10. Hypothesis Testing for Two Samples
Two Means - Matched Pairs (Dependent Samples)
3:16 minutesProblem 13.3.13c
Textbook Question
Rank Sums Exercise 12 uses Data Set 33 “Disney World Wait Times” in Appendix B, and the sample size for the 5:00 PM Tower of Terror wait times is n = 50.
c. If we have sample paired data with 50 nonzero differences and the sum of the positive ranks is 165, find the absolute value of the sum of the negative ranks.
Verified step by step guidance1
Understand the context: This problem involves paired data and the Wilcoxon signed-rank test, which is a non-parametric test used to compare paired samples. The test ranks the absolute differences between paired observations, assigning positive or negative signs based on the direction of the difference.
Recall the property of ranks: The sum of all ranks (positive and negative) for a sample of size n is given by the formula \( \text{Sum of all ranks} = \frac{n(n+1)}{2} \). For this problem, \( n = 50 \), so calculate \( \frac{50(50+1)}{2} \) to find the total sum of ranks.
Use the given information: The sum of the positive ranks is provided as 165. Let the sum of the negative ranks be denoted as \( S_{-} \). Since the total sum of ranks is the sum of the positive and negative ranks, set up the equation \( S_{+} + S_{-} = \text{Total Sum of Ranks} \). Substitute the known values into this equation.
Solve for the sum of the negative ranks: Rearrange the equation to isolate \( S_{-} \), which gives \( S_{-} = \text{Total Sum of Ranks} - S_{+} \). Substitute the calculated total sum of ranks and the given positive rank sum (165) into this formula.
Find the absolute value: Since the problem asks for the absolute value of the sum of the negative ranks, take the absolute value of \( S_{-} \) after solving the equation. This ensures the result is a positive number, as required.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rank Sums
Rank sums refer to the total of the ranks assigned to a set of data points, often used in non-parametric tests like the Wilcoxon signed-rank test. In this context, ranks are assigned to the differences between paired observations, allowing for the analysis of the direction and magnitude of changes. The sum of positive ranks and the sum of negative ranks are crucial for determining the overall effect in paired data.
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Paired Data
Paired data consists of two related observations for each subject, often used to assess the effect of a treatment or intervention. In this case, the wait times before and after a specific event or condition are compared. Analyzing paired data helps control for variability between subjects, providing a clearer understanding of the treatment effect.
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Wilcoxon Signed-Rank Test
The Wilcoxon signed-rank test is a non-parametric statistical test used to compare two related samples or matched observations. It evaluates whether their population mean ranks differ, making it suitable for analyzing paired data. The test calculates the sum of the ranks of the positive and negative differences, which is essential for determining statistical significance in the context of the given problem.
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