10. Hypothesis Testing for Two Samples
Two Means - Matched Pairs (Dependent Samples)
3:23 minutesProblem 13.3.13a
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.
a. If we have sample paired data with 50 nonzero differences, what are the smallest and largest possible values of T?
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Understand the context: The problem involves paired data with 50 nonzero differences, and we are tasked with finding the smallest and largest possible values of the test statistic T, which is used in the Wilcoxon signed-rank test.
Recall that the Wilcoxon signed-rank test involves ranking the absolute values of the differences, assigning ranks from 1 to n (where n is the number of nonzero differences). In this case, n = 50, so the ranks range from 1 to 50.
The test statistic T is the sum of the ranks associated with either the positive or negative differences. To find the smallest and largest possible values of T, consider the extreme cases where all differences are either positive or negative.
For the smallest value of T, assume all differences are negative. In this case, T would be the sum of the ranks associated with the positive differences, which is 0 because there are no positive differences.
For the largest value of T, assume all differences are positive. In this case, T would be the sum of all ranks from 1 to 50. Use the formula for the sum of the first n integers: \( \text{Sum} = \frac{n(n+1)}{2} \), where n = 50, to calculate the total sum of ranks.
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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 the observations in a dataset. In non-parametric statistics, particularly in tests like the Wilcoxon rank-sum test, rank sums are used to compare two independent samples. Understanding how to calculate and interpret rank sums is crucial for analyzing paired data and determining statistical significance.
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Paired Data
Paired data consists of observations that are linked or matched in some way, often representing two measurements taken on the same subjects. In the context of the question, the paired differences are essential for calculating the test statistic T. Recognizing the nature of paired data helps in applying the appropriate statistical methods for analysis.
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Test Statistic T
The test statistic T in the context of rank sums is a value calculated from the ranks of the differences between paired observations. It is used to assess the significance of the differences observed. Understanding how to determine the smallest and largest possible values of T is important for interpreting the results of the statistical test and making inferences about the population.
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