat 7.7108 14.1140 9.8573 Significance F 0.0000 Pvalue 0.0000 0.0000 0.0000 Lower 95% 1691.2751 10.8906 68.0099 Upper 95% 982.1690 14.5818 103.6204 The multiple regression equation is Y = 1336.7221 + 12.7362 Age + 85.8151 Bidders b) The slope of age is 12.7362. For every additional increase in age, the price increases by 12.7362. The slope of bidders is 85.8151, for every additional increase in bidders, the price increases by 85.8151. c) The regression coefficient b0 is 1336.7221. This indicates that when age and bidders are zero, the price is 1336.7221. d) Since the pvalue of bidder is approximately zero, null hypothesis is rejected. There is sufficient evidence to conclude that the variable bidder is statistically significant. e) 95% confidence interval is 68.0099 to 103.6204. Since the interval does not contain zero, we are 95% confident that the population slope of bidder is not equal to zero. Using the pvalue, they are not correct about the claim. f) The coefficient of multiple determination is 0.8927. 89.27% of the total variation is explained by the regression model. g) Residual Plot for age PMBA 305 Spring '13 Module #5 Since it does not appear to follow particular pattern, the assumptions of linearity and equal variance are not violated. The normal probability plot is close to the straight line, therefore normality assumption is not violated. PMBA 305 Spring '13 Module #5 Since it does not appear to follow particular pattern, the assumptions of linearity and equal variance are not violated. The assumption of independence error is not violated. h) PMBA 305 Spring '13 Module #5 It does not appear to follow any pattern and hence there is a not major difference in the variability of the residuals. i) Durbin-Watson Calculations Sum of Squared Difference of Residuals Sum of Squared Residuals Durbin-Watson Statistic 958303.3768 514034.5153 1.86427827 Since DurbinWatson Statistic is greater than 1.57, therefore there is no evidence of positive autocorrelation among the residuals. j) Regression Statistics Multiple R 0.9769 R Square 0.9544 PMBA 305 Spring '13 Adjusted R Square Standard Error Observations Module #5 0.9495 88.3674 32 ANOVA df Regression Residual 3 28 Total 31 SS 4572547.987 2 218646.2316 4791194.218 8 Coefficien ts 322.7544 0.8733 -93.4099 1.2979 Standard Error 293.3251 2.0197 29.7077 0.2110 Intercept Age Bidders Interaction term MS 1524182.66 24 7808.7940 F 195.188 0 t Stat 1.1003 0.4324 -3.1443 6.1504 P-value 0.2806 0.6688 0.0039 0.0000 Significanc eF 0.0000 Lower 95% -278.0950 -3.2638 -154.2633 0.8656 Upper 95% 923.6037 5.0104 -32.5565 1.7302 Since the pvalue of bidders and interaction term is almost zero, therefore they are statistically significant. The pvalue of age is greater than 0.05, and hence is not statistically significant. The variable age does not contribute significantly to the model. Therefore, we remove the variable age. Regression Statistics Multiple R 0.9768 R Square 0.9541 Adjusted R Square 0.9509 Standard Error 87.1199 Observations 32 ANOVA df Regression Residual 2 29 Total 31 SS 4571088.010 3 220106.2085 4791194.218 8 Coefficien ts 447.0965 -105.6313 1.3850 Standard Error 57.0238 9.0184 0.0617 Intercept Bidders Interaction term MS 2285544.00 51 7589.8693 F 301.130 9 t Stat 7.8405 -11.7128 22.4488 P-value 0.0000 0.0000 0.0000 Significanc eF 0.0000 Lower 95% 330.4698 -124.0760 1.2589 Upper 95% 563.7232 -87.1865 1.5112 PMBA 305 Spring '13 Module #5 Since the pvalue of bidder and interaction term is less than 0.05, both the variables are statistically significant. It does not appear to follow any particular pattern and hence the assumptions of linearity and equal variance are not violated. PMBA 305 Spring '13 Module #5 It does not appear to follow any particular pattern and hence the assumptions of linearity and equal variance are not violated. PMBA 305 Spring '13 Since the data are not timedependent, the assumption of independence errors is not violated. Module #5 PMBA 305 Spring '13 We have two models Y = 1336.7221 + 12.7362 Age + 85.8151 Bidders Y = 447.0965 - 105.6313 Bidders + 1.3850 Interaction term Hence, the interaction term should be included and remove the variable age from the data. Module #5 PMBA 305 Spring '13 Quantitative Aspects of Decision Making Module #5: Multiple Regression Draft Due: