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.RE.7

8장, 문제 8.RE.7

In Exercises 5–8, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1<μ2; α=0.10

Population statistics: σ1= 0.11 and σ2= 0.10

Sample statistics: x̅1 = 0.28, n1 = 41, and x̅2= 0.33, n2= 34

검증된 단계별 안내

Identify the null and alternative hypotheses based on the claim μ1 < μ2. The null hypothesis (H0) is μ1 ≥ μ2, and the alternative hypothesis (H1) is μ1 < μ2.

Determine the significance level α = 0.10, which will be used to find the critical value for the test.

Since the population standard deviations σ1 and σ2 are known, use the z-test for the difference between two means. Calculate the test statistic using the formula: \[\text{z} = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}\] Note that under the null hypothesis, (μ1 - μ2) = 0.

Calculate the standard error of the difference between the sample means using: \[\text{SE} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}\]

Compare the calculated z-test statistic to the critical z-value for a left-tailed test at α = 0.10. If the test statistic is less than the critical value, reject the null hypothesis; otherwise, do not reject it.

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주요 개념

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Hypothesis Testing for Two Population Means

This involves formulating null and alternative hypotheses about the difference between two population means (μ1 and μ2). The goal is to determine if there is enough evidence to support the claim (μ1 < μ2) using sample data, by comparing a test statistic to a critical value or p-value at a given significance level.

Sampling Distribution and Test Statistic

The test statistic measures how far the observed difference between sample means (x̅1 - x̅2) is from the hypothesized difference under the null hypothesis. When population standard deviations (σ1, σ2) are known and samples are independent and normal, the test statistic follows a normal distribution, allowing calculation of z-scores.

Significance Level and Decision Rule

The significance level (α = 0.10) defines the probability of rejecting the null hypothesis when it is true (Type I error). It sets the critical value(s) for the test statistic. If the test statistic falls into the rejection region determined by α, the null hypothesis is rejected in favor of the alternative claim.

관련 실천

교과서 질문

In Exercises 11–16, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1>= μ2; α=0.01. Assume (σ1)^2 = (σ2)^2

Sample statistics: x̅1= 44.5, s1= 5.85, n1= 17 and x̅2= 49.1, s2= 5.25, n2= 18

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교과서 질문

In Exercises 25–28, determine whether a normal sampling distribution can be used. If it can be used, test the claim about the difference between two population proportions p1 and p2 at the level of significance α. Assume the samples are random and independent.

Claim: p1<p2; α=0.05

Sample statistics: x1 = 86, n1=900 and x2 = 107, n2 = 1200

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교과서 질문

Claim: μ1≠ μ2; α=0.01. Assume (σ1)^2 = (σ2)^2

Sample statistics: x̅1= 61, s1= 3.3, n1= 5 and x̅2= 55.1, s2= 1.2, n2= 7

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교과서 질문

Claim: μ1< μ2; α=0.10. Assume (σ1)^2 ≠ (σ2)^2

Sample statistics: x̅1=0.015, s1=0.011, n1= 8 and x̅2=0.019, s2=0.004, n2= 6

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교과서 질문

"In Exercises 9 and 10, (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.

A career counselor claims that the mean annual salaries of entry level paralegals in Dayton, Ohio, and Coventry, Rhode Island, are the same. The mean annual salary of 40 randomly selected entry level paralegals in Dayton is \$58,180. Assume the population standard deviation is \$10,990. The mean annual salary of 35 randomly selected entry level paralegals in Coventry is \$61,120. Assume the population standard deviation is \$11,850. At α=0.10, is there enough evidence to reject the counselor’s claim? (Adapted from Salary.com)"

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교과서 질문

In Exercises 17 and 18, (a) identify the claim and state Ho and Ha, Assume the samples are random and independent, and the populations are normally distributed.

A real estate agent claims that there is no difference between the mean household incomes of two neighborhoods. The mean income of 12 randomly selected households from the first neighborhood is \$52,750 with a standard deviation of \$2900. In the second neighborhood, 10 randomly selected households have a mean income of \$51,200 with a standard deviation of \$2225. At α=0.01, can you reject the real estate agent’s claim? Assume the population variances are equal.

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