10. Hypothesis Testing for Two Samples

Two Means - Matched Pairs (Dependent Samples)

10. Hypothesis Testing for Two Samples

Two Means - Matched Pairs (Dependent Samples)

3:04 minutes

Problem 13.3.13d

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.

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

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.

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.

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.

Substitute the formula for \( T \) into the expression for the other rank sum. This gives: \( \text{Other Rank Sum} = \frac{n(n+1)}{2} - k \).

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.

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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 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.

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Related Practice

Textbook Question

Using the Wilcoxon Signed-Ranks Test

In Exercises 5–8, refer to the sample data for the given exercises in Section 13-2. Use the Wilcoxon signed-ranks test to test the claim that the matched pairs have differences that come from a population with a median equal to zero. Use a 0.05 significance level.

Exercise 6 “Do Men and Women Talk the Same Amount?”

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?

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.

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?

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.

Textbook Question

In Exercises 1–10, use a 0.05 significance level with the indicated test. If no particular test is specified, use the appropriate nonparametric test from this chapter.

The Freshman 15 The “Freshman 15” refers to the belief that college students gain 15 lb (or 6.8 kg) during their freshman year. Listed below are weights (kg) of randomly selected male college freshmen (from Data Set 13 “Freshman 15” in Appendix B). The weights were measured in September and later in April. Use the sign test to test the claim that for the population of freshman male college students there is not a significant difference between the weights in September and the weights in the following April. What do you conclude about the Freshman 15 belief?

Textbook Question

In Exercises 1–10, use a 0.05 significance level with the indicated test. If no particular test is specified, use the appropriate nonparametric test from this chapter.

Wilcoxon Signed-Ranks Test Repeat the preceding exercise using the Wilcoxon signed-ranks test.

Textbook Question

In Exercises 1–10, use a 0.05 significance level with the indicated test. If no particular test is specified, use the appropriate nonparametric test from this chapter.

Rank Correlation Use the paired sample data from Exercise 1 and analyze the data using the rank correlation coefficient. What does the result tell us about the belief that college students gain 15 lb (or 6.8 kg) during their freshman year?

Multiple Choice

A personal trainer is studying whether a new stretching routine improves flexibility. She records the forward reach (in cm) of 6 clients before and after a 4-week program. Calculate the difference (after - before) for each client, the mean difference, and standard deviation.

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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.

Key Concepts

Rank Sums

Paired Data

Nonzero Differences

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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.