Ch. 8 - Hypothesis Testing with Two Samples

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470당신이 사용하는 게 아니라요?교과서 변경

Larson 8th Edition

Ch. 8 - Hypothesis Testing with Two Samples

문제 8.1.2

8장, 문제 8.1.2

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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Step 1: State the null and alternative hypotheses. The null hypothesis (H₀) typically states that there is no difference between the population means (μ₁ = μ₂), while the alternative hypothesis (H₁) states that there is a difference (μ₁ ≠ μ₂, μ₁ > μ₂, or μ₁ < μ₂ depending on the context).

Step 2: Identify the sample statistics and population parameters. Gather the sample means (x̄₁ and x̄₂), sample sizes (n₁ and n₂), and the population standard deviations (σ₁ and σ₂) for both groups. Ensure the samples are independent and the population standard deviations are known.

Step 3: Calculate the test statistic (z). Use the formula:

z = \frac{(x_{1} - x_{2})}{\sqrt{\frac{σ_{1} ²}{n_{1}} + \frac{σ_{2} ²}{n_{2}}}}

. This formula accounts for the difference in sample means and the variability of the two populations.

Step 4: Determine the critical value or p-value. Based on the significance level (α) and the type of test (one-tailed or two-tailed), find the critical z-value from the standard normal distribution table or calculate the p-value corresponding to the test statistic.

Step 5: Make a decision. Compare the test statistic to the critical value or use the p-value. If the test statistic exceeds the critical value or the p-value is less than α, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the results in the context of the problem.

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Two-Sample Z-Test

A two-sample z-test is a statistical method used to determine if there is a significant difference between the means of two independent groups. This test is applicable when the population variances are known and the sample sizes are sufficiently large (typically n > 30). It compares the means by calculating a z-score, which indicates how many standard deviations the observed difference is from the expected difference under the null hypothesis.

Independent Samples

Independent samples refer to groups that are not related or paired in any way. In the context of a two-sample z-test, this means that the data collected from one sample does not influence or affect the data collected from the other sample. This independence is crucial for the validity of the test, as it ensures that the results are not biased by any relationship between the groups.

Population Means and Variances

Population means are the average values of a characteristic in a population, while population variances measure the spread of data points around the mean. In a two-sample z-test, knowing the population variances allows for the calculation of the standard error of the difference between the two means. This information is essential for determining the z-score and ultimately assessing whether the observed difference is statistically significant.

관련 실천

교과서 질문

Constructing Confidence Intervals for μ1-μ2. You can construct a confidence interval for the difference between two population means μ1-μ2 , as shown below, when both population standard deviations are known, and either both populations are normally distributed or both n1>= 30 and n2>=30 . Also, the samples must be randomly selected and independent.

[Image]

In Exercises 29 and 30, construct the indicated confidence interval for μ1-μ2 .

Software Engineer Salaries Construct a 95% confidence interval for the difference between the mean annual salaries of entry level software engineers in Santa Clara, California, and Greenwich, CT, using the data from Exercise 27.

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교과서 질문

What conditions are necessary to use the dependent samples t-test for the mean of the differences for a population of paired data?

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교과서 질문

[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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교과서 질문

"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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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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교과서 질문

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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