Aa Aa Getting a good night's sleep is important for overall health and well-being. Suppose that a social psychologist is interested in determining whether a female's marital status has an effect on the quality of her nighttime sleep. She 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, she 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. She obtains behavioral data on each subject's sleep-wake patterns from wrist activity monitors that the subjects wore for 1 month. She 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 2345 3.4 2 7 4.7 6 7.6 4.4 9.3 959 8 7.3 8 9 5.1 10 10 3.5 11 3.7 734 The two groups of females (stably married and stably unmarried) are examples of The Mann-Whitney test is the nonparametric counterpart of the samples. The Mann-Whitney test is the nonparametric counterpart of the The psychologist intends to use the Mann-Whitney test to test the null hypothesis that stably married and stably unmarried women do not differ in their nighttime sleep quality versus the alternative hypothesis that stably married and stably unmarried women differ in their nighttime sleep quality. 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 her rankings such that the highest sleep quality gets a rank of 1, the test statistic for the Mann-Whitney test, W,' = 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. Select n1 At the a=0.05 level of significance, the psychologist will reject the null hypothesis if W, (or W,') is Thus, she the null hypothesis, and she conclude that stably married and stably unmarried women differ in their nighttime sleep quality.