April 6, 2012 Name Darrin Williams ID# 0337239 Professional Master of Business Administration Program The Ageno School of Business GOLDEN GATE UNIVERSITY [1] A consumer products company wants to measure the effectiveness of different types of advertising media in the promotion of its products. Specially, two types of advertising media are to be considered: radio and television advertising and newspaper advertising (including the cost of discount coupons). A sample of 22 cities with approximately equal population is selected for study during a test period of one month. Each city is allocated a specific expenditure level both for radio and television advertising and for newspaper advertising. The sales of the product (in thousands of dollars) and also the levels of media expenditure (in thousands of dollars) during the test month are recorded, with the data set ADVERTISE.XLS. Using EXCEL or PHStat2, answer the following: a) State the multiple regression equation. b) Interpret the meaning of the slopes b1 and b2 in this problem. c) Interpret the meaning of the regression coefficient b0 in this problem. d) Predict the average sales for a city in which radio and television advertising is $20,000 and newspaper advertising is $20,000. e) Set up a 95% confidence interval estimate for the average sales for cities in which radio and television advertising is $20,000 and newspaper advertising is $20,000. f) Set up a 95% confidence interval estimate for the sales for an individual city in which radio and television advertising is $20,000 and newspaper advertising is $20,000. g) Determine whether there is a significant relationship between sales and the two independent variables (radio advertising and newspaper advertising) at the 0.05 level of significance. h) Interpret the meaning of the pvalue. i) Determine the coefficient of multiple determination, r2. j) Determine the adjusted r2. k) Perform a residual analysis on your results and determine the adequacy of the fit of the model. l) Set up a 95% confidence interval estimate of the population slope between sales and radio and television advertising. m) At the 0.05 level of significance, determine whether each explanatory variable makes a significant contribution to the regression model. On the basis of these results, indicate the independent variables that should be included in this model. n) Compute the coefficients of partial determination, r2Y1.2.and r2Y2.1.and interpret their meaning. Solution: a) Regression Statistics Multiple R 0.8993 R Square 0.8087 Adjusted R Square 0.7886 Standard Error 158.9041 Observations 22 ANOVA df Regression SS MS F 2 2028032.6896 1014016.3448 40.1582 Residual 19 479759.9014 Total Significanc eF 0.0000 21 2507792.5909 Coefficients Standard Error Intercept 25250.5211 t Stat P-value Lower 95% Upper 95% 156.4304 126.7579 1.2341 0.2322 -108.8768 421.7377 Radio 13.0807 1.7594 7.4349 0.0000 9.3983 16.7631 Newspaper 16.7953 2.9634 5.6676 0.0000 10.5929 22.9977 The multiple regression equation is Y = 156.43 + 13.08 (Radio) + 16.80 (Newspaper) b) The regression coefficient b1 is 13.08. For every additional increase in radio, the sales increase by 13.08. What units are you using? The regression coefficient b2 is 16.80. For every additional increase in newspaper, the sales increase by 16.80. What units are you using? c) The regression coefficient b0 is 156.43. When there is no change in radio and no change in newspaper, the sales is 156.43. What units are you using? Does this value make sense to you? d) Average sales = 156.43 + 13.08 (20) + 16.80 (20) = 753.95 What units are you using? e) For Average Predicted Y (YHat) Interval Half Width 130.9113 Confidence Interval Lower Limit 623.0383 Confidence Interval Upper Limit 884.8609 95% confidence interval estimate for the average sales for cities in which radio and television advertising is $20,000 and newspaper advertising is $20,000 is 623.04 to 884.86. f) For Individual Response Y Interval Half Width 357.4269 Prediction Interval Lower Limit 396.5227 Prediction Interval Upper Limit 1111.376 95% confidence interval estimate for the sales for an individual city in which radio and television advertising is $20,000 and newspaper advertising is $20,000 is 396.52 to 1111.38. What units are you using? g) Since pvalue of Fstatistics is less than 0.05, we reject the null hypothesis. There is significant relationship between sales and the two independent variables (radio advertising and newspaper advertising). h) If pvalue is less than significance level, null hypothesis is rejected or accepted otherwise. i) The coefficient of multiple determination, r2 is 0.8993. 