Lesson 15 - 5

Inferences about the Differences between Two Medians:

Independent Samples

Objectives

• Test a claim about the difference between the medians of two independent samples

Vocabulary

• Mann–Whitney Test – nonparametric procedure used to test the equality of two population medians from independent samples

• Population “X” – is the population of interest from the problem statement

Mann-Whitney Test

• Similar in structure to Wilcoxon Matched-Pairs Signed-Ranks Test

• Because samples are independent different test statistics and critical values are used

Hypothesis Tests Using Mann–Whitney TestStep 0 Requirements:

1. the samples are independent random samples and2. the shape of the distributions are the same (assume to met in our problems)

Step 1 Box Plots: Draw a side-by-side boxplot to compare the sample data from the two populations. This helps to visualize the difference in the medians.

Step 2 Hypotheses:

Step 3 Ranks: Rank all sample observations from smallest to largest. Handle ties by finding the mean of the ranks for tied values. Find the sum of the ranks for the sample from population X.

Step 4 Level of Significance: (level of significance determines the critical value) Determine a level of significance, based on the seriousness of making a Type I error Small-sample case: Use Table XII. Large-sample case: Use Table IV.

Step 5 Compute Test Statistic:

Step 6 Critical Value Comparison: Reject H0 if test statistic more extreme (further away from 0) than the critical value

Step 7 Conclusion: Reject or Fail to Reject

Left-Tailed Two-Tailed Right-Tailed

H0: Mx = My H1: Mx < My

H0: Mx = My H1: Mx ≠ My

H0: Mx= My H1: Mx > My

Test Statistic for the Mann–Whitney Test

The test statistic will depend on the size of the samples from each population. Let n1 represent the sample size for population X and represent the n2 sample size for population Y.

Small-Sample Case: (n1 ≤ 20 and n2 ≤ 20 )If S is the sum of the ranks corresponding to the sample from population X, then the test statistic, T, is given by

n1 (n1 - 1) T = S – --------------

2

Note: The value of S is always obtained by summing the ranks of the sample data that correspond to Mx , the median of population X, in the hypothesis.

Large-Sample Case: (n1 > 20 or n2 > 20 )From the Central Limit Theorem, the test statistic is given by

where T is the test statistic from the small-sample case.

n1 n2 T – ---------- 2z0 = ----------------------------- n1n2 (n1 + n2 + 1) ------------------------ 12

Small-Sample Case: (n1 ≤ 20 and n2 ≤ 20 )Using α as the level of significance, the critical value is obtained from Table XII in Appendix A.

Large-Sample Case: (n1 > 20 or n2 > 20 )Using α as the level of significance, the critical value is obtained from Table IV in Appendix A.

Left-Tailed Two-Tailed Right-Tailed

-zα zα/2

-zα/2

zα

Left-Tailed Two-Tailed Right-Tailed

wα wα/2

w1- α/2 = n1n2 - wα/2

w1-α = n1n2 - wα

Critical Value for Mann–Whitney Test

Example 1 from 15.5

H I Order Rank Order Rank

10 640 10 1.5 320 12.5

320 80 10 1.5 320 12.5

320 1280 80 3.5 320 12.5

320 160 80 3.5 320 12.5

320 640 160 7 640 18

80 640 160 7 640 18

160 1280 160 7 640 18

10 640 160 7 640 18

640 160 160 7 640 18

160 320 320 12.5 1280 21.5

320 160 320 12.5 1280 21.5

101 152 = S

Example Cont.

• Hypothesis: H0: Med ill = Med healthy

Ha: Med ill > Med healthy

• Test Statistic: T = S – ½ n1(n1+1) = 152 – ½ 11(12) = 86

where S is the sum of the ranks obtained from the sample observations from ill population and n1 is the number of observations from sample of ill people

• Critical Value: w1-α = n1n2 – wα = 121 – 41 = 80

• Conclusion: Since T > w1-α , we reject the null hypothesis and conclude that there is a difference between the median values for ill and healthy people

Summary and Homework

• Summary– The Mann-Whitney test is a nonparametric test for

comparing the median of two independent samples– This test is a sum of ranks of the combined dataset– The critical values for small samples are given in

tables– The critical values for large samples can be

approximated by a calculation with the normal distribution

• Homework– problems 4, 5, 9, 10, 12 from the CD