Mann-Whitney U Test Solutions for PSTAT 120C Midterm, Exams of Asian literature

Download Mann-Whitney U Test Solutions for PSTAT 120C Midterm and more Exams Asian literature in PDF only on Docsity! PSTAT 120C: Midterm Solutions May 20, 2009 1. (a) For W = 99, 235 the appropriate U is U = 99, 235− 298(299) 2 = 54, 684 The expected value of this statistic is n1n2/2 = 49, 021 so we really should use U = 298(329)− 54, 684 = 43, 358 (b) The standard deviation of the U statistic here is√ n1n2(n1 + n2 + 1)/12 = √ (298)(329)(628)/12 = 2, 265.141 Thus the test statistic Z = 43, 358− 49, 021 2, 265.141 = −2.50 which gives a P value of P = 2P{Z < −2.50} = 2(.0062) = 0.0124. (c) The advantage of the Mann–Whitney test is that we do not have to assume that the observations are normally distributed. (d) The disadvantage is that the Mann–Whitney test has less power than the t-test when the data is normally distributed. 2. A survey asked people to rate their opinion on the economy on a scale from 1 to 10 where 10 meant they were “extremely optimistic”, 1 meant they were “extremely pessimistic”, and 5 indicated that they were “neutral.” Here are the results in the order they were collected: 1, 5, 8, 6, 4, 9, 10, 5, 5 We wish to test H0 : µ = 5.3 versus H0 : µ 6= 5.3. (a) We want to test whether this data is different than 5.3 so we subtract 5.3 from all the data and then rank the magnitudes of the error. Data 1 5 8 6 4 9 10 5 5 Difference -4.3 -0.3 2.7 0.7 -1.3 3.7 4.7 -0.3 -0.3 Sign − − + + − + + − − Rank 8 2 6 4 5 7 9 2 2 The sum of the ranks of the negative observations is T− = 8 + 2 + 5 + 2 + 2 = 19. The sum of the ranks of the positive observations is T+ = 6 + 4 + 7 + 9 = 26. Thus, T = 19, the smaller of the two. (b) For this set of ranks, there are four ways that T− ≤ 3 Rank 8 2 6 4 5 7 9 2 2 1 + + + + + + + + + 2 + − + + + + + + + 3 + + + + + + + − + 4 + + + + + + + + − 1