Stat 371 Assignment #7 Due Friday, April 8, by 4pm *Submit your homework to your Support TA (Hao Zhou)'s mailbox anytime prior to the due date/time. The mailboxes are to the left as you enter the Medical Science Center (1300 University Ave.) from the main University Ave. entrance. *No late homework will be accepted for credit! *If a problem asks you to use R, include a copy of the code and output. Please edit your code and output to be only the relevant portions. *If a problem does not specify how to compute the answer, you may use any appropriate method. 1. A dairy scientist is testing a new feed additive. She chooses 13 cows at random from a large population of cows. She randomly assigns nold = 8 to get the old diet, and nnew = 5 to get the new diet including the additive. The cows are housed in 13 separated pens and each gets separate feed, with or without additive as appropriate. After two weeks, she picks a day and milks each cow using standard procedures and records the milk produced in pounds. The data are below: Old Diet: 43, 51, 44, 47, 38, 46, 40, 35 New Diet: 47, 75, 85, 100, 58 Let new and old be the population mean milk productions for the new and old diets, respectively. She wishes to test: H0 : new old = 0 vs. HA : new old = 0, using = 0.05. (a) Are the two populations paired or independent? Explain your answer. (b) Graph the data as you see t. Why did you choose the graph(s) that you did and what does it (do they) tell you? (c) Choose a test appropriate for the hypotheses above, and justify your choice based on your answers to parts (a) and (b). Then perform the test by computing a p-value, and making a reject or not reject decision. Do not use R for this, and show your work. Finally, state your conclusion in the context of the problem. 1 Stat 371 2. Two new mathematics learning techniques are being tested. Twenty students were randomly selected from a population. nA = 9 of them were randomly assigned to use technique A, and nB = 11 of them were randomly assigned to use technique B. Each student spent 30 minutes learning the technique to which they were assigned, and then were asked to complete a task. The time to complete the task was recorded, in seconds. A shorter time indicates better mastery of the task. The data are below: Technique A: 23.1, 21.4, 20.6, 15.5, 21.9, 36.0, 30.2, 33.1, 33.4 Technique B: 32.7, 36.8, 39.1, 37.3, 40.3, 46.8, 75.5, 53.0, 55.6, 54.1, 55.7 We wish to test: H0 : A B = 0 vs. HA : A B = 0, using = 0.05. (a) Graph the data as you see t. Why did you choose the graph(s) that you did and what does it (do they) tell you? (b) Use the bootstrap to perform the test, using B = 10000 resamplings. Compute a p-value, and make a reject or not reject decision. Finally, state your conclusion in the context of the problem. (c) Use the Wilcoxon Rank Sum test to perform the test. You may use R. Compute a p-value, and make a reject or not reject decision. Finally, state your conclusion in the context of the problem. 2 Stat 371 3. A shoe manufacturer compared two new materials for the soles of shoes, call them A and B. Twelve adult volunteers, from locations spread around the USA, each got two shoes. The left was made with material A, and the right was made with material B. On both shoes, the material was exactly 1 inch thick. They were instructed to wear the shoes as they would normal shoes, and ship them back to the manufacturer after 2 months. Technicians then re-measured the thickness of the soles, and recorded the amount of wear (in microns). The data is below: Participant Sole A Sole B 1 379 372 2 378 376 3 328 328 4 372 368 5 325 283 6 304 252 7 356 369 8 309 321 9 354 379 10 318 303 11 355 328 12 392 411 They wish to test: H0 : A B = 0 vs. HA : A B = 0, using = 0.05. (a) Are the two populations paired or independent? Explain your answer. (b) Graph the data as you see t. Why did you choose the graph(s) that you did and what does it (do they) tell you? (c) Choose a test appropriate for the hypotheses above, and justify your choice based on your answers to parts (a) and (b). Then perform the test by computing a p-value, and making a reject or not reject decision. Do not use R for this, and show your work. Finally, state your conclusion in the context of the problem. R Notes See the le, 'Two Ind Pops R Code.txt' for R Code to do top to bottom histograms, side by side boxplots, the Wilcoxon Rank Sum test, and the bootstrap for two populations. 3