PSTAT 120C: Assignment #2 - Nonparametric Testing, Assignments of Asian literature

Download PSTAT 120C: Assignment #2 - Nonparametric Testing and more Assignments Asian literature in PDF only on Docsity! 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 H0 : µ = 0 vs. HA : µ > 0 Let M = the number of observations that are greater than 0, and x̄ = the average of the 150 observations. (a) Find the critical value z∗ for the critical region C1 = {x̄ > z∗} such that the level of the test is α = 0.05 (b) Find the critical value m∗ for the critical region C2 = {M > m∗} such that the level of the test is still α = 0.05. (c) Calculate the power of C1 when µ = 2. (d) Calculate the power of C2 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. (a) P0 {T− = 2} (b) P0 {T− = 5} (c) P0 {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 H0 : θ = 0 versus HA : θ 6= 0 where θ is the median of the distribution, we calculated the signed-rank statistics T− = 25, 850. 1