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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.RE.10
Chapter 8, Problem 8.RE.10

"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)"

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Step 1: Identify the claim and state the null hypothesis (H\_0) and alternative hypothesis (H\_a). The claim is that the mean annual salaries of entry level paralegals in Dayton and Coventry are the same. Therefore, the null hypothesis is H\_0: \(\mu\)_1 = \(\mu\)_2, and the alternative hypothesis is H\_a: \(\mu\)_1 \(\neq\) \(\mu\)_2, where \(\mu\)_1 is the mean salary in Dayton and \(\mu\)_2 is the mean salary in Coventry. This is a two-tailed test because the claim is about equality.
Step 2: Find the critical value(s) and identify the rejection region(s). Since the significance level \(\alpha\) = 0.10 and the test is two-tailed, split \(\alpha\) into two tails of 0.05 each. Use the standard normal distribution (Z-distribution) to find the critical values \(\pm\) z_{\(\alpha\)/2}. These critical values define the rejection regions: reject H\_0 if the test statistic is less than -z_{\(\alpha\)/2} or greater than +z_{\(\alpha\)/2}.
Step 3: Calculate the standardized test statistic z. Use the formula for the difference between two means with known population standard deviations:
\[z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}\]
where \(\bar{x}\)_1 = 58180, \(\bar{x}\)_2 = 61120, \(\sigma\)_1 = 10990, \(\sigma\)_2 = 11850, n_1 = 40, and n_2 = 35. Since under H\_0, \(\mu\)_1 - \(\mu\)_2 = 0, substitute these values into the formula to compute z.
Step 4: Compare the calculated z value to the critical values found in Step 2. If z falls into the rejection region (less than -z_{\(\alpha\)/2} or greater than +z_{\(\alpha\)/2}), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
Step 5: Interpret the decision in the context of the original claim. If you rejected H\_0, conclude that there is sufficient evidence at the 0.10 significance level to say the mean salaries are different. If you failed to reject H\_0, conclude that there is not sufficient evidence to say the mean salaries differ, supporting the counselor's 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 decide whether there is enough evidence to reject a claim about a population parameter. It involves stating a null hypothesis (Ho) representing no effect or difference, and an alternative hypothesis (Ha) representing the claim to be tested. The test evaluates sample data to determine if the observed effect is statistically significant.
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Two-Sample Z-Test for Means

A two-sample z-test compares the means of two independent populations when population standard deviations are known. It calculates a standardized test statistic (z) to measure the difference between sample means relative to the variability expected by chance. This test helps determine if the difference in means is statistically significant.
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Critical Value and Rejection Region

The critical value is a threshold determined by the significance level (α) that defines the rejection region(s) for the test statistic. If the calculated test statistic falls within the rejection region, the null hypothesis is rejected. This concept helps control the probability of making a Type I error, or falsely rejecting a true null hypothesis.
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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=0.10. Assume (σ1)^2 ≠ (σ2)^2


Sample statistics: x̅1= 520, s1= 25, n1= 7 and x̅2= 500, s2= 55, n2= 6

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

In Exercises 23 and 24, (a) identify the claim and state Ho and Ha , (b) find the critical value(s) and identify the rejection region(s), (c) calculate d̄ and sd, (d) find the standardized test statistic t, (e) decide whether to reject or fail to reject the null hypothesis, and (f) interpret the decision in the context of the original claim. Assume the samples are random and dependent, and the populations are normally distributed.


A physical fitness instructor claims that a weight loss supplement will help users lose weight after two weeks. The table shows the weights (in pounds) of 9 adults before using the supplement and two weeks after using the supplement. At α=0.10, is there enough evidence to support the physical fitness instructor’s claim?


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=0.10. Assume (σ1)^2 ≠ (σ2)^2


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

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.


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.