89.93% of the total variation is explained by the regression model. j) The adjusted rsquare is 0.7886. k) Since it does not appear to follow particular pattern, the assumptions of linearity and equal variance are not violated. Are you sure about this? There is a pattern to me. Since it does not appear to follow particular pattern, the assumptions of linearity and equal variance are not violated. l) Coefficients Intercept Standard Error t Stat P-value Lower 95% Upper 95% 156.4304 126.7579 1.2341 0.2322 -108.8768 421.7377 Radio 13.0807 1.7594 7.4349 0.0000 9.3983 16.7631 Newspaper 16.7953 2.9634 5.6676 0.0000 10.5929 22.9977 95% confidence interval for the population slope between sales and radio is 9.3983 to 16.7631. 95% confidence interval for the population slope between sales and advertising is 10.5929 to 22.9977. m) Since the pvalue of both the variables is approximately zero and hence are significant to the model. n) Coefficient of partial determination cannot be calculated for this data according to phstat. [2] A collector of antique grandfather clocks believes that the price (in dollars) received for the clocks at an antique auction increases with the age of the clocks and with the number of bidders. Thus the model is hypothesized is where y = auction price, x1 = age of clock (years) and x2 = number of bidders. A sample of 32 auction prices of grandfather clocks, along with their ages and the number of bidders, is given below (clock_bidders.xls). Age (x1)Bidders (x2) Price (y) Age (x1) Bidders (x2) Price (y) 127 13 1235 170 14 2131 115 12 1080 182 8 1550 127 7 845 162 11 1884 150 9 1522 184 10 2041 156 6 1047 143 6 854 182 11 1979 159 9 1483 156 12 1822 108 14 1055 132 10 1253 175 8 1545 137 9 1297 108 6 729 113 9 946 179 9 1792 137 15 1713 111 15 1175 117 11 1024 187 8 1593 137 8 1147 111 7 785 153 6 1092 115 7 744 117 13 1152 194 5 1356 126 10 1336 168 7 1262 a) State the multiple regression equation. b) Interpret the meaning of the slopes b1 and b2 in the model. c) Interpret the meaning of the regression coefficient b0. d) Test H0: b = 0 against H1: b 2 2 0. Interpret your finding. e) Use a 95% confidence interval to estimate b . Interpret the pvalue corresponding to the estimate b . Does 2 2 the confidence interval support your interpretation in d)? f) Determine the coefficient of multiple determination r2 and interpret its meaning. g) Perform a residual analysis on your results and determine the adequacy of the fit of the model. h) Plot the residuals against the prices. Is there evidence of a pattern in the residuals? Explain. i) At a = 0.05, is there evidence of positive autocorrelation in the residuals? j) Suppose the collector, having observed many auctions, believes that the rate of increase of the auction price with age will be driven upward by a large number of bidders. In other words, the collector believes that the age of clock and the number of bidders should interact. Is there evidence to support his claim that the rate of change in the mean price of the clocks with age increases as the number of bidders increases? Should the interaction term (x1x x2) be included in the model? If so, what is the multiple regression equation? Solution: a) PhSTAT OUTPUT Regression Statistics Multiple R 0.9448 R Square 0.8927 Adjusted R Square 0.8853 Standard Error 133.1365 Observations 32 ANOVA df Regression SS 2 F 4277159.703 2138579.851 120.651 4 7 1 Residual 29 514034.5153 Total 31 Coefficient s MS Significanc e F 0.0000 17725.3281 4791194.218 8 Standard Error t Stat Pvalue Lower 95% Upper 95% 1336.7221 173.3561 7.7108 0.0000 1691.2751 982.1690 Age 12.7362 0.9024 14.1140 0.0000 10.8906 14.5818 Bidders 85.8151 8.7058 9.8573 0.0000 68.0099 103.6204 Intercept The multiple regression equation is Y = 1336.7221 + 12.7362 Age + 85.8151 Bidders b) The slope of age is 12.7362. For every additional increase in age, the price increases by 12.7362. The slope of bidders is 85.8151, for every additional increase in bidders, the price increases by 85.8151. c) The regression coefficient b0 is 1336.7221. This indicates that when age and bidders are zero, the price is 1336.7221. d) Since the pvalue of bidder is approximately zero, null hypothesis is rejected. There is sufficient evidence to conclude that the variable bidder is statistically significant. e) 95% confidence interval is 68.0099 to 103.6204. Since the interval does not contain zero, we are 95% confident that the population slope of bidder is not equal to zero. Using the pvalue, they are not correct about the claim. f) The coefficient of multiple determination is 0.8927. 