Ch. 8 - Hypothesis Testing with Two Samples

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 8 - Hypothesis Testing with Two Samples

Problem 8.4.11

Chapter 8, Problem 8.4.11

Seat Belt Use In a survey of 1000 drivers from the West, 934 wear a seat belt. In a survey of 1000 drivers from the Northeast, 909 wear a seat belt. At α=0.05, can you support the claim that the proportion of drivers who wear seat belts is greater in the West than in the Northeast? (Adapted from National Highway Traffic Safety Administration)

Verified step by step guidance

Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). Here, H₀: p_West ≤ p_Northeast (the proportion of seat belt users in the West is less than or equal to that in the Northeast), and H₁: p_West > p_Northeast (the proportion in the West is greater).

Calculate the sample proportions for each region:

\(\hat{p}\)_{West} = \(\frac{934}{1000}\)

and

\(\hat{p}\)_{Northeast} = \(\frac{909}{1000}\)

Compute the pooled proportion

\(\hat{p}\)

since the null hypothesis assumes the proportions are equal:

\(\hat{p}\) = \(\frac{934 + 909}{1000 + 1000}\)

Calculate the standard error (SE) of the difference between the two sample proportions using the formula:

SE = \(\sqrt{\hat{p}\)(1 - \(\hat{p}\)) \(\left\)( \(\frac{1}{n_{West}\)} + \(\frac{1}{n_{Northeast}\)} \(\right\))}

, where

n_{West} = 1000

and

n_{Northeast} = 1000

Compute the test statistic (z) using:

z = \(\frac{\hat{p}\)_{West} - \(\hat{p}\)_{Northeast}}{SE}

. Then, compare this z-value to the critical z-value for α = 0.05 in a one-tailed test to decide whether to reject H₀.

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Key Concepts

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

Hypothesis Testing for Two Proportions

This involves comparing two population proportions to determine if there is a statistically significant difference between them. We set up a null hypothesis (no difference) and an alternative hypothesis (one proportion is greater), then use sample data to test these claims at a chosen significance level.

Significance Level (α) and p-value

The significance level, α, is the threshold for rejecting the null hypothesis, commonly set at 0.05. The p-value measures the probability of observing the sample data if the null hypothesis is true. If the p-value is less than α, we reject the null hypothesis, supporting the alternative claim.

Calculation of Test Statistic for Two Proportions

The test statistic compares the difference between sample proportions relative to the variability expected under the null hypothesis. It is calculated using the pooled proportion and standard error, then converted to a z-score to assess significance against the standard normal distribution.

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Guided course

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Difference in Proportions: Hypothesis Tests

Related Practice

Textbook Question

Describe another way you can perform a hypothesis test for the difference between the means of two populations using independent samples with and known that does not use rejection regions.

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Textbook Question

Independent and Dependent Samples In Exercises 5–8, classify the two samples as independent or dependent and justify your answer.

Sample 1: The commute times of 10 workers when they use their own vehicles

Sample 2: The commute times of the same 10 workers when they use public transportation

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Textbook Question

Getting at the Concept Explain why the null hypothesis Ho: μ1=μ2 is equivalent to the null hypothesis .Ho: μ1-μ2=0

168

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Textbook Question

Constructing Confidence Intervals for μ1-μ2. When the sampling distribution for x̅1-x̅2 is approximated by a t-distribution and the populations have equal variances, you can construct a confidence interval for μ1-μ2, as shown below.

Construct the indicated confidence interval for μ1-μ2 . Assume the populations are approximately normal with equal variances.

10K Race

To compare the mean ages of male and female participants in a 10K race, you randomly select several ages from both sexes. The results are shown below. Construct a 95% confidence interval for the difference in mean ages of male and female participants in the race. (Adapted from Great Race)

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Textbook Question

In Exercises 11–14, test the claim about the difference between two population means and at the level of significance . Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1<μ2; α=0.05

Population statistics:σ1=75 and σ2=105

Sample Statistics: x̅1=2435, n1=35, x̅2=2432, n2=90

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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)

Socializing and Taking Classes At α=0.05, can you support the claim that the proportion of 18- to 24-year-olds who benefit mentally from socializing in parks is different from the proportion who benefit mentally from taking classes in parks?

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