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UW-Madison STAT 571 - Final Exam

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Stat/For/Hort 571 Clayton and Lin December 16, 2002Final ExamName:For the lecture that you attend please indicate:Instructor:(circle one) Clayton LinTA: (circle one) Gaffigan Tang ZhengInstructions:1. This exam is open book. You may use textbooks, notebooks, class notes, and a calculator.2. You do not need to check the assumptions of the procedures that you use unless you are specificallydirected to do so. In checking for normality it is sufficient to construct a stem and leaf display. It isnot necessary to make a normal scores plot.3. Do all your work in the spaces provided. If you need additional space, use the back of the precedingpage, indicating clearly that you have done so.4. To get full credit, you must show your work. Partial credit will be awarded.5. Some partial computations have been provided on some questions. You may find some but not neces-sarily all of these computations useful. You may assume that these computations are correct.6. Do not dwell too long on any one question. Answer as many questions as you can.7. Note that some questions have multiple parts. For some questions, these parts are independent; insuch cases you can work, for example, on part (b) or (c) separately from part (a).For graders’ use:Question Possible Points Score1 252 253 154 205 15Total 1001Stat/For/Hort 571 Clayton and Lin December 16, 20021. An experiment was conducted to evaluate the design of a cultivator in terms of the effect of the designon the yield of corn plants. Five different designs of cultivator were used; for our purposes we will referto them by abbreviated codes: A, B, C, D, and E. In the experiment, 3 plots were randomly chosenfor each cultivator, and the yield was measured for each plot. The data are summarized below:Cultivator: A B C D ESample mean yield: 14.06 11.26 13.59 19.44 17.68Sample variance: 2.911 5.273 3.164 2.755 3.236Let µArepresent the population mean yield for corn that has received Treatment A; let µBrepresentthe population mean yield for Treatment B, etc.(a) Complete the ANOVA table for this experiment, perform the associated F test, and show thatthe p-value is less than 0.05.Source df SS MSTreatmentsErrorTotal 165.278(b) State, in symbols, the null hypothesis that is being tested in part (a).2Stat/For/Hort 571 Clayton and Lin December 16, 2002(c) Perform a test of H0:12(µB+ µC) =13(µA+ µD+ µE) versus the two-sided alternative.(d) By using Fisher’s LSD method, perform a comparison of all pairs of means, and summarize yourresults using the display used in class.3Stat/For/Hort 571 Clayton and Lin December 16, 20022. A landscape ecology study investigated the characteristics of a number of plots of prairie land. Foreach plot they measured the “length of edge” and the number of species. The data are presented below,along with some summary statistics.Length of edge (x): 18 10 13 29 16 13 22 9 32Number of species (y): 7 10 14 28 12 9 10 11 25Pxi= 162Pyi= 126Pxiyi= 2648Px2i= 3448Py2i= 2200(a) Consider the regression line relating y to x. Calculate the least squares estimates of the slope andintercept for the regression line relating number of species to length of edge (i.e. relating y to x).(b) Construct the ANOVA Table for this regression problem. (Indicate Source, df, SS, and MS. Youdo not need to perform any tests.)4Stat/For/Hort 571 Clayton and Lin December 16, 2002(c) Perform a test of the hypothesis H0: b0= 10 versus the two-sided alternative.(d) The investigators plan to continue to examine additional plots of prairie land. If, tomorrow, theyshould find a plot whose length of edge is 13, give a prediction of the number of species in thatplot, and give a 90% confidence interval for your prediction.5Stat/For/Hort 571 Clayton and Lin December 16, 20023. (a) Suppose we have a random sample of size 32 from a normal distribution with mean µ1and varianceσ21= 40. Independent of that sample, we have another random sample of size 8 from a normaldistribution with mean µ2and variance σ22= 16. Provide a test of H0: µ1− µ2= 0 versus thetwo sided alternative, if we observe a sample mean of 25 for the first sample, and a sample meanof 29 for the second sample.6Stat/For/Hort 571 Clayton and Lin December 16, 2002(b) Repeat part (a), adding in the additional assumption that µ1is known to be 26.7Stat/For/Hort 571 Clayton and Lin December 16, 20024. This question concerns the comparison of two different rejection rules for evaluating a hypothesis. Thedata consist of measurements of the breaking strength of wooden boards. For each board, an increasingforce is applied until the board breaks. The measurement is the amount of force required to break theboard.A random sample of 100 observations of breaking force is available. Assume that these observationsfollow a N(µ, 800) distribution. Of interest is the null hypothesis H0: µ = 56 versus the alternativeH0: µ > 56. Two different rejection rules are being considered; each has roughly the same value of α.Rule A Reject H0if¯X > 60.Rule B For each observation, Xi, declare the observation to be “defective” if Xi> 70. Reject H0ifthe number of defective observations is ≥ 36.If, in fact, µ = 62, find the power for each rule and determine which is more powerful.8Stat/For/Hort 571 Clayton and Lin December 16, 20025. Some researchers have collected a random sample of 120 observations from an unknown distribution.They would like to perform a test to see whether the data come from a standard normal distribution(their null hypothesis). To proceed, they have taken each observation and have recorded whether itwas larger than 1.4, between −1.2 and 1.4, or smaller than −1.2. The observed data are:Number of observationsLess than −1.2 24Between −1.2 and 1.4 85Larger than 1.4 11Based on this information, conduct a test of their null


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UW-Madison STAT 571 - Final Exam

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