89.27% of the total variation is explained by the regression model. g) Residual Plot for age Since it does not appear to follow particular pattern, the assumptions of linearity and equal variance are not violated. The normal probability plot is close to the straight line, therefore normality assumption is not violated. Since it does not appear to follow particular pattern, the assumptions of linearity and equal variance are not violated. The assumption of independence error is not violated. h) It does not appear to follow any pattern and hence there is a not major difference in the variability of the residuals. i) Durbin-Watson Calculations Sum of Squared Difference of Residuals 958303.3768 Sum of Squared Residuals 514034.5153 Durbin-Watson Statistic 1.86427827 Since DurbinWatson Statistic is greater than 1.57, therefore there is no evidence of positive autocorrelation among the residuals. j) Regression Statistics Multiple R 0.9769 R Square 0.9544 Adjusted R Square 0.9495 Standard Error 88.3674 Observations 32 ANOVA df Regression SS MS F 3 4572547.9872 1524182.6624 195.1880 Residual 28 218646.2316 Total 31 4791194.2188 7808.7940 Significanc eF 0.0000 Coefficients Standard Error Intercept Age Bidders Interaction term t Stat P-value Lower 95% Upper 95% 322.7544 293.3251 1.1003 0.2806 -278.0950 923.6037 0.8733 2.0197 0.4324 0.6688 -93.4099 29.7077 -3.1443 1.2979 0.2110 6.1504 -3.2638 5.0104 0.0039 -154.2633 -32.5565 0.0000 0.8656 1.7302 Since the pvalue of bidders and interaction term is almost zero, therefore they are statistically significant. The p value of age is greater than 0.05, and hence is not statistically significant. The variable age does not contribute significantly to the model. Therefore, we remove the variable age. Regression Statistics Multiple R 0.9768 R Square 0.9541 Adjusted R Square 0.9509 Standard Error 87.1199 Observations 32 ANOVA df Regression SS MS F 2 4571088.0103 2285544.0051 301.1309 Residual 29 220106.2085 Total Significanc eF 0.0000 31 4791194.2188 Coefficients Standard Error 7589.8693 t Stat Intercept 447.0965 57.0238 7.8405 Bidders -105.6313 9.0184 -11.7128 P-value Lower 95% 0.0000 Upper 95% 330.4698 563.7232 0.0000 -124.0760 -87.1865 Interaction term 1.3850 0.0617 22.4488 0.0000 1.2589 1.5112 Since the pvalue of bidder and interaction term is less than 0.05, both the variables are statistically significant. It does not appear to follow any particular pattern and hence the assumptions of linearity and equal variance are not violated. It does not appear to follow any particular pattern and hence the assumptions of linearity and equal variance are not violated. Since the data are not timedependent, the assumption of independence errors is not violated. We have two models Y = 1336.7221 + 12.7362 Age + 85.8151 Bidders Y = 447.0965 - 105.6313 Bidders + 1.3850 Interaction term Hence, the interaction term should be included and remove the variable age from the data. PMBA 305 Spring '13 Quantitative Aspects of Decision Making Module #5: Multiple Regression Draft Due: April 6, 2012 Name ID# Professional Master of Business Administration Program The Ageno School of Business GOLDEN GATE UNIVERSITY PMBA 305 Spring '13 Module #5 [1] A consumer products company wants to measure the effectiveness of different types of advertising media in the promotion of its products. Specially, two types of advertising media are to be considered: radio and television advertising and newspaper advertising (including the cost of discount coupons). A sample of 22 cities with approximately equal population is selected for study during a test period of one month. Each city is allocated a specific expenditure level both for radio and