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rev1midW11 - STAT 3504 Review Problems I Midterm 1 A...

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STAT 3504 Review Problems I: Midterm 1. A marketing researcher, having collected data on breakfast cereal expenditures by families with 1, 2, 3, 4 and 5 children living at home, plans to use an ordinary regression model to estimate the mean expenditures at each of these five family size levels. However, the researcher is undecided between fitting a linear or a quadratic model, and the data not do give clear evidence in favour of one model or the other. A colleague suggests: “For your purposes you might simply use an ANOVA model”. Is this a useful suggestion? Explain. 2. Text #16.5 3. A student asks: “Why is the F test for inequality of factor level means not a two-tailed test, since any differences among the factor level means can occur in either direction?” Explain using the expected mean squares for MSE and MSTR. 4. A rehabilitation center researcher was interested in examining the relationship between physical fitness prior to surgery of persons undergoing corrective knee surgery and time required in physical therapy until successful rehabilitation. Patient records in the center were examined and 24 male subjects ranging in age from 18 to 30 years who had undergone similar corrective knee surgery during the past year were selected for the study. The number of days required for successful completion of physical therapy and the prior physical fitness status (below average, average, above average) for each patient are given below. j i (level) 1 2 3 4 5 6 7 8 9 10 Below average 29 42 38 40 43 40 30 42 Average 30 35 39 28 31 31 29 35 29 33 Above average 26 32 21 20 23 22 Y 2 ij n j=1 r =1 i i = 25664, Y 1. = 304, Y 2. = 320, Y 3. = 144 Do this problem by hand - not using ANY software (you will not have Excel or anything else on the midterm). Assume all assumptions are satisfied. a) Obtain the fitted values. b) Obtain the residuals for the "high average" factor level. For this factor level find E{e 3j } (up to a proportionality constant) assuming errors are normally distributed. c) Obtain the analysis of variance table. Include the expected mean squares . d) Test at a significance level of 0.01 whether the mean number of days required for rehabilitation differs between the 3 fitness groups. e) Estimate with a 99% confidence interval the mean number of days in rehabilitation required for persons of average physical fitness. Assume that it had been decided in advance of looking at the data that this was the C.I. of interest. f) Use the Bonferroni procedure with 95% family confidence to obtain confidence intervals for μ μ 3 2 - and μ μ 2 1 - . Interpret your results. What is the per comparison level of significance here?
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g) Would the Tukey procedure have been more efficient in part (f)? Explain. h) Under what conditions are the confidence intervals of (f) valid? i) If the researcher had wished to estimate μ μ 3 1 - as well as the other 2 comparisons in (f), would the t-value per confidence interval need to be modified? Would this also be the case if the Tukey procedure had been used?
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