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Ch. 8 - Hypothesis Testing with Two Samples
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 8 - Hypothesis Testing with Two SamplesProblem 8.3.8
Chapter 8, Problem 8.3.8

Test the claim about the mean of the differences for a population of paired data at the level of significance α. Assume the samples are random and dependent, and the populations are normally distributed.
Claim: μd≠0 , α=0.10, Sample statistics: d̄ =-1, sd=2.75, n=20

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Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: μd = 0 (the mean difference is 0), and the alternative hypothesis is H₁: μd ≠ 0 (the mean difference is not 0). This is a two-tailed test because the claim is μd ≠ 0.
Step 2: Calculate the test statistic using the formula for a t-test for paired data: t = (d̄ - μd) / (sd / √n). Here, d̄ = -1, μd = 0 (from H₀), sd = 2.75, and n = 20. Substitute these values into the formula.
Step 3: Determine the degrees of freedom (df) for the t-distribution. The degrees of freedom for a paired t-test is given by df = n - 1. In this case, df = 20 - 1 = 19.
Step 4: Find the critical t-value(s) for a two-tailed test at the significance level α = 0.10 and df = 19. Use a t-distribution table or statistical software to find the critical values. These values will define the rejection region for the test.
Step 5: Compare the calculated t-value from Step 2 to the critical t-values from Step 4. If the calculated t-value falls in the rejection region (i.e., it is greater than the positive critical value or less than the negative critical value), reject the null hypothesis H₀. Otherwise, fail to reject H₀. Interpret the result in the context of the claim.

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Key Concepts

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

Paired Data

Paired data refers to two sets of related observations, often collected from the same subjects under different conditions. This type of data is used in statistical tests to determine if there is a significant difference between the two conditions. In this context, the differences between paired observations are analyzed to test the claim about their mean.
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Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample statistics to determine whether to reject H0. In this case, the null hypothesis would state that the mean of the differences (μd) equals zero, while the alternative claims it does not.
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Step 1: Write Hypotheses

Level of Significance (α)

The level of significance, denoted as α, is the threshold for determining whether to reject the null hypothesis. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. In this scenario, α is set at 0.10, indicating a 10% risk of concluding that a difference exists when there is none.
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Related Practice
Textbook Question

Yellowfin Tuna

A marine biologist claims that the mean fork length (see figure at the left) of yellowfin tuna is different in two zones in the eastern tropical Pacific Ocean. A sample of 26 yellowfin tuna collected in Zone A has a mean fork length of 76.2 centimeters and a standard deviation of 16.5 centimeters. A sample of 31 yellowfin tuna collected in Zone B has a mean fork length of 80.8 centimeters and a standard deviation of 23.4 centimeters. At ,α=0.01 can you support the marine biologist’s claim? Assume the population variances are equal. (Adapted from Fishery Bulletin)

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Textbook Question

Testing the Difference Between Two Means In Exercises 15–24, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

Home Prices A real estate agency says that the mean home sales price in Casper, Wyoming, is the same as in Cheyenne, Wyoming. The mean home sales price for 35 homes in Casper is \$349,237. Assume the population standard deviation is \$158,005. The mean home sales price for 41 homes in Cheyenne is \$435,244. Assume the population standard deviation is \$154,716. At α=0.01, is there enough evidence to reject the agency’s claim? (Adapted from Realtor.com)

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Textbook Question

Testing the Difference Between Two Means In Exercises 15–24, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.

Repair Costs: Washing Machines You want to buy a washing machine, and a salesperson tells you that the mean repair costs for Model A and Model B are equal. You research the repair costs. The mean repair cost of 24 Model A washing machines is \$208. Assume the population standard deviation is \$18. The mean repair cost of 26 Model B washing machines is \$221. Assume the population standard deviation is \$22. At α=0.01, can you reject the salesperson’s claim?

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Textbook Question

Parks and Mental Health In Exercises 13–18, use the figure, which shows the percentages from a survey of two hundred 18- to 24-year-olds in the United States who say that various park and recreation activities have a positive impact on their mental health. (Adapted from National Recreation and Park Association)



Taking Classes and Enjoying Nature At α=0.05, can you support the claim that the proportion of 18- to 24-year-olds who benefit mentally from taking classes in parks is less than the proportion who benefit mentally from enjoying nature in parks?

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