3. The Mann-Whitney test, unequal sample sizes example Aa Aa Getting a good night's sleep is important for overall health and well-being. Suppose that a social psychologist believes that married females have higher nighttime sleep quality than unmarried females who are in long-term relationships. He conducts a study to compare the quality of nighttime sleep for stably married women and for stably unmarried women (women who are in longtime stable relationships but are not married). Because the sleep quality variable in the study is measured on an ordinal scale, he uses the Mann-Whitney test to analyze the data. (The scenario for this problem is inspired by an actual 8-year study on the association between women's relationship histories and their sleep conducted at the University of Pittsburgh School of Medicine. Wendy Troxel was the lead author of the study.) The psychologist randomly selects five stably married females and six stably unmarried females. He obtains behavioral data on each subject's sleep-wake patterns from wrist activity monitors that the subjects wore for 1 month. He converts the wrist monitor data to sleep quality scores. The higher the sleep quality score, the more restful the subject's nighttime sleep. The sleep quality scores and their corresponding ranks for each of the eleven females in the study are shown in the following table. Stably Married Females Female Sleep Quality Rank Stably Unmarried Females Female Sleep Quality Rank 1 9.8 11 6 2.9 1 2 3.4 2 7 4.7 6 1345 7.6 9 8 7.3 8 4.4 5 9 5.1 7 9.3 10 10 3.5 3 11 3.7 4 The two groups of females (stably married and stably unmarried) are examples of [ The Mann-Whitney test is the nonparametric counterpart of the samples. independent matched The Mann-Whitney test is the nonparametric counterpart of the t test for independent samples t test for related samples one-way ANOVA F test two-way ANOVA F test ed females have The psychologist intends to use the Mann-Whitney test to test th the same nighttime sleep quality relative to stably unmarried females versus the alternative hypotesis that The psychologist intends to use the Mann-Whitney test to test the null hypothesis that stably married females have the same nighttime sleep quality relative to stably unmarried females versus the alternative hypothesis that nighttime sleep quality is higher for stably married women relative to stably unmarried women. The sum of the ranks for the sample of stably married females is [ The sum of the ranks for the sample of stably unmarried females is [ The test statistic for the Mann-Whitney test, W. = Had the psychologist reversed his rankings such that the highest sleep quality gets a rank of 1, the test statistic for the Mann-Whitney test, w₂ =[ Considering the values of two test statistics, W, and W', the psychologist should compare the value of to the values in a table of critical values for the Mann-Whitney test in order to decide whether or not to reject the null hypothesis. The tables below provide some critical one-tailed values for W. for different values of n₁ and n₂. n1 = 4 One-Tailed Significance Level for n₁ = 4 2456 12 0.025 0.05 0.10 10 11 13 11 12 14 12 13 15 n1 5 X One-Tailed Significance Level for n₁ = 5 2557 0.025 0.05 0.10 17 19 20 6 18 20 22 20 21 23 n1 = 6 One-Tailed Significance Level for n₁ = 6 n2 0.025 0.05 0.10 670 27 8 29 222 26 28 30 29 32 31 34 At the a = 0.025 level of significance, the psychologist will reject the null hypothesis if W, (or W₂') is the null hypothesis, and he less than or equal to greater than less than greater than or equal to Thus, he is higher for stably married women relative to stably unmarried women. conclude At the a = 0.025 level of significance, the psychologist will reject the null hypothesis if W. (or W₂') is Thus, he that nighttime sleep quality is higher for stably m rejects does not reject the null hypothesis, and he ive to stably unmarried women. conclude At the α = 0.025 level of significance, the psychologist will reject the null hypothesis if W. (or W.) is . Thus, he the null hypothesis, and he that nighttime sleep quality is higher for stably married women relative to stably unmarried wom cannot can conclude