television advertising and for newspaper advertising. The sales of the product (in thousands of dollars) and also the levels of media expenditure (in thousands of dollars) during the test month are recorded, with the data set ADVERTISE.XLS. Using EXCEL or PHStat2, answer the following: a) State the multiple regression equation. b) Interpret the meaning of the slopes b1 and b2 in this problem. c) Interpret the meaning of the regression coefficient b0 in this problem. d) Predict the average sales for a city in which radio and television advertising is $20,000 and newspaper advertising is $20,000. e) Set up a 95% confidence interval estimate for the average sales for cities in which radio and television advertising is $20,000 and newspaper advertising is $20,000. f) Set up a 95% confidence interval estimate for the sales for an individual city in which radio and television advertising is $20,000 and newspaper advertising is $20,000. g) Determine whether there is a significant relationship between sales and the two independent variables (radio advertising and newspaper advertising) at the 0.05 level of significance. h) Interpret the meaning of the pvalue. i) Determine the coefficient of multiple determination, r2. j) Determine the adjusted r2. k) Perform a residual analysis on your results and determine the adequacy of the fit of the model. l) Set up a 95% confidence interval estimate of the population slope between sales and radio and television advertising. m) At the 0.05 level of significance, determine whether each explanatory variable makes a significant contribution to the regression model. On the basis of these results, indicate the independent variables that should be included in this model. n) Compute the coefficients of partial determination, r2Y1.2.and r2Y2.1.and interpret their meaning. Solution: a) Regression Statistics Multiple R 0.8993 R Square 0.8087 Adjusted R Square 0.7886 Standard Error 158.9041 Observations 22 ANOVA df Regression 2 SS 2028032.689 6 MS 1014016.34 48 F 40.158 2 Significanc eF 0.0000 PMBA 305 Spring '13 Module #5 Residual 19 Total 21 479759.9014 2507792.590 9 Coefficien ts 156.4304 13.0807 16.7953 Standard Error 126.7579 1.7594 2.9634 Intercept Radio Newspaper 25250.5211 t Stat 1.2341 7.4349 5.6676 Pvalue 0.2322 0.0000 0.0000 Lower 95% -108.8768 9.3983 10.5929 Upper 95% 421.7377 16.7631 22.9977 The multiple regression equation is Y = 156.43 + 13.08 (Radio) + 16.80 (Newspaper) b) The regression coefficient b1 is 13.08 (in thousands of dollars). For every additional increase in radio (what is increased?), the sales increase by 13.08 (in thousands of dollars). The regression coefficient b2 is 16.80 (in thousands of dollars). For every additional increase in newspaper (what is increased?), the sales increase by 16.80 (in thousands of dollars). c) The regression coefficient b0 is 156.43 ((in thousands of dollars). When there is no change in radio and no change in newspaper, the sales is 156.43. What kind of change are you talking about here? Will this make sense to you? d) Average sales = 156.43 + 13.08 (20) + 16.80 (20) = 753.95 (in thousands of dollars) e) For Average Predicted Y (YHat) Interval Half Width 130.9113 Confidence Interval Lower Limit 623.0383 Confidence Interval Upper Limit 884.8609 95% confidence interval estimate for the average sales for cities in which radio and television advertising is $20,000 and newspaper advertising is $20,000 is 623.04 to 884.86. f) For Individual Response Y Interval Half Width 357.4269 Prediction Interval Lower Limit 396.5227 Prediction Interval Upper Limit 1111.376 95% confidence interval estimate for the sales for an individual city in which radio and television advertising is $20,000 and newspaper advertising is $20,000 is 396.52 to 1111.38. g) Since pvalue of Fstatistics is less than 0.05, we reject the null hypothesis. There is significant relationship between sales and the two independent variables (radio advertising and newspaper advertising). What are you testing here? What is your null hypothesis? h) If pvalue is less than significance level, null hypothesis is rejected or accepted otherwise. PMBA 305 Spring '13 Module #5 This is not the meaning of the pvalue. This is how to use the pvalue for a hypothesis test. i) The coefficient of multiple determination, r 2 is 0.8993. 