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 8.2.17

Chapter 8, Problem 8.2.17

Annual Income
A politician claims that the mean household income in a recent year is greater in York County, South Carolina, than it is in Elmore County, Alabama. In York County, a sample of 23 residents has a mean household income of \$64,900 and a standard deviation of \$16,000. In Elmore County, a sample of 19 residents has a mean household income of \$59,500 and a standard deviation of \$23,600. At , α= 0.05can you support the politician’s claim? Assume the population variances are not equal. (Adapted from U.S. Census Bureau)

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Identify the type of hypothesis test needed: Since we are comparing the means of two independent samples with unequal variances, we will use a two-sample t-test with unequal variances (Welch's t-test).

Set up the null and alternative hypotheses: The null hypothesis (H₀) states that the mean household income in York County is less than or equal to that in Elmore County (μ₁ ≤ μ₂). The alternative hypothesis (H₁) states that the mean income in York County is greater than in Elmore County (μ₁ > μ₂), reflecting the politician's claim.

Calculate the test statistic using the formula for Welch's t-test: t = rac{ar{x}_1 - ar{x}_2}{ oot{2}rac{s_1^2}{n_1} + rac{s_2^2}{n_2}} where ar{x}_1 and ar{x}_2 are the sample means, s_1 and s_2 are the sample standard deviations, and n_1 and n_2 are the sample sizes for York and Elmore counties respectively.

Determine the degrees of freedom for the test using the Welch-Satterthwaite equation: df = rac{ig(rac{s_1^2}{n_1} + rac{s_2^2}{n_2}ig)^2}{rac{(rac{s_1^2}{n_1})^2}{n_1 - 1} + rac{(rac{s_2^2}{n_2})^2}{n_2 - 1}} This accounts for the unequal variances and sample sizes.

Find the critical t-value from the t-distribution table at significance level α = 0.05 for a one-tailed test with the calculated degrees of freedom. Compare the calculated t-statistic to this critical value to decide whether to reject the null hypothesis and support the politician's claim.

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Key Concepts

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

Two-Sample t-Test for Means with Unequal Variances

This test compares the means of two independent samples when population variances are unknown and assumed unequal. It uses the sample means, standard deviations, and sizes to calculate a test statistic, which helps determine if there is a significant difference between the two population means.

Hypothesis Testing and Significance Level (α)

Hypothesis testing involves setting up a null hypothesis (no difference) and an alternative hypothesis (difference exists). The significance level α (0.05 here) is the threshold for rejecting the null hypothesis, indicating a 5% risk of concluding a difference when none exists.

Degrees of Freedom in Unequal Variance t-Test

When variances are unequal, degrees of freedom are calculated using the Welch-Satterthwaite equation, which adjusts for sample size and variance differences. This affects the critical t-value and p-value, ensuring accurate inference about the difference between means.

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Related Practice

Textbook Question

[APPLET] Teaching Methods

Two teaching methods and their effects on science test scores are being reviewed. A group of students is taught in traditional lab sessions. A second group of students is taught using interactive simulation software. The science test scores for the two groups are shown in the back-to-back stem-and-leaf plot.

At , α=0.01 can you support the claim that the mean science test score is lower for students taught using the traditional lab method than it is for students taught using the interactive simulation software? Assume the population variances are equal.

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Textbook Question

Explain how to perform a two-sample z-test for the difference between two population means using independent samples with and known.

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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.

[APPLET] Precipitation A climatologist claims that the precipitation in Seattle, Washington, was greater than in Birmingham, Alabama, in a recent year. The daily precipitation amounts (in inches) for 30 days in a recent year in Seattle are shown below. Assume the population standard deviation is 0.25 inch.

0.00 0.00 0.05 0.01 0.21 0.00 0.00 0.52 0.00 0.010.00 0.19 0.00 0.18 0.02 0.02 0.13 0.00 0.03 0.000.04 0.00 0.41 0.23 0.00 0.80 0.15 0.00 0.00 0.79

The daily precipitation amounts (in inches) for 30 days in a recent year in Birmingham are shown below. Assume the population standard deviation is 0.52 inch.

0.00 0.96 0.84 0.00 0.10 0.00 0.00 0.20 0.00 0.54 0.97 0.00 0.35 0.02 0.04 0.70 0.00 0.00 0.00 0.00 0.03 0.01 0.15 0.27 0.00 0.00 0.93 0.00 0.89 0.01

At α=0.05, can you support the climatologist’s claim? (Source: NOAA)"

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Textbook Question

Constructing Confidence Intervals for p1-p2 You can construct a confidence interval for the difference between two population proportions p1-p2 by using the inequality below.

$({\hat{p}}_{1} - {\hat{p}}_{2}) - z_{c} \sqrt{\frac{{\hat{p}}_{1} {\hat{q}}_{1}}{n_{1}} + \frac{{\hat{p}}_{2} {\hat{q}}_{2}}{n_{2}}} < p_{1} - p_{2} < ({\hat{p}}_{1} - {\hat{p}}_{2}) + z_{c} \sqrt{\frac{{\hat{p}}_{1} {\hat{q}}_{1}}{n_{1}} + \frac{{\hat{p}}_{2} {\hat{q}}_{2}}{n_{2}}}$

In Exercises 23–26, construct the indicated confidence interval for p1-p2. Assume the samples are random and independent.

Students Planning to Study Visual and Performing Arts In a survey of 10,000 students taking the SAT, 7% were planning to study visual and performing arts in college. In another survey of 8000 students taken 10 years before, 8% were planning to study visual and performing arts in college. Construct a 95% confidence interval for p1-p2, where p1 is the proportion from the recent survey and p2 is the proportion from the survey taken 10 years ago. (Adapted from College Board)

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Textbook Question

What conditions are necessary to use the z-test for testing the difference between two population proportions?

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Textbook Question

Testing the Difference Between Two Means (a) identify the claim and state Ho and Ha , (b) find the critical value(s) and identify the rejection region(s), (c) calculate d̄ and Sd, (d) find the standardized test statistic t, (e) decide whether to reject or fail to reject the null hypothesis, and (f) interpret the decision in the context of the original claim. Assume the samples are random and dependent, and the populations are normally distributed.

[APPLET] Therapeutic Taping

A physical therapist claims that the use of a specific type of therapeutic tape reduces pain in patients with chronic tennis elbow. The table shows the pain levels on a scale of 0 to 10, where 0 is no pain and 10 is the worst pain possible, for 15 patients with chronic tennis elbow when holding a 1 kilogram weight. At , α=0.05 is there enough evidence to support the therapist’s claim? (Adapted from BioMed Central, Ltd.)

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