final-sol - 1 MAR 5621 Advanced Managerial Statistics Final...

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1 MAR 5621: Advanced Managerial Statistics: Final Exam Solutions Please make sure that your answers are legible and understandable. Show your work. There are 50 points total; each part is worth 2 points unless otherwise noted. Good luck! I. Cafeteria Coffee (13 pts) A staff analyst for a cafeteria chain wants to investigate the relation between the number of self-service coffee dispensers in a cafeteria (variable name: disp ) and sales of coffee (variable name: sales ). n=14 different cafeterias were studied, with the number of coffee dispensers ranging from 1 to 14; each cafeteria was randomly assigned to get a particular number of dispensers. Coffee sales are measured in gallons per week. Several different models are examined, including a simple straight line model, and a quadratic model. Selected computer output is given below; note that the variable dispsq is equal to disp squared (i.e., dispsq = disp 2 ). Straight-Line Model: R 2 = .9201 R 2 (Adjusted) = .9134 SSE=18073 Residual SD = 38.8 Coefficients Standard Error t Stat Intercept 441 21.0 20.1 disp 30.2 2.57 11.8 Quadratic Model: R 2 = .9917 R 2 (Adjusted) = .9901 SSE=1885 Residual SD = 13.1 Coefficients Standard Error t Stat Intercept 347 12.20 28.43 disp 65.6 3.74 17.54 dispsq -2.36 .243 -9.72 1. Determine the predicted coffee sales (in gallons) for a cafeteria with 10 coffee dispensers for the straight-line model , and also for the quadratic model . Straight line mode: predicted sales = 441 + 30.2 * 10 = 743 Quadratic model: predicted sales = 347 + 65.6*10 -2.36 * 10^2 = 767 2. Which of the following could be the overall standard deviation of the 14 observations of coffee sales in the sample? (circle just one; Hint: no elaborate calculations are required) 132 gallons 45 gallons 39 gallons 27 gallons 13 gallons 0 gallons The overall SD of Y needs to be greater than the residual SD for any model that explains part of Y. So we now the overallSD must be greater than 38.8. But will it be a little bit bigger, or a lot bigger? Well, notice that the straight line model is explaining a whole ot of the variability in Y (92%). Thus, the leftover variability is a pretty small fraction of the original variability. Therefore the original variability is quite a bit bigger than 38.8. The residual variability is 8% of the original variability, based on R^2. The ratio of the residual variance to the original variance (38.8^2 / 132^2) is about .08. In more detail: SSResidual = 12*38.8^2; SSTotal = 13*132^2; 1-R2 = SSResidual / SSTotal = .0799
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