89.93% of the total variation is explained by the regression model. j) The adjusted rsquare is 0.7886. k) Since it does not appear to follow particular pattern, the assumptions of linearity and equal variance are not violated. You don't think that there is a pattern. PMBA 305 Spring '13 Module #5 Since it appears to follow particular pattern, the assumptions of linearity and equal variance are violated. You need more plots for the residual analysis. See 14.3. l) Intercept Radio Newspap er Coefficients 156.4304 13.0807 Standard Error 126.7579 1.7594 16.7953 2.9634 t Stat P-value 1.2341 0.2322 7.4349 0.0000 5.6676 0.0000 Lower 95% -108.8768 9.3983 Upper 95% 421.7377 16.7631 10.5929 22.9977 95% confidence interval for the population slope between sales and radio is 9.3983 to 16.7631. 95% confidence interval for the population slope between sales and advertising is 10.5929 to 22.9977. m) Since the pvalue of both the variables is approximately zero and hence are significant to the model. What are you testing here? n) Coefficient of partial determination cannot be calculated for this data according to phstat. What version of the PHStat are you using? I can't calculate them without any problem. PMBA 305 Spring '13 Module #5 [2] A collector of antique grandfather clocks believes that the price (in dollars) received for the clocks at an antique auction increases with the age of the clocks and with the number of bidders. Thus the model is hypothesized is Y 0 1 x1 2 x 2 where y = auction price, x1 = age of clock (years) and x2 = number of bidders. A sample of 32 auction prices of grandfather clocks, along with their ages and the number of bidders, is given below (clock_bidders.xls). Age (x1) Bidders (x2) Price (y) Age (x1) Bidders (x2) Price (y) 127 115 127 150 156 182 156 132 137 113 137 117 137 153 117 126 13 12 7 9 6 11 12 10 9 9 15 11 8 6 13 10 1235 1080 845 1522 1047 1979 1822 1253 1297 946 1713 1024 1147 1092 1152 1336 170 182 162 184 143 159 108 175 108 179 111 187 111 115 194 168 14 8 11 10 6 9 14 8 6 9 15 8 7 7 5 7 2131 1550 1884 2041 854 1483 1055 1545 729 1792 1175 1593 785 744 1356 1262 a) State the multiple regression equation. b) Interpret the meaning of the slopes b1 and b2 in the model. c) Interpret the meaning of the regression coefficient b0. d) Test H0: b = 0 against H1: b 0. Interpret your finding. 2 2 e) Use a 95% confidence interval to estimate b . Interpret the pvalue corresponding to the estimate b . 2 2 Does the confidence interval support your interpretation in d)? f) Determine the coefficient of multiple determination r2 and interpret its meaning. g) Perform a residual analysis on your results and determine the adequacy of the fit of the model. h) Plot the residuals against the prices. Is there evidence of a pattern in the residuals? Explain. i) At a = 0.05, is there evidence of positive autocorrelation in the residuals? j) Suppose the collector, having observed many auctions, believes that the rate of increase of the auction price with age will be driven upward by a large number of bidders. In other words, the collector believes that the age of clock and the number of bidders should interact. Is there evidence to support his claim that the rate of change in the mean price of the clocks with age increases as the number of bidders increases? Should the interaction term (x1x x2) be included in the model? If so, what is the multiple regression equation? Solution: a) PhSTAT OUTPUT PMBA 305 Spring '13 Module #5 Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.9448 0.8927 0.8853 133.1365 32 ANOVA df Regression Residual Total Intercept Age Bidders SS 2 29 31 4277159.7034 514034.5153 4791194.2188 Coefficient s 1336.7221 12.7362 85.8151 Standard Error 173.3561 0.9024 8.7058 MS 2138579.851 7 17725.3281 F 120.651 1 t Stat 7.7108 14.1140 9.8573 Significance F 0.0000 Pvalue 0.0000 0.0000 0.0000 Lower 95% 1691.2751 10.8906 68.0099 Upper 95% 982.1690 14.5818 103.6204 The multiple regression equation is Y = 1336.7221 + 12.7362 Age + 85.8151 Bidders b) The slope of age is 12.7362. For every additional increase in age, the price increases by 12.7362. The slope of bidders is 85.8151, for every additional increase in bidders, the price increases by 85.8151. c) The regression coefficient b0 is 1336.7221. This indicates that when age and bidders are zero, the price is 1336.7221. d) Since the pvalue of bidder is approximately zero, null hypothesis is rejected. There is sufficient evidence to conclude that the variable bidder is statistically significant. e) 95% confidence interval is 68.0099 to 103.6204. Since the interval does not contain zero, we are 95% confident that the population slope of bidder is not equal to zero. Using the pvalue, they are not correct about the claim. f) The coefficient of multiple determination is 0.8927. 