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 8.4.23

Chapter 8, Problem 8.4.23

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)

Verified step by step guidance

Step 1: Identify the given values from the problem. From the recent survey, the sample size is n1 = 10,000 and the sample proportion is p̂1 = 0.07. From the survey taken 10 years ago, the sample size is n2 = 8,000 and the sample proportion is p̂2 = 0.08.

Step 2: Calculate the complement proportions for each sample. For the recent survey, q̂1 = 1 - p̂1 = 1 - 0.07. For the older survey, q̂2 = 1 - p̂2 = 1 - 0.08.

Step 3: Compute the standard error (SE) for the difference in proportions using the formula: SE = sqrt((p̂1 * q̂1 / n1) + (p̂2 * q̂2 / n2)). Substitute the values of p̂1, q̂1, n1, p̂2, q̂2, and n2 into the formula.

Step 4: Determine the critical value (z_c) for a 95% confidence level. For a 95% confidence interval, z_c is approximately 1.96.

Step 5: Construct the confidence interval for p1 - p2 using the formula: (p̂1 - p̂2) - z_c * SE < p1 - p2 < (p̂1 - p̂2) + z_c * SE. Substitute the values of p̂1, p̂2, z_c, and SE into the formula to find the interval.

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Key Concepts

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

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. It is expressed with a certain level of confidence, such as 95%, indicating the probability that the interval includes the parameter. In the context of proportions, it helps estimate the difference between two population proportions based on sample data.

Proportion and Sample Size

A proportion is a statistical measure that represents the fraction of a whole, often expressed as a percentage. In this case, p1 and p2 represent the proportions of students planning to study visual and performing arts from two different surveys. The sample size, denoted as n1 and n2, is crucial because larger samples generally provide more reliable estimates of the population proportions.

Z-score and Standard Error

The Z-score is a statistical measurement that describes a value's relation to the mean of a group of values, expressed in terms of standard deviations. In constructing confidence intervals, the Z-score (z_c) is used to determine the margin of error. The standard error, calculated from the sample proportions and sizes, quantifies the variability of the sample proportion estimates, which is essential for determining the width of the confidence interval.

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Related Practice

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

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

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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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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