Intro to Business Statistics: Nonparametric Tests Wilcoxon SignedRank Test Paired Data Single Sample Median

What This Is

The Wilcoxon Signed-Rank Test is a non-parametric test used to compare paired data or a single sample median to a known population median. A retail chain wants to know if average daily sales exceed $10,000. They collect data on daily sales for a month and want to determine if the median daily sales is greater than $10,000.

Key Formulas & Symbols

• W = Σ |R| where R = signed ranks, |R| = absolute value of ranks.
• n = sample size.
• T = (W + 1) / 2.
• T = (n + 1) / 2 for large samples (n > 20).
• p-value = 2P(T ≤ t) where t = test statistic.
• Critical value (T) depends on α and sample size.
• H₀: M = μ₀ where M = population median, μ₀ = known population median.
• H₁: M ≠ μ₀ or H₁: M < μ₀ or H₁: M > μ₀.

Step-by-Step Procedure

1. State hypotheses: Define H₀ and H₁ based on the research question.
2. Choose test: Select the Wilcoxon Signed-Rank Test for paired data or a single sample median.
3. Compute test statistic: Calculate W, T, or p-value using the formulas above.
4. Find p-value or critical value: Determine the p-value or critical value using a Wilcoxon Signed-Rank Test table or calculator.
5. Compare to α: Compare the p-value or critical value to α (0.05).
6. Conclude: Reject H₀ if p-value < α or critical value < T; fail to reject H₀ otherwise.

Common Mistakes

• Mistake: Misinterpreting the p-value as the probability that H₀ is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if H₀ is true.
• Mistake: Failing to check assumptions (e.g., normality, equal variances).
• Correction: Check assumptions before selecting a test.
• Mistake: Using the Wilcoxon Signed-Rank Test for independent samples.
• Correction: Use the Wilcoxon Rank-Sum Test for independent samples.

Quick Practice Problems

1. A company wants to know if the median employee salary is greater than $50,000. They collect data on employee salaries and calculate a median of $45,000. What is the p-value if the sample size is 20 and α = 0.05? Answer: p-value = 0.03 (The p-value is calculated using a Wilcoxon Signed-Rank Test table or calculator).
2. A retail chain wants to know if the median daily sales exceed $10,000. They collect data on daily sales for a month and calculate a median of $12,000. What is the critical value if the sample size is 30 and α = 0.05? Answer: Critical value = 15 (The critical value is determined using a Wilcoxon Signed-Rank Test table or calculator).
3. A company wants to know if the median customer satisfaction rating is less than 4. They collect data on customer satisfaction ratings and calculate a median of 4.5. What is the test statistic if the sample size is 25? Answer: T = 13 (The test statistic is calculated using the formula T = (W + 1) / 2).

Last-Minute Cram Sheet

1. Wilcoxon Signed-Rank Test: Non-parametric test for paired data or single sample median.
2. W = Σ |R|: Test statistic for Wilcoxon Signed-Rank Test.
3. T = (W + 1) / 2: Test statistic for large samples (n > 20).
4. p-value = 2P(T ≤ t): p-value for Wilcoxon Signed-Rank Test.
5. Critical value (T): Depends on α and sample size.
6. H₀: M = μ₀: Null hypothesis for Wilcoxon Signed-Rank Test.
7. H₁: M ≠ μ₀: Alternative hypothesis for Wilcoxon Signed-Rank Test.
8. ⚠️ p-value is NOT the probability that H₀ is true: p-value is the probability of observing the data (or more extreme) if H₀ is true.
9. ⚠️ Wilcoxon Signed-Rank Test is for paired data or single sample median: Not for independent samples.
10. ⚠️ Check assumptions before selecting a test: Normality, equal variances, etc.