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Name ST 711 Midterm 2 11/5/2004 You may not communicate with or receive information from any other person (except the instructor) during this exam. Please sign the following honor pledge. I have neither given nor received unauthorized aid on this test. ________________________________________ If you have questions about terminology during the exam, ask the instructor. Show your work. Partial credit is given. 1. A consumer magazine wants to compare 10 different models of home charcoal grills for grilling food outdoors. They contract with three chefs, who were selected randomly from a large pool of volunteers. They give each chef one grill of each model to test, and instruct each chef to cook one piece of salmon, one piece of trout, and a half dozen shrimp on each grill. The response variable is a measure of moistness of the cooked seafood. Some SAS output from Procs GLM and Mixed is given on page 3. The model, F-tests, and so forth may or may not be correct for this problem, but you can compute all needed quantities from the given output. a. (10 pts) What type of experimental design is this? b. (10 pts) Write down a statistical model matching the design. Define all terms and subscripts. Write down the common assumptions for this model. c. (10 pts) Compute the standard error for the mean difference between grill models 1 and 4. d. (10 pts) What proportion does variation among chefs contribute to the total variance of y? --------------------------------------------------------------------------------------------Problem 1. Compare Grill Models 1 The GLM Procedure Class Level Information Class chef grillmodel fish Levels 3 10 3 Values 1 2 3 1 2 3 4 5 6 7 8 9 10 salmon shrimp trout Number of observations 90 ------------------------------------------------------------------------------------------------Dependent Variable: y Sum of Source DF Squares Mean Square F Value Pr > F Model 49 3712.291737 75.761056 58.92 <.0001 Error 40 51.432448 1.285811 Corrected Total 89 3763.724185 R-Square 0.986335 Coeff Var -51.14246 Root MSE 1.133936 y Mean -2.217211 Source chef grillmodel chef*grillmodel fish grillmodel*fish DF 2 9 18 2 18 Type I SS 914.500692 125.823946 121.034928 2440.882800 110.049371 Mean Square 457.250346 13.980438 6.724163 1220.441400 6.113854 F Value 355.61 10.87 5.23 949.16 4.75 Pr > F <.0001 <.0001 <.0001 <.0001 <.0001 Source chef grillmodel chef*grillmodel fish grillmodel*fish DF 2 9 18 2 18 Type III SS 914.500692 125.823946 121.034928 2440.882800 110.049371 Mean Square 457.250346 13.980438 6.724163 1220.441400 6.113854 F Value 355.61 10.87 5.23 949.16 4.75 Pr > F <.0001 <.0001 <.0001 <.0001 <.0001 ------------------------------------------------------------------------------------------------The GLM Procedure Least Squares Means chef 1 2 3 grillmodel 1 2 3 4 5 6 7 8 9 10 y LSMEAN 2.28119177 -4.21154419 -4.72128050 y LSMEAN -3.23064228 -3.22636717 -1.97915027 -2.31673740 -3.34912816 -3.73283947 -1.82386441 -2.05503707 0.09054120 -0.54888472 Standard Error 0.20702747 0.20702747 0.20702747 Standard Error 0.37797872 0.37797872 0.37797872 0.37797872 0.37797872 0.37797872 0.37797872 0.37797872 0.37797872 0.37797872 Pr > |t| <.0001 <.0001 <.0001 Pr > |t| <.0001 <.0001 <.0001 <.0001 <.0001 <.0001 <.0001 <.0001 0.8119 0.1543 Standard fish y LSMEAN Error Pr > |t| salmon 3.31376088 0.20702747 <.0001 shrimp -9.19430500 0.20702747 <.0001 trout -0.77108880 0.20702747 0.0006 ------------------------------------------------------------------------------------------------The Mixed Procedure Covariance Parameter Estimates Cov Parm chef chef*grillmodel Residual Estimate 15.0175 1.8128 1.2858 Fit Statistics -2 Res Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better) 259.8 265.8 266.3 263.1 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value grillmodel 9 18 2.08 fish 2 40 949.16 grillmodel*fish 18 40 4.75 Effect grillmodel grillmodel grillmodel grillmodel grillmodel grillmodel grillmodel grillmodel grillmodel grillmodel fish fish fish 2. fish salmon shrimp trout Least Squares Means Standard grillmodel Estimate Error 1 -3.2306 2.3985 2 -3.2264 2.3985 3 -1.9792 2.3985 4 -2.3167 2.3985 5 -3.3491 2.3985 6 -3.7328 2.3985 7 -1.8239 2.3985 8 -2.0550 2.3985 9 0.09054 2.3985 10 -0.5489 2.3985 3.3138 2.2603 -9.1943 2.2603 -0.7711 2.2603 Pr > F 0.0890 <.0001 <.0001 DF 18 18 18 18 18 18 18 18 18 18 40 40 40 t Value -1.35 -1.35 -0.83 -0.97 -1.40 -1.56 -0.76 -0.86 0.04 -0.23 1.47 -4.07 -0.34 Pr > |t| 0.1947 0.1953 0.4201 0.3469 0.1796 0.1370 0.4569 0.4028 0.9703 0.8216 0.1505 0.0002 0.7348 A chemistry teacher wants his students to compare 7 methods of measuring volume. In one class period there is time to complete just 3 of the measurement procedures. The teacher divides the students into 7 groups and asks each group to measure a set volume of water using 3 of the methods. He considers two different cyclic designs for assigning the methods to the groups. a. (10 pts) Construct a cyclic design for 7 treatments with starting block (0 1 3). b. (10 pts) Construct a cyclic design for 7 treatments with starting block (0 2 4). c. (10 pts) Which design do you recommend and why? d. (10 pts) For the design of part (a), what is the variance of the mean difference between method 1 and method 7? e. (10 pts) Compute the efficiency factor for the design of part (a). f. (10 pts) Relative efficiency compares two variances. For part (e) of this problem what two variances would be compared? Be explicit about the type of design and number of replicates in the designs that are compared.