May 3, 2004Statistics 512 Final Exam Spring 2005Statistics 512Final ExamMay 3, 20044:40-6:30 p.m.The following rules apply.1. You may bring with 3 sheets of paper, double-sided with any information you need.2. You may use a calculator.3. You may not collaborate or copy.4. You may not use any device, such as a cell-phone or PDA that allows you to access the internet or other outside information.5. Failure to comply with item 3 could lead to reduction in your grade, ordisciplinary action.I have read the rules above and agree to comply with them.Signature ________________________________________________Name (printed) ___________________________________________problem your score total points1 15 2 43 204 16Total 551Statistics 512 Final Exam Spring 2005All the problems refer to the following situation:A large computer services company wishes to estimate the amount of time it takes to service disk drives, so that it can better compute the cost of service contracts. There are 3main brands. The company wishes to determine whether the time differs among brands. Because a supervisor needs to be on hand during the study, the quality control engineer determines that at most 36 disk drive repairs can be used in the study.1. The quality control engineer needs to design the study. For each of the designs below, explain how the randomization of disks should be done for a completely balanced design, and fill in the relevant parts of the ANOVA table.a) Completely randomized designRandomization method: ANOVA TableSource d.f. F-test denominatorbranderror XXXXXXXXXXXXXXXX2Statistics 512 Final Exam Spring 2005b) There are 3 service centers which handle the repairs. The company wishes to use a randomized complete block design and consider the “service center effect” to be random.Randomization method:ANOVA TableSource d.f. F-test denominatorbrandcenterbrand*centererror XXXXXXXXXXXXXXXX3Statistics 512 Final Exam Spring 2005c) The company is evaluating 2 training programs for technicians, and wishes to add training program as a factor to the experiment in a two-way factorial design. (They will keep the blocks.) They will not include the 2-way interactions with center in the model.Randomization method:ANOVA TableSource d.f. F-test denominatorbrandprogramcenterbrand*program center*treatmenterror XXXXXXXXXXXXXXXX4Statistics 512 Final Exam Spring 2005d) The company determines that the technician effect might be important. They thereforedecide to use a split plot design as follows: in each service center, 4 technicians are chosen and 2 are assigned at random each of the 2 training programs. Each technician is then assigned to repair disks of all 3 types. For this problem, do not compute the ANOVA table. Instead, fill in the table on the structure of the blocks and experimental units. Remember that in a split plot design, a type I split plot does not have blocks at the whole plot level, but a type II split plot has blocks at both levels. So, one of the entries inblocks could be “none”Randomization method:DesignWhole plot treatment:Whole plot experimental unit:Whole plot blocking factor:Subplot treatment:Subplot experimental unit:Subplot blocking factor:5Statistics 512 Final Exam Spring 20052. Suppose that the biggest effect in the study turns out to be the technician effect. How would you do the randomization so that the brand effect could be estimated with the variance due to technician? (You need to consider the “service center effect” but for this problem you can assume that the training program effect is no longer of interest.)6Statistics 512 Final Exam Spring 20053. In the end, the company uses the randomized complete block design (b) with 5 observations per brand in each block. The quality control engineer says that the “service center effect” should be random because it is considered to account for covariance amongrepairs done at the same service center. The business manager, who also studied statistics, claims that the service center effect should be fixed because there are only 3 service centers and so there is no “larger population of service centers” for which we want to draw inference. (We will not try to resolve the issue here, but we will look at the differences between the 2 models.)a) What is the statistical model for the observations, taking into account both the brand and the service center? Be sure to write the constraints and distribution assumptions.Use the following notation:effect descriptionYijktime for service for the kth disk of brand I at center j the mean time for service over all effects in the studyi the effect of brand ijthe effect of center jijthe interaction between the brand and the centerijkrandom errorModel when blocks are fixed effects:Model when blocks are random effects:7Statistics 512 Final Exam Spring 2005b) Briefly explain the difference between analysis of the 2-way complete factorial model with effects brand and service center and the analysis of the randomized complete block design.Difference when blocks are fixed effects:Difference when blocks are random effects:c) What is the covariance structure for the observations for the 2 models? Don’t forget toinclude the case i=r, j=s, t=k (i.e. the variance)blocks are fixed effectsCov(Yijk, Yrst) = blocks are random effectsCov(Yijk, Yrst) = 8Statistics 512 Final Exam Spring 2005d) How do you compute the F-test for the brand main effect for each of the 2 randomized complete block design models?blocks are fixed effectsF* = MS /MSblocks are fixed effectsF* = MS /MS9Statistics 512 Final Exam Spring 2005e) The engineer is most interested in contrasts between the brand means (i·) of the form i· - r·. It turns out that for these balanced data, the contrast can be estimated by----riYY for both models.What is the variance of the contrast? (Hint: If you get the same answer for both models you have done this incorrectly. Use the fact that ------iiiiY)( where-3131jj, -31)(31)(jijiand  --3151151j kijki)blocks are fixed effectsvar(----riYY) = blocks are random effectsvar(----riYY) = 10Statistics 512 Final Exam Spring 20054. The data were collected for the study, and some of the SAS output is attached. Center was treated as a random block.a) What are the statistically significant effects in the model? Do Type 3 statistical tests