10. Hypothesis Testing for Two Samples

Two Means - Matched Pairs (Dependent Samples)

10. Hypothesis Testing for Two Samples

Two Means - Matched Pairs (Dependent Samples)

2:31 minutes

Problem 13.RE.3

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?

Verified step by step guidance

Step 1: Understand the problem. The goal is to analyze the paired sample data using the rank correlation coefficient (Spearman's rank correlation coefficient). This test is used to measure the strength and direction of the association between two ranked variables. The significance level is 0.05.

Step 2: Rank the data. For each variable in the paired data, assign ranks to the values. If there are tied values, assign the average of the ranks that would have been assigned to those values.

Step 3: Calculate the difference in ranks (d) for each pair of observations. For each pair, subtract the rank of one variable from the rank of the other variable.

Step 4: Compute the squared differences (d²) for each pair. Then, sum up all the squared differences to get Σd².

Step 5: Use the formula for Spearman's rank correlation coefficient:

r_s = 1 - (6Σd²) / (n(n² - 1))

, where n is the number of pairs. Compare the calculated r_s value to the critical value for Spearman's rank correlation at the 0.05 significance level to determine if there is a significant correlation. Interpret the result in the context of the belief about weight gain during the freshman year.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rank Correlation Coefficient

The rank correlation coefficient, often represented by Spearman's rho, measures the strength and direction of association between two ranked variables. It assesses how well the relationship between the variables can be described using a monotonic function. This is particularly useful when the data does not meet the assumptions of parametric tests, making it a robust choice for nonparametric analysis.

Significance Level

The significance level, commonly denoted as alpha (α), is the threshold for determining whether a result is statistically significant. In this context, a significance level of 0.05 indicates that there is a 5% risk of concluding that a difference exists when there is none (Type I error). It helps researchers decide whether to reject the null hypothesis based on the p-value obtained from the test.

Nonparametric Tests

Nonparametric tests are statistical methods that do not assume a specific distribution for the data, making them suitable for ordinal data or when sample sizes are small. These tests are particularly useful when the assumptions of parametric tests, such as normality, are violated. Examples include the Wilcoxon signed-rank test and the Kruskal-Wallis test, which are often used in place of t-tests or ANOVA when data does not meet required conditions.

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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?

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.

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.

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?

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