89.27% of the total variation is explained by the regression model. g) Residual Plot for age PMBA 305 Spring '13 Module #5 Since it does not appear to follow particular pattern, the assumptions of linearity and equal variance are not violated. The normal probability plot is close to the straight line, therefore normality assumption is not violated. PMBA 305 Spring '13 Module #5 Since it does not appear to follow particular pattern, the assumptions of linearity and equal variance are not violated. The assumption of independence error is not violated. h) PMBA 305 Spring '13 Module #5 It does not appear to follow any pattern and hence there is a not major difference in the variability of the residuals. i) Durbin-Watson Calculations Sum of Squared Difference of Residuals Sum of Squared Residuals Durbin-Watson Statistic 958303.3768 514034.5153 1.86427827 Since DurbinWatson Statistic is greater than 1.57, therefore there is no evidence of positive autocorrelation among the residuals. j) Regression Statistics Multiple R 0.9769 R Square 0.9544 PMBA 305 Spring '13 Adjusted R Square Standard Error Observations Module #5 0.9495 88.3674 32 ANOVA df Regression Residual 3 28 Total 31 SS 4572547.987 2 218646.2316 4791194.218 8 Coefficien ts 322.7544 0.8733 -93.4099 1.2979 Standard Error 293.3251 2.0197 29.7077 0.2110 Intercept Age Bidders Interaction term MS 1524182.66 24 7808.7940 F 195.188 0 t Stat 1.1003 0.4324 -3.1443 6.1504 P-value 0.2806 0.6688 0.0039 0.0000 Significanc eF 0.0000 Lower 95% -278.0950 -3.2638 -154.2633 0.8656 Upper 95% 923.6037 5.0104 -32.5565 1.7302 Since the pvalue of bidders and interaction term is almost zero, therefore they are statistically significant. The pvalue of age is greater than 0.05, and hence is not statistically significant. The variable age does not contribute significantly to the model. Therefore, we remove the variable age. Regression Statistics Multiple R 0.9768 R Square 0.9541 Adjusted R Square 0.9509 Standard Error 87.1199 Observations 32 ANOVA df Regression Residual 2 29 Total 31 SS 4571088.010 3 220106.2085 4791194.218 8 Coefficien ts 447.0965 -105.6313 1.3850 Standard Error 57.0238 9.0184 0.0617 Intercept Bidders Interaction term MS 2285544.00 51 7589.8693 F 301.130 9 t Stat 7.8405 -11.7128 22.4488 P-value 0.0000 0.0000 0.0000 Significanc eF 0.0000 Lower 95% 330.4698 -124.0760 1.2589 Upper 95% 563.7232 -87.1865 1.5112 PMBA 305 Spring '13 Module #5 Since the pvalue of bidder and interaction term is less than 0.05, both the variables are statistically significant. It does not appear to follow any particular pattern and hence the assumptions of linearity and equal variance are not violated. PMBA 305 Spring '13 Module #5 It does not appear to follow any particular pattern and hence the assumptions of linearity and equal variance are not violated. PMBA 305 Spring '13 Since the data are not timedependent, the assumption of independence errors is not violated. Module #5 PMBA 305 Spring '13 We have two models Y = 1336.7221 + 12.7362 Age + 85.8151 Bidders Y = 447.0965 - 105.6313 Bidders + 1.3850 Interaction term Hence, the interaction term should be included and remove the variable age from the data. Module #5 PMBA 305 Spring '13 Quantitative Aspects of Decision Making Module #5: Multiple Regression Draft Due: April 6, 2012 Name ID# Professional Master of Business Administration Program The Ageno School of Business GOLDEN GATE UNIVERSITY PMBA 305 Spring '13 Module #5 [1] A consumer products company wants to measure the effectiveness of different types of advertising media in the promotion of its products. Specially, two types of advertising media are to be considered: radio and television advertising and newspaper advertising (including the cost of discount coupons). A sample of 22 cities with approximately e