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.RE.12

Chapter 8, Problem 8.RE.12

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=0.10. Assume (σ1)^2 ≠ (σ2)^2

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

Verified step by step guidance

Identify the null hypothesis (H₀) and the alternative hypothesis (H₁) based on the claim μ₁ < μ₂. Here, H₀: μ₁ ≥ μ₂ and H₁: μ₁ < μ₂, since the claim is that the first mean is less than the second.

Since the population variances are not equal (σ₁² ≠ σ₂²) and the sample sizes are small, use the two-sample t-test with unequal variances (Welch's t-test).

Calculate the test statistic using the formula:

t = x̅

₁ - x̅₂s₁2n₁ + s₂2n₂, where

x̅

₁, x̅₂ are sample means,

s

₁, s₂ are sample standard deviations, and

n_1, n_2

are sample sizes.

Determine the degrees of freedom (df) for the test using the Welch-Satterthwaite equation:

df = s

₁2n₁ + s₂2n₂2>s₁2n₁2n₁ - 1 + s₂2n₂2n₂ - 1.

Find the critical t-value from the t-distribution table for a left-tailed test at significance level α = 0.10 with the calculated degrees of freedom, then compare the test statistic to this critical value to decide whether to reject H₀.

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Key Concepts

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

Hypothesis Testing for Two Population Means

Hypothesis testing involves making a claim about population parameters and using sample data to assess its validity. Here, the claim is that μ1 < μ2, which is a one-tailed test comparing two means. The goal is to determine if the observed difference in sample means provides enough evidence to support this claim at the given significance level α.

Unequal Variances (Welch's t-test)

When the population variances are unknown and assumed unequal, Welch's t-test is used instead of the pooled t-test. This test adjusts the degrees of freedom and does not assume equal variances, making it more reliable for samples with different variances and sizes, as in this problem.

Level of Significance (α) and Decision Rule

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

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Related Practice

Textbook Question

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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Textbook Question

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

Textbook Question

In Exercises 17 and 18, (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 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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Textbook Question

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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Textbook Question

"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)"

Textbook Question

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.