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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 8.2.6a
Chapter 8, Problem 8.2.6a

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=0.01 , n1=19 , n2=22

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 degrees of freedom formula: \(df = n_1 + n_2 - 2\).
Find the level of significance for each tail. Since \(\alpha = 0.01\) and the test is two-tailed, divide \(\alpha\) by 2 to get \(\alpha/2 = 0.005\) for each tail.
Use the \(t\)-distribution table or a calculator to find the critical \(t\)-value corresponding to \(df\) degrees of freedom and the upper tail probability of \(0.005\).
The critical values will be \(\pm t_{\alpha/2, df}\), 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) indicates a two-tailed test, meaning we are checking if the two population means are different in either direction.
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Level of Significance (α)

The level of significance, α, represents the probability of rejecting the null hypothesis when it is actually true (Type I error). Here, α = 0.01 means there is a 1% risk of a false positive, which affects the critical value and the rejection region 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 determines the critical values for the test.
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Related Practice
Textbook Question

The mean room rate for two adults for a random sample of 26 three-star hotels in Cincinnati has a sample standard deviation of \$31. Assume the population is normally distributed. (Adapted from Expedia)


Construct a 99% confidence interval for the population standard deviation.

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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

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.

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A researcher claims that sprint interval training improves running performance in trained athletes. The table shows the maximum aerobic speed (MAS), in kilometers per hour, of trained athletes before and after six sessions of sprint interval training. At , α=0.10 is there enough evidence to support the researcher’s claim? (Adapted from National Strength and Conditioning Association)

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Textbook Question

Testing the Difference Between Two Means (b) find the critical value(s) and identify the rejection region(s), 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=0.10 , n1=30 , n2=32

33
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Textbook Question

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

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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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