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 8.4.1

Chapter 8, Problem 8.4.1

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

Verified step by step guidance

Understand that the z-test for testing the difference between two population proportions is used to determine if there is a significant difference between the proportions of two populations.

Ensure that the data comes from two independent random samples. Independence is a key assumption for the z-test to be valid.

Verify that the sample sizes are sufficiently large. Specifically, for both populations, the expected counts of successes and failures should be at least 5. This means that for each sample, the conditions n₁p₁ ≥ 5, n₁(1-p₁) ≥ 5, n₂p₂ ≥ 5, and n₂(1-p₂) ≥ 5 must be satisfied, where n₁ and n₂ are the sample sizes, and p₁ and p₂ are the sample proportions.

Confirm that the sampling distribution of the difference between the sample proportions is approximately normal. This is typically ensured by the large sample size condition mentioned above.

Check that the populations are large enough relative to the sample sizes to satisfy the assumption of independence. Specifically, the population sizes should be at least 10 times larger than the sample sizes (N₁ ≥ 10n₁ and N₂ ≥ 10n₂).

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Key Concepts

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

Z-Test

The z-test is a statistical method used to determine if there is a significant difference between the means of two groups or the proportions of two populations. It assumes that the sampling distribution of the sample proportion is approximately normal, which is valid under certain conditions, particularly when sample sizes are large enough.

Normal Approximation

For the z-test to be applicable when comparing two population proportions, the sample sizes must be sufficiently large to ensure that the sampling distribution of the proportion is approximately normal. This is typically satisfied if both np and n(1-p) are greater than 5, where n is the sample size and p is the sample proportion.

Independence of Samples

The samples being compared in a z-test for proportions must be independent of each other. This means that the selection of one sample does not influence the selection of the other, which is crucial for the validity of the test results and ensures that the observed differences are due to actual population differences rather than sampling bias.

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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

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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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)

101

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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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Textbook Question

Constructing Confidence Intervals for μ1-μ2, When the sampling distribution for x̅1-x̅2 is approximated by a t-distribution and the population variances are not equal, 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 unequal variances.

10K Race

To compare the mean finishing times of male and female participants in a 10K race, you randomly select several finishing times from both sexes. The results are shown at the left. Construct an 80% confidence interval for the difference in mean finishing times of male and female participants in the race. (Adapted from Great Race)

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Textbook Question

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 =6.5, sd=9.54, n=16

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