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 8.2.10

Chapter 8, Problem 8.2.10

Test the claim about the difference between two population means and 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=0.345, s1=0.305 , n1=11 and x̅2=0.515, s2=0.215, n2=9

Verified step by step guidance

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, since the claim is that the first mean is less than the second.

Since the population variances are assumed equal (σ1² = σ2²), calculate the pooled sample variance using the formula: \[ S_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2} \] where \(s_1\) and \(s_2\) are the sample standard deviations, and \(n_1\) and \(n_2\) are the sample sizes.

Calculate the test statistic for the difference between two means using the pooled variance: \[ t = \frac{\bar{x}_1 - \bar{x}_2}{S_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \] where \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means.

Determine the degrees of freedom for the t-test, which is \(df = n_1 + n_2 - 2\).

Find the critical t-value from the t-distribution table for a left-tailed test at significance level \(\alpha = 0.10\) with \(df\) degrees of freedom. Compare the calculated t-statistic to this critical value to decide whether to reject or fail to reject the null hypothesis.

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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

This involves comparing the means of two independent populations to determine if there is statistical evidence supporting a specific claim about their difference. The null hypothesis typically states no difference (μ1 = μ2), while the alternative reflects the claim (μ1 < μ2). The test uses sample data to decide whether to reject the null at a given significance level.

Pooled Variance and Equal Population Variances Assumption

When population variances are assumed equal, sample variances are combined into a pooled variance estimate to improve accuracy. This pooled variance is used to calculate the standard error of the difference between means, which is essential for the test statistic in a two-sample t-test.

Significance Level and One-Tailed Test

The significance level (α = 0.10) defines the threshold for rejecting the null hypothesis, representing a 10% risk of Type I error. Since the claim is directional (μ1 < μ2), a one-tailed test is used, focusing on the left tail of the distribution to determine if the sample evidence supports the claim.

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Performing Hypothesis Tests: Proportions

Related Practice

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.

ACT Mathematics and Science Scores The mean ACT mathematics score for 60 high school students is 20.2. Assume the population standard deviation is 5.7. The mean ACT science score for 75 high school students is 20.6. Assume the population standard deviation is 5.9. At α=0.01, can you reject the claim that ACT mathematics and science scores are equal? (Source: ACT, Inc.)

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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.

[Image] Complicated mathematical formula.

In Exercises 23–26, construct the indicated confidence interval for p1-p2. Assume the samples are random and independent.

Critical Threats Repeat Exercise 25 but with a 99% confidence interval. Describe the likelihood that equal proportions of the population see cyberterrorism and the spread of infectious diseases as critical threats in the next 10 years.

150

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Textbook Question

In Exercises 11–14, test the claim about the difference between two population means and at the level of significance . Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1<μ2; α=0.03

Population statistics:σ1=136 and σ2=215

Sample Statistics: x̅1=5004, n1=144, x̅2=4895, n2=156

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Textbook Question

Independent and Dependent Samples In Exercises 5–8, classify the two samples as independent or dependent and justify your answer.

Sample 1: The IQ scores of 60 females

Sample 2: The IQ scores of 60 males

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Textbook Question

ACT English and Reading Scores The mean ACT English score for 120 high school students is 19.9. Assume the population standard deviation is 7.2. The mean ACT reading score for 150 high school students is 21.2. Assume the population standard deviation is 7.1. At α=0.10, can you support the claim that ACT reading scores are higher than ACT English scores? (Source: ACT, Inc.)

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Textbook Question

Find the critical value(s) for the alternative hypothesis, level of significance , and sample sizes and . Assume that the samples are random and independent, the populations are normally distributed, and the population variances are (a) equal and (b) not equal.

Ha:μ1>μ2 , α=0.01 , n1=12 , n2=15

128

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