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Unformatted text preview: 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 2 3. (3 pts) Does there appear to be substantial curvature in the data? Circle all the appropriate statements below. (a) Yes, there appears to be substantial curvature in the relationship between coffee sales and number of dispensers, because the dispsq coefficient is significantly different from 0 in the quadratic model....
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This document was uploaded on 06/05/2011.
- Spring '09