Table of contents

1. Intro to Stats and Collecting Data1h 14m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically2h 5m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables3h 6m
6. Normal Distribution and Continuous Random Variables2h 11m
7. Sampling Distributions & Confidence Intervals: Mean3h 23m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample3h 29m
10. Hypothesis Testing for Two Samples4h 50m
11. Correlation1h 6m
12. Regression1h 50m
13. Chi-Square Tests & Goodness of Fit1h 57m
14. ANOVA1h 57m

10. Hypothesis Testing for Two Samples

Two Means - Matched Pairs (Dependent Samples)

Statistics

10. Hypothesis Testing for Two Samples

Two Means - Matched Pairs (Dependent Samples)

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3:23 minutes

Problem 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?

Verified step by step guidance

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.

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.

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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Related Practice

Multiple Choice

Construct a 95% confidence interval for the mean difference of the population given the following information. Would you reject or fail to reject the claim that there is no difference in the mean?

$\overset{ˉ}{d} = - 0.728$

$s sub d equals 1.34$

$n equals 10$

Textbook Question

In Exercises 17–24, use the indicated Data Sets from Appendix B. The complete data sets can be found at www.TriolaStats.com. Assume that the paired sample data are simple random samples and the differences have a distribution that is approximately normal.

Heights of Presidents Repeat Exercise 12 “Heights of Presidents” using all of the sample data from Data Set 22 “Presidents” in Appendix B.

Textbook Question

Sign Test vs. Wilcoxon Signed-Ranks Test Using the data in Exercise 1, we can test for no difference between hospital admissions on Friday 6th and Friday 13th by using the sign test or the Wilcoxon signed-ranks test. In what sense does the Wilcoxon signed-ranks test incorporate and use more information than the sign test?

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.

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

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.

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?

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