Intro to Business Statistics: Nonparametric Tests MannWhitney U Test Wilcoxon RankSum Two Independent Samples

What This Is

The Mann-Whitney U test, also known as the Wilcoxon Rank-Sum test, is a non-parametric test used to compare the distributions of two independent samples. This test is useful when the data do not meet the assumptions of the t-test, such as normality or equal variances. A retail chain wants to know if the average daily sales of its new store location exceed $10,000, compared to its existing locations.

Key Formulas & Symbols

• U = n1n2 + (n1(n1+1))/2 + (n2(n2+1))/2 - (n1n2 + 1)/2 where U = Mann-Whitney U statistic, n1 = sample size of group 1, n2 = sample size of group 2.
• U1 = (n1(n1+1))/2 - R1 where U1 = U statistic for group 1, R1 = sum of ranks for group 1.
• U2 = (n2(n2+1))/2 - R2 where U2 = U statistic for group 2, R2 = sum of ranks for group 2.
• n = n1 + n2 where n = total sample size.
• R = (n1n2 + 1)/2 where R = expected value of U under the null hypothesis.
• T = U - R where T = test statistic for the Mann-Whitney U test.
• p-value = 2 * min(P(T ≤ t), P(T ≥ t)) where p-value = probability of observing the data (or more extreme) if the null hypothesis is true, t = observed test statistic.

Step-by-Step Procedure

1. State hypotheses: Formulate the null and alternative hypotheses. For example, H₀: μ1 = μ2 (the two population means are equal) vs. H₁: μ1 ≠ μ2 (the two population means are not equal).
2. Choose test: Select the Mann-Whitney U test as the appropriate non-parametric test for comparing two independent samples.
3. Compute test statistic: Calculate the U statistic using the formula U = n1n2 + (n1(n1+1))/2 + (n2(n2+1))/2 - (n1n2 + 1)/2.
4. Find p-value or critical value: Determine the p-value using the formula p-value = 2 * min(P(T ≤ t), P(T ≥ t)) or find the critical value from a table or calculator.
5. Compare to α: Compare the p-value to the significance level α (default = 0.05) or compare the test statistic to the critical value.
6. Conclude: Reject the null hypothesis if the p-value is less than α or the test statistic is more extreme than the critical value.

Common Mistakes

• Mistake: Misinterpreting the p-value as the probability that the null hypothesis is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if the null hypothesis is true. It does not provide information about the probability of the null hypothesis being true.
• Mistake: Failing to check the assumptions of the Mann-Whitney U test, such as independence and continuity.
• Correction: Verify that the samples are independent and the data are continuous before applying the Mann-Whitney U test.
• Mistake: Using the Mann-Whitney U test for paired samples.
• Correction: Use the Wilcoxon signed-rank test for paired samples instead.

Quick Practice Problems

1. A marketing firm wants to compare the average response rates of two different email campaigns. The first campaign had a response rate of 5% with a sample size of 100, while the second campaign had a response rate of 3% with a sample size of 150. Calculate the U statistic.
   Answer: U = 175.5, This is calculated using the formula U = n1n2 + (n1(n1+1))/2 + (n2(n2+1))/2 - (n1n2 + 1)/2.
2. A quality control engineer wants to compare the average defect rates of two different manufacturing processes. The first process had a defect rate of 2% with a sample size of 50, while the second process had a defect rate of 1% with a sample size of 75. What is the p-value? Answer: p-value = 0.012, This is calculated using the formula p-value = 2 * min(P(T ≤ t), P(T ≥ t)).
3. A sales manager wants to compare the average sales of two different product lines. The first product line had an average sales of $10,000 with a sample size of 20, while the second product line had an average sales of $12,000 with a sample size of 30. What is the critical value for a two-tailed test with α = 0.05 and n1 = 20, n2 = 30? Answer: Critical value = 124.5, This is found using a table or calculator.

Last-Minute Cram Sheet

• The Mann-Whitney U test is a non-parametric test used to compare two independent samples.
• U = n1n2 + (n1(n1+1))/2 + (n2(n2+1))/2 - (n1n2 + 1)/2 is the formula for the U statistic.
• p-value = 2 * min(P(T ≤ t), P(T ≥ t)) is the formula for the p-value.
• The Mann-Whitney U test assumes independence and continuity.
• ⚠️ p-value is NOT the probability that H₀ is true – it’s the probability of observing the data (or more extreme) if H₀ is true.
• ⚠️ Use the Wilcoxon signed-rank test for paired samples, not the Mann-Whitney U test.
• ⚠️ Check the assumptions of the Mann-Whitney U test before applying it.
• The critical value for a two-tailed test with α = 0.05 and n1 = 20, n2 = 30 is 124.5.
• The p-value for a two-tailed test with α = 0.05 and n1 = 20, n2 = 30 is 0.012.