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.


d. If we have sample paired data with n nonzero differences and one of the two rank sums is k, find an expression for the other rank sum.

Verified step by step guidance

1. Understand that the problem involves paired data and rank sums. In the Wilcoxon signed-rank test, the rank sums are derived from the absolute differences of paired data, with positive and negative signs assigned based on the direction of the differences.
2. Recall that the total sum of ranks for n nonzero differences is given by the formula: \( T = \frac{n(n+1)}{2} \). This is because the ranks are assigned sequentially from 1 to n.
3. If one of the rank sums is denoted as \( k \), the other rank sum can be expressed as \( T - k \), where \( T \) is the total sum of ranks.
4. Substitute the formula for \( T \) into the expression for the other rank sum. This gives: \( \text{Other Rank Sum} = \frac{n(n+1)}{2} - k \).
5. This expression allows you to calculate the other rank sum if you know the value of \( k \) and the sample size \( n \). Ensure that \( n \) corresponds to the number of nonzero differences in the paired data.

Key Concepts

Rank Sums

Rank sums are a statistical method used to analyze ordinal data by assigning ranks to data points and summing these ranks for different groups. In the context of paired data, each pair is ranked, and the sums of these ranks are calculated separately for each group. This method is particularly useful in non-parametric tests, such as the Wilcoxon signed-rank test, where the focus is on the ranks rather than the actual data values.

Paired Data

Paired data refers to two sets of related observations, where each observation in one set is uniquely matched with an observation in the other set. This type of data is often used in experiments where the same subjects are measured under different conditions, allowing for a direct comparison. Analyzing paired data helps control for variability between subjects, making it easier to detect differences due to the treatment or condition being tested.

Nonzero Differences

Nonzero differences in paired data indicate that there is a measurable change or effect between the two conditions being compared. When analyzing paired data, it is essential to focus on these differences, as they provide the basis for statistical tests. The presence of nonzero differences allows for the calculation of rank sums and other statistics, which can help determine if the observed changes are statistically significant.