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assign3 - (a P T-= 2(b P T-= 5(c P T-≤ 3 4 In order to...

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PSTAT 120C: Assignment # 2 Due April 23, 2009 These problems will be due in lecture next Thursday. 1. Exercise 15.3 from page 748 in WMS. 2. I want to compare the power of a nonparametric test to a typical normal test. Suppose I have 150 independent observations from a normal distribution with mean μ and standard deviation 10. I want to test H 0 : μ = 0 vs. H A : μ > 0 Let M = the number of observations that are greater than 0, and ¯ x = the average of the 150 observations. (a) Find the critical value z * for the critical region C 1 = { ¯ x > z * } such that the level of the test is α = 0 . 05 (b) Find the critical value m * for the critical region C 2 = { M > m * } such that the level of the test is still α = 0 . 05. (c) Calculate the power of C 1 when μ = 2. (d) Calculate the power of C 2 when μ = 2. 3. Suppose that we have 6 observations, and we want to test whether the median= 0 or if it is greater than 0. Let T - be the Wilcoxon signed-rank statistic for the observations that are less than 0. Calculate the probability of these events under the null hypothesis.

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Unformatted text preview: (a) P { T-= 2 } (b) P { T-= 5 } (c) P { T-≤ 3 } 4. In order to test the eﬃciency of some new statistical software, it was tested against the old software using some real data. Each program was run on 10 data sets and timed to see how long it would take to produce the output. Test whether or not the new software runs signiﬁcantly faster using a signed-rank test. Use α = 0 . 05. Data set Old program New Program 1 71.6 79.9 2 35.5 15.2 3 37.7 25.2 4 35.4 21.0 5 37.5 27.3 6 75.8 67.1 7 41.6 36.3 8 80.1 98.5 9 61.3 87.9 10 37.9 32.2 5. For the 300 observations in the data set to test H : θ = 0 versus H A : θ 6 = 0 where θ is the median of the distribution, we calculated the signed-rank statistics T-= 25 , 850. 1 (a) Calculate T + . (b) Calculate E T when the null hypothesis is true. (c) Is there suﬃcient evidence to reject the null hypothesis at a α = 0 . 05? Calculate the appropriate P value. 2...
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assign3 - (a P T-= 2(b P T-= 5(c P T-≤ 3 4 In order to...

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