1 - A new hamburger outlet is unsure of the best price to sell its burgers...

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1 - A new hamburger outlet is unsure of the best price to sell its burgers for maximum profit. To gather data on what the optimal price should be, each week they change the hamburger price slightly using special deals of various kinds. The quantities sold and the corresponding prices are shown in the table below. Note that you need to use the natural logarithm for this question. Week Number of Hamburgers Sold (qi) Price (pi) 1 892 1.23 23456780222 1012 1.15 1060 1.10 987 1.20 680 1.35 739 1.25 809 1.28 1275 0.99 9 946 1.22 10 874 1.25 11 720 1.30 12 1096 1.05 - (a) Assume the statistical model is given by: y = o + Bxi + i Where: Yi = = log(qi) xi = log(pi) ~ 3 N(0,02) Calculate As x is log(pi) then xi xi = 0.08990.0607 +0.0414 + 0.0792 + 0.13 + 0.0969 + 0.107+(-)0.00436 + 0.0864 + 0.0969 + 0.114 +0.0212 = 0.919 As Yi 2.982.94 + 2.86 + 3.04 Calculate Yi is log(q) then Yi 2.953.01 +3.03 +2.99 +2.83 +2.87+2.91 +3.11+ 35.5 x = - xi = 0.919 12 = 0.0766 and y = Y = = 35.5 12 = 2.96 Week Xi Yi (xi - x) (Yi - y) (xi x) (Yi ) (2 x) 1 0.0899 2.95 0.0133 -0.01 -0.000133 0.000177 0.0607 3.01 -0.0159 0.05 -0.000795 0.000253 3 0.0414 3.03 -0.0352 0.07 -0.00246 0.00124 0.0792 2.99 0.0026 0.03 0.000078 0.00000676 0.13 2.83 0.0534 -0.13 -0.00694 0.00285 0.0969 2.87 0.0203 -0.09 -0.00183 0.000412 0.107 2.91 0.0304 -0.05 -0.00152 0.000924 -0.00436 3.11 -0.0810 0.15 -0.0121 0.00656 9 0.0864 2.98 0.0098 0.02 0.000196 0.000096 10 0.0969 2.94 0.0203 -0.02 -0.000406 0.000412 11 0.114 2.86 0.0374 -0.1 -0.00374 0.00140 12 0.0212 3.04 -0.0554 0.08 -0.00443 0.00307 (b). WW (xi x) = 0.000 (y)=0.000 ((xi x)(Yi ) = -0.03408 (xi) = 0.0257 Use least squares to estimate o and B. Show all your working. B = (xi)(yi-) (ix)2 -0.03408 0.0257 = -1.33 B = y Box = 2.96 - (-)1.33 * 0.0766 - == 3.062 (d) Interpret the estimated parameters Bo and 1. Bo is the y intercept if the regression is mapped (the value of y when x is zero) B is the slope if the regression is mapped or how y changes as x does. - (e) - 2 Week Xi Yi i ei e 1 0.0899 2.95 -1.055 2 0.0607 3.01 -1.14 3 0.0414 3.03 -1.203 4 0.0792 2.99 -1.087 5 0.13 2.83 -0.932 6 0.0969 2.87 -1.033 7 0.107 2.91 -1.0024 8 -0.00436 3.11 -1.34 9 0.0864 2.98 -1.065 10 0.0969 2.94 -1.033 11 0.114 2.86 -0.981 12 0.0212 3.04 -1.27 (f) Estimate the standard deviation of the errors, . 02 19.85 (1-1)2 n-2 = = 19.85 = 4.46 - 16.7+17.2+17.9+16.6+14.2+15.2+15.3+19.8+16.4+15.8+14.8+18.6 12-2 (g) Find a 95% confidence interval for and interpret the result. : 198.5 10 (h) Test the hypotheses that Ho = 0 & H 0 = 0.05 and interpret the result. (i) Predict the quantity of hamburgers purchased when the price of one ham- burger is $1.25, construct a prediction interval and interpret the results. Note that the formula for computing the prediction interval in a simple linear regression is: Var (yo) = o (1 + 1 ) + (xo - x) Var (B) - (j) Construct a 95% confidence interval for (1 - o) 1 - A new hamburger outlet is unsure of the best price to sell its burgers for maximum profit. To gather data on what the optimal price should be, each week they change the hamburger price slightly using special deals of various kinds. The quantities sold and the corresponding prices are shown in the table below. Note that you need to use the natural logarithm for this question. Week Number of Hamburgers Sold (qi) Price (pi) 1 892 1.23 23456780222 1012 1.15 1060 1.10 987 1.20 680 1.35 739 1.25 809 1.28 1275 0.99 9 946 1.22 10 874 1.25 11 720 1.30 12 1096 1.05 - (a) Assume the statistical model is given by: y = o + Bxi + i Where: Yi = = log(qi) xi = log(pi) ~ 3 N(0,02) Calculate As x is log(pi) then xi xi = 0.08990.0607 +0.0414 + 0.0792 + 0.13 + 0.0969 + 0.107+(-)0.00436 + 0.0864 + 0.0969 + 0.114 +0.0212 = 0.919 As Yi 2.982.94 + 2.86 + 3.04 Calculate Yi is log(q) then Yi 2.953.01 +3.03 +2.99 +2.83 +2.87+2.91 +3.11+ 35.5 x = - xi = 0.919 12 = 0.0766 and y = Y = = 35.5 12 = 2.96 Week Xi Yi (xi - x) (Yi - y) (xi x) (Yi ) (2 x) 1 0.0899 2.95 0.0133 -0.01 -0.000133 0.000177 0.0607 3.01 -0.0159 0.05 -0.000795 0.000253 3 0.0414 3.03 -0.0352 0.07 -0.00246 0.00124 0.0792 2.99 0.0026 0.03 0.000078 0.00000676 0.13 2.83 0.0534 -0.13 -0.00694 0.00285 0.0969 2.87 0.0203 -0.09 -0.00183 0.000412 0.107 2.91 0.0304 -0.05 -0.00152 0.000924 -0.00436 3.11 -0.0810 0.15 -0.0121 0.00656 9 0.0864 2.98 0.0098 0.02 0.000196 0.000096 10 0.0969 2.94 0.0203 -0.02 -0.000406 0.000412 11 0.114 2.86 0.0374 -0.1 -0.00374 0.00140 12 0.0212 3.04 -0.0554 0.08 -0.00443 0.00307 (b). WW (xi x) = 0.000 (y)=0.000 ((xi x)(Yi ) = -0.03408 (xi) = 0.0257 Use least squares to estimate o and B. Show all your working. B = (xi)(yi-) (ix)2 -0.03408 0.0257 = -1.33 B = y Box = 2.96 - (-)1.33 * 0.0766 - == 3.062 (d) Interpret the estimated parameters Bo and 1. Bo is the y intercept if the regression is mapped (the value of y when x is zero) B is the slope if the regression is mapped or how y changes as x does. - (e) - 2 Week Xi Yi i ei e 1 0.0899 2.95 -1.055 2 0.0607 3.01 -1.14 3 0.0414 3.03 -1.203 4 0.0792 2.99 -1.087 5 0.13 2.83 -0.932 6 0.0969 2.87 -1.033 7 0.107 2.91 -1.0024 8 -0.00436 3.11 -1.34 9 0.0864 2.98 -1.065 10 0.0969 2.94 -1.033 11 0.114 2.86 -0.981 12 0.0212 3.04 -1.27 (f) Estimate the standard deviation of the errors, . 02 19.85 (1-1)2 n-2 = = 19.85 = 4.46 - 16.7+17.2+17.9+16.6+14.2+15.2+15.3+19.8+16.4+15.8+14.8+18.6 12-2 (g) Find a 95% confidence interval for and interpret the result. : 198.5 10 (h) Test the hypotheses that Ho = 0 & H 0 = 0.05 and interpret the result. (i) Predict the quantity of hamburgers purchased when the price of one ham- burger is $1.25, construct a prediction interval and interpret the results. Note that the formula for computing the prediction interval in a simple linear regression is: Var (yo) = o (1 + 1 ) + (xo - x) Var (B) - (j) Construct a 95% confidence interval for (1 - o)