STAT 516 HW 7 Please write your answers neatly and clearly! You MUST make sure your answers to these questions are written in the same order as the questions are listed in the assignment. If this instruction is not followed, your homework WILL NOT be graded. HAND CALCULATIONS & CONCEPT QUESTIONS: 1. The table below gives the results of an experiment about the effect of salt in the soil on grass growth. The response values (Y) given in the table below are the grass emergence percentages for each plot. The four treatments considered were the levels of salt (0, 8, 16, and 24). The experiment was done on three separate fields, so each field served as a separate block. Salt Field Emergence (Y) ---- ----- ------------- 0 1 68 0 2 79 0 3 74 0 1 75 0 2 89 0 3 81 0 1 75 0 2 89 0 3 82 0 1 75 0 2 89 0 3 82 8 1 70 8 2 55 8 3 74 8 1 84 8 2 73 8 3 87 8 1 87 8 2 74 8 3 88 8 1 87 8 2 75 8 3 80 16 1 40 16 2 43 16 3 36 16 1 78 16 2 81 16 3 70 16 1 82 16 2 85 16 3 74 16 1 83 16 2 87 16 3 75 24 1 11 24 2 18 24 3 12 24 1 62 24 2 75 24 3 50 24 1 72 24 2 82 24 3 62 24 1 72 24 2 86 24 3 66 Do the following by hand, SHOWING WORK. (a) What is the value of t (# of treatments)? What is the value of b (# of blocks)? (b) Explain why this is an example of what we call a "Randomized Block Design with Sampling". (c) Note that for this experiment, SS(Trts) = 4340.917; SS(Blocks) = 246.542; SS(Exp. error) = 1408.458; and SS(Samp. error) = 11790.0. Calculate the correct mean squares for each of these sources of variation. (d) Calculate the correct F-statistic for testing whether salt level has a significant effect on the mean grass emergence. At the 5% level, is this result significant? (e) Calculate the correct F-statistic for testing whether there is significant variation in grass emergence across fields. At the 5% level, is this result significant? 2. You are asked to design the following experiment. You want to compare the mean time it takes to bake a cake across five different ovens: Amana, GE, Kenmore, Maytag, and Vulcan. There are five brands of cake (Betty Crocker, Duncan Hines, Pepperidge Farm, Pillsbury, Southern Home), and five flavors for each brand (chocolate, yellow, marble, carrot, orange) so that a total of 25 cakes can be baked. You want to control the variation due to cake brand and the variation due to flavor. The main response variable is time needed for baking (in minutes) and the factor of interest is the oven type. Set up a Latin Square Design for this experiment, explaining your notation and the design carefully, so that a non-statistician could carry out the experiment according to your instructions. COMPUTER CALCULATIONS: 3. Forty architecture students were each asked to judge 5 different building structures. The response variable of interest is the judge's overall satisfaction (SAT), where a higher score is better. We wish to compare the mean satisfaction rating across the five buildings, so the factor of interest is BLDG. The data are in Table 8.33 of the book on page 414. The data are also given as buildingsdata.txt (for SAS) or buildingsdataR.txt (for R) on the course web page. (a) Why does it make sense to use the judge (denoted SUBJ in the data set) as a blocking variable? Why should we treat this blocking factor as having random effects? (b) Analyze the data as a RBD, where SAT is the response, BLDG is the treatment factor, and SUBJ is the blocking factor. Based on the appropriate F-test, is there a significant difference in mean satisfaction rating across the five buildings? NOTE: Use a 10% significance level. Give the test statistic value and P-value of your test. (c) Based on the appropriate F-test, is there significant variation among the judges? NOTE: Use a 10% significance level. Give the test statistic value and P-value of your test. (d) Of particular interest to the investigators is whether the mean satisfaction for building 1 differs significantly from the mean satisfaction for the other four buildings. What is the appropriate contrast here? Based on the file buildingsdataoutput.txt on the course web page, answer the investigators' question. NOTE: Use a 10% significance level. Give the test statistic value and P-value of your test. 4. Table 10.31 on p. 507 gives the wheat yield for three different varieties of wheat (A, B, and C). Two blocking factors were considered: the orientation of the plantings and the locations. A Latin Square Design was used, so we will analyze this experiment as a Latin Square. (Treat Location and Orientation as FIXED effects.) The data are also given as wheatdataLS.txt (for SAS) or wheatdataLSR.txt (for R) on the course web page. (a) Use the appropriate F-test to test whether there is a significant difference in mean yield for the different varieties. NOTE: Use a 10% significance level. Give the test statistic value and P-value of your test. (b) Use Tukey's procedure to determine which pairs of varieties have significantly different mean yields. NOTE: Use a 10% (experimentwise) significance level. (c) Based on the SAS/R output, would it have been reasonable to conduct this experiment as a RBD, with only a single blocking factor? If so, which variable would you use for the blocks?