EXAMPLE 14 Performing the Wilcoxon rank sum test

We are interested in testing whether the population median pulse rate for women (Population 1) is less than that for men (Population 2). We use the data from Example 13 supplemented with an additional seven women and seven men, sampled randomly and independently. The data are presented below.8 Perform the Wilcoxon rank sum test at level of significance .

Women 66 77 57 62 68 78 73 81 84 69 62 79
Men 79 71 68 71 68 86 73 58 68 74 78

Solution

The data were obtained using random samples. Also, we assume that the distributions of the populations have the same shape. Also, we have and , so the conditions for performing the Wilcoxon rank sum test are satisfied.

  • Step 1 State the hypotheses. The key words “less than” indicate that we have a left-tailed test, from Table 13:

    where and represent the population median pulse rates of the first (women) and second (men) samples, respectively.

  • Step 2 Find the critical value and state the rejection rule. The level of significance is , so from Table 14 our critical value is , and our rejection rule is to reject if .

  • Step 3 Find the value of the test statistic. We combine the two samples and arrange in increasing order. We then rank the data values from smallest to largest, as shown in the following table, assigning ties to the mean rank value.

    Combined data 57 58 62 62 66 68 68 68 68 69 71 71
    Rank 1 2 3.5 3.5 5 7.5 7.5 7.5 7.5 10 11.5 11.5
    Combined data 73 73 74 77 78 78 79 79 81 84 86
    Rank 13.5 13.5 15 16 17.5 17.5 19.5 19.5 21 22 23

    The sum of the ranks for the women is

    14-32

    We have

    So that

  • Step 4 State the conclusion and the interpretation. We said in Step 2 that we would reject if . But , which is not ≤ −1.645. Therefore, our conclusion is to not reject . There is insufficient evidence that the population median pulse rate for women is less than that for men.

NOW YOU CAN DO

Exercises 11–14.