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.


b. If we have sample paired data with 50 nonzero differences, what is the expected value of T if the population consists of matched pairs with differences having a median of 0?

Verified step by step guidance

1. Understand the context of the problem: The question involves paired data with 50 nonzero differences, and we are tasked with finding the expected value of T under the assumption that the population of differences has a median of 0. This suggests the use of the Wilcoxon signed-rank test, a nonparametric test for matched pairs.
2. Recall the key property of the Wilcoxon signed-rank test: If the population median of the differences is 0, the signed ranks are symmetrically distributed around 0. The test statistic T is the sum of the positive ranks.
3. Determine the total number of ranks: Since there are 50 nonzero differences, the ranks will range from 1 to 50. The sum of all ranks can be calculated using the formula for the sum of the first n integers: \( \text{Sum of ranks} = \frac{n(n+1)}{2} \), where \( n = 50 \).
4. Understand the expected value of T: If the population median of the differences is 0, the positive and negative ranks are equally likely. Therefore, the expected value of T (the sum of positive ranks) is half of the total sum of ranks. This can be expressed as \( E(T) = \frac{1}{2} \times \text{Sum of ranks} \).
5. Substitute the total sum of ranks into the formula for \( E(T) \): Use the formula \( \text{Sum of ranks} = \frac{n(n+1)}{2} \) to calculate the total sum of ranks, and then divide it by 2 to find the expected value of T. This will give you the final expression for \( E(T) \).

Key Concepts

Paired Data

Paired data refers to two sets of related observations, often collected from the same subjects under different conditions. In this context, it involves measuring wait times for the same ride at different times or conditions, allowing for a direct comparison of differences. This method helps control for variability between subjects, making it easier to detect effects or differences.

Median of Differences

The median of differences is a measure of central tendency that indicates the middle value of a set of differences when arranged in order. In the context of paired data, if the median of the differences is 0, it suggests that there is no systematic difference between the paired observations. This is crucial for hypothesis testing, as it helps determine if the observed differences are statistically significant.

Expected Value of T

The expected value of T in the context of rank sums or paired data refers to the average value of the test statistic under the null hypothesis. If the population of differences has a median of 0, the expected value of T would reflect this condition, indicating that, on average, the ranks of the differences would balance out around zero. This concept is essential for understanding the distribution of the test statistic and for making inferences about the population.