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Ch. 8 - Hypothesis Testing with Two Samples
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 8 - Hypothesis Testing with Two SamplesProblem 8.1.17
Chapter 8, Problem 8.1.17

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=0.01, Assume (σ1)^2=(σ2)^2

Verified step by step guidance
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Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (Hₐ). Since the claim is μ₁ = μ₂, the null hypothesis is H₀: μ₁ = μ₂, and the alternative hypothesis is Hₐ: μ₁ ≠ μ₂ (two-tailed test).
Step 2: Determine the test statistic to use. Since the population variances are assumed to be equal (σ₁² = σ₂²), use the pooled t-test formula for the test statistic: t = (x1 - x2)(n1 - 1)s1² + (n2 - 1)s2²n1 + n2 - 2, where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.
Step 3: Calculate the degrees of freedom (df) for the test. Since the population variances are equal, use the formula: df = n1 + n2 - 2.
Step 4: Determine the critical t-value(s) from the t-distribution table for a two-tailed test at the significance level α = 0.01 and the calculated degrees of freedom. These critical values will define the rejection region.
Step 5: Compare the calculated test statistic (t) to the critical t-value(s). If the test statistic falls in the rejection region, reject the null hypothesis (H₀). Otherwise, fail to reject the null hypothesis. Interpret the result in the context of the claim.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to determine whether to reject H0. In this case, the null hypothesis states that the means of two populations are equal (μ1=μ2), while the alternative suggests they are not.
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Step 1: Write Hypotheses

Level of Significance (α)

The level of significance, denoted as α, is the threshold for determining whether to reject the null hypothesis. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. In this scenario, α is set at 0.01, indicating a 1% risk of concluding that there is a difference between the population means when there is none.
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Assumption of Equal Variances

The assumption of equal variances, often referred to as homoscedasticity, is crucial when comparing two population means. It implies that the variances of the two populations are the same (σ1^2 = σ2^2), which allows for the use of specific statistical tests, such as the pooled t-test. This assumption simplifies the analysis and increases the reliability of the test results.
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Related Practice
Textbook Question

Testing the Difference Between Two Proportions In Exercises 7–12, (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.


Multiple Sclerosis Drug In a study to determine the effectiveness of using a drug to treat multiple sclerosis, 488 subjects were given the drug and 244 subjects were given a placebo. The numbers of subjects who had 12-week confirmed disability progression were tracked. The results are shown at the left. At α=0.01, can you support the claim that there is a difference in the proportion of subjects who had no 12-week confirmed disability progression? (Adapted from The New England Journal of Medicine)


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Textbook Question

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.

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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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Textbook Question

Young Adults In a survey of 3500 males ages 20 to 24 whose highest level of education is some college, but no bachelor’s degree, 80.2% were employed. In a survey of 2000 males ages 20 to 24 whose highest level of education is a bachelor’s degree or higher, 86.4% were employed. At α=0.01, can you support the claim that there is a difference in the proportion of those employed between the two groups? (Adapted from National Center for Education Statistics)

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Textbook Question

What conditions are necessary to use the t-test for testing the difference between two population means?

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Textbook Question

[APPLET] Teaching Methods

A new method of teaching reading is being tested on third grade students. A group of third grade students is taught using the new curriculum. A control group of third grade students is taught using the old curriculum. The reading test scores for the two groups are shown in the back-to-back stem-and-leaf plot.

At , α=0.10 is there enough evidence to support the claim that the new method of teaching reading produces higher reading test scores than the old method does? Assume the population variances are equal.

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Textbook Question

Blue Crabs A marine researcher claims that the stomachs of blue crabs from one location contain more fish than the stomachs of blue crabs from another location. The stomach contents of a sample of 25 blue crabs from Location A contain a mean of 320 milligrams of fish and a standard deviation of 60 milligrams. The stomach contents of a sample of 15 blue crabs from Location B contain a mean of 280 milligrams of fish and a standard deviation of 80 milligrams. At , α= 0.01can you support the marine researcher’s claim? Assume the population variances are equal.

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