EXAM I Stat 514 Name Stat 514 (6 pts) Problem Points Score 1 32 2 30 3 32 USE YOUR TIME WISELY USE CLOSEST DF AVAILABLE IN TABLE SHOW YOUR WORK TO RECEIVE PARTIAL CREDIT WRITE LEGIBLY. ANYTHING UNREADABLE WILL NOT BE GRADED Good Luck!!!! 1. An instructor wants to evaluate the eectiveness of his teaching assistants. In one class period, the students were randomly divided into equal-sized groups and each group was taught power calculations from one of the assistants. At the beginning of the next class, each student took a quiz on power calculations and these scores (Y ) were compared. The SAS output is shown below. Source Model Error Corrected Total DF 2 33 35 Sum of Squares 1194.000000 4599.000000 5793.000000 Source trt DF 2 Type I SS 1194.000000 Mean Square 597.000000 F Value 4.28 Pr > F 0.0222 Source trt DF 2 Type III SS 1194.000000 Mean Square 597.000000 F Value 4.28 Pr > F 0.0222 Mean Square 597.000000 139.363636 F Value 4.28 Pr > F 0.0222 a (3 pts) How many teaching assistants and how many students were involved in this experiment? b (4 pts) Write down the null and alternative hypothesis associated with the F test above. Using = .05, what is your conclusion? c (3 pts) In addition to the quiz score, the instructor had results on a statistical aptitude test (X) taken at the beginning of the semester. These summaries are presented below. Level of trt 1 2 3 -------y---------Mean Std Dev 35.00 10.80 21.00 15.20 29.50 8.39 -------x--------Mean Std Dev 38.75 16.05 34.50 15.61 37.08 10.47 Did the random assignment result in reasonably equivalent groups in terms of statistical aptitude? Explain. d (5 pts) The instructor would like to do analysis of covariance. What additional model assumptions must the instructor be aware of when using this method of analysis? Why are they important in terms of inference (i.e., comparing treatments)? e (7 pts) Partial information from the analysis of covariance are shown below. Use these results (and previous information) to assess if there are assistant dierences (after adjusting for aptitude). Source Error Sum of Squares 1369.135804 Source trt x Type I SS 1194.000000 3229.864196 Source trt x Type III SS 732.159007 3229.864196 Parameter Intercept trt 1 trt 2 trt 3 x Estimate 3.783600516 4.344206765 -6.708520485 0.000000000 0.693475941 B B B B Error 3.51084504 2.67368875 2.67832598 . 0.07981556 Standard t Value Pr > |t| 1.08 0.2892 1.62 0.1140 -2.50 0.0175 . . 8.69 <.0001 f (6 pts) Compute the adjusted means using the output above and previous information. How do they compare with those from the original analysis? g (3 pts) In what ways is this second analysis an improvement over the original analysis of variance? 2. An experimenter is interested in investigating the eects of two stimulant drugs (A and B) on rats. She equally divided up a total of 20 rats into 5 groups (placebo, Drug A low, Drug A high, Drug B low, and Drug B high) and 20 minutes after injecting the drug recorded each rat's activity level (higher score is more active). Use the results from the following table to answer the following questions. 12345- trt Placebo Low A High A Low B High B mean 14.00 15.25 18.25 16.75 22.50 var 8.00 12.25 12.25 6.25 11.00 a (10 pts) Construct the ANOVA table, perform the overall F test and state your conclusion (use = .05). b (5 pts) State the analysis of variance assumptions and describe one diagnostic (test procedure or plot) that can be used to assess each assumption. c (6 pts) Construct a set of coecients that will provide the following comparisons: i) Low versus high dosage for Drug A ii) Low versus high dosage for Drug B iii) Drug A versus Drug B iv) Control versus the average of the experimental groups d (3 pts) Which pairs of comparisons are mutually orthogonal? Why? e (6 pts) Complete the following table and, using the Bonferroni correction, state your conclusions. Contrast a b c d DF 1 1 1 1 Contrast SS 56.1125 33.0625 66.1250 F Value 5.64 3.32 Pr > F 0.0313 0.0883 6.65 0.0210 3. For a and b, indicate whether the statement is True or False by circling the appropriate letter. For the remaining parts, provide a clear, concise answer. a. (5 pts) Heidi Seeke performs a randomization test (paired data) and the P-value is 0.03. T T T T T T F F F F F F : : : : : : If = .01, she accepts the Null hypothesis If = .05, she rejects the Null hypothesis If Ha is one-sided, the P-value for the two-sided alternative is 0.015 If = .01, a Type II error is possible There is a 3.0% chance the Null hypothesis is true If = .05, a Type II error is possible b. (6 pts) Ivana Noe is designing her experiment involving six dierent dose levels of a new drug. She is interested in determining the number of mice needed at each dose level. T T T T T T F F F F F F : : : : : : The model degrees of freedom will be 6 More mice will result in a more powerful experiment A larger will result in a more powerful experiment Using Bonferroni for pairwise comparisons is more powerful than Tukey Using Bonferroni over Tukey will result in fewer Type II errors Using a covariate that only decreases the MSE will decrease the power c. (4 pts) Suppose an investigator is primarily interested in comparing two treatment conditions labeled A and B but adds a third condition C to nd out where it falls relative to the other two. He runs the experiment but then seeks your advice because the overall F test comes back insignicant (P > .05) but the comparison of A versus B is signicant (P < .05). What do you recommend he do? Explain. d. (4 pts) We briey discussed in class that analysis of variance is more susceptible to the problems of unequal variance when the sample sizes dier. Assuming the sample sizes are dierent, would the problem be more or less serious if the small sample groups have the greatest variability? Explain. e. (6 pts) Nondigestible carbohydrates can be used in diet food but they may have eects on colonic hydrogen production in humans. You decide to test and see if insulin, fructooligosaccharide, and lactulose are equivalent in their hydrogen production. You want to construct the test ( = .05) such that it will detect a dierence of 15 between any two treatments 95% of the time. If you decide to use n = 6 replicates and preliminary data suggests an error variance of 35, does this meet your goal? Show your work. f. (6 pts) The following ANOVA table is from an experiment where ve identically equipped Subaru Outbacks were chosen at random from a dealership and each tested four times for gas mileage. Source Car Error Sum of Squares 1280 2400 Find estimates of the variance components and the intraclass correlation coecient