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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 8.2.3a
Chapter 8, Problem 8.2.3a

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
Ha:μ1≠μ2 , α=0.10 , n1=11 , n2=14

Verified step by step guidance
1
Identify the type of test based on the alternative hypothesis. Since the alternative hypothesis is \(H_a: \mu_1 \neq \mu_2\), this is a two-tailed test.
Determine the degrees of freedom for the test. Because the population variances are assumed equal, use the pooled variance approach. The degrees of freedom (df) is calculated as \(df = n_1 + n_2 - 2\).
Find the level of significance for each tail. Since \(\alpha = 0.10\) and the test is two-tailed, split the significance level equally between the two tails: \(\alpha/2 = 0.05\) for each tail.
Use the \(t\)-distribution table or a calculator to find the critical \(t\)-value corresponding to \(\alpha/2 = 0.05\) and the calculated degrees of freedom. This will give you the positive critical value \(t_{\alpha/2, df}\).
The critical values for the test are then \(\pm t_{\alpha/2, df}\), representing the rejection regions in both tails of the \(t\)-distribution.

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Key Concepts

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

Hypothesis Testing and Alternative Hypothesis

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis. The alternative hypothesis (Ha: μ1 ≠ μ2) suggests that the two population means are different, indicating a two-tailed test where critical values lie on both ends of the distribution.
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Level of Significance (α)

The level of significance, α, represents the probability of rejecting the null hypothesis when it is true (Type I error). Here, α = 0.10 means there is a 10% risk of such an error, which determines the critical values that define the rejection regions in the test.
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Two-Sample t-Test with Equal Variances

When population variances are assumed equal, a pooled variance estimate is used in the two-sample t-test. The test statistic follows a t-distribution with degrees of freedom calculated from the sample sizes (n1 + n2 - 2), which is essential for finding the critical t-values.
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Related Practice
Textbook Question

Take this test as you would take a test in class.For each exercise, perform the steps below.

b.Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed, and whether to use a z-test or a t-test. Explain your reasoning.


A real estate agency says that the mean home sales price in Olathe, Kansas, is greater than in Rolla, Missouri. The mean home sales price for 39 homes in Olathe is \$392,453. Assume the population standard deviation is \$224,902. The mean home sales price for 38 homes in Rolla is \$285,787. Assume the population standard deviation is \$330,578. At α=0.05, is there enough evidence to support the agency’s claim? (Adapted from Realtor.com)

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Textbook Question

Testing the Difference Between Two Means (a) identify the claim and state Ho and Ha .Assume the samples are random and dependent, and the populations are normally distributed.

[APPLET] Passing Play Percentages The passing play percentages of 10 randomly selected NCAA Division 1A college football teams for home and away games in the 2020–2021 season are shown in the table. At , α=0.20 is there enough evidence to support the claim that passing play percentage is different for home and away games? (Source: TeamRankings)

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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

Ha:μ1<μ2 , α=0.05 , n1=7 , n2=11

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Textbook Question

In Exercises 4 and 5, use technology to perform a two-sample t-test to determine whether there is a difference in the mint dates and in the values of coins found on a street from 1985 through 1996 for the two mint locations. Write your conclusion as a sentence. Use α = 0.05.



Mint dates of coins (years)


Philadelphia: x̅1=1984.8, s1=8.6


Denver: x̅2=1983.4, s2=8.4



Assume population variances are equal.

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Textbook Question

Confounding Variables A pharmaceutical company has applied for approval to market a new arthritis medication. The research involved a test group that was given the medication and another test group that was given a placebo. Describe some possible confounding variables that could influence the results of the study.

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Textbook Question

Testing the Difference Between Two Means (a) identify the claim and state Ho and Ha

[APPLET] Migraines

A researcher claims that injections of onabotulinumtoxinA reduce the number of days per month that chronic migraine sufferers have headaches. The table shows the number of days chronic migraine sufferers suffered migraines before and after using the treatment. At , α= 0.01 is there enough evidence to support the researcher’s claim? (Adapted from Journal of Headache and Pain)

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