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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 8.2.13
Chapter 8, Problem 8.2.13

Testing the Difference Between Two Means, (a) identify the claim and state H0 and Ha , (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic t, (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.
Pet Food
A pet association claims that the mean annual costs of food for dogs and cats are the same. The results for samples of the two types of pets are shown at the left. At , α=0.10 can you reject the pet association’s claim? Assume the population variances are equal. (Adapted from American Pet Products Association)
Table displaying sample statistics for annual pet food costs: dogs' mean \$255, cats' mean \$231, with respective standard deviations and sample sizes.

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Step 1: Identify the claim and state the null hypothesis (H0) and alternative hypothesis (Ha). The claim is that the mean annual costs of food for dogs and cats are the same. Thus, H0: μ1 = μ2 (the means are equal), and Ha: μ1 ≠ μ2 (the means are not equal).
Step 2: Determine the critical value(s) and rejection region(s). Since α = 0.10 and the test is two-tailed (due to Ha: μ1 ≠ μ2), use a t-distribution table to find the critical t-value(s) for the degrees of freedom (df). The degrees of freedom can be calculated using the formula: df = n1 + n2 - 2.
Step 3: Calculate the standardized test statistic t. Use the formula for the t-test for two independent samples with equal variances: t = (x̄1 - x̄2) / √[sp²(1/n1 + 1/n2)], where sp² = [(n1 - 1)s1² + (n2 - 1)s2²] / (n1 + n2 - 2). Substitute the given values for x̄1, x̄2, s1, s2, n1, and n2 into the formula.
Step 4: Compare the calculated t-value to the critical t-value(s) to decide whether to reject or fail to reject the null hypothesis. If the calculated t-value falls in the rejection region, reject H0; otherwise, fail to reject H0.
Step 5: Interpret the decision in the context of the original claim. If H0 is rejected, conclude that there is sufficient evidence to support the claim that the mean annual costs of food for dogs and cats are not the same. If H0 is not rejected, conclude that there is insufficient evidence to reject the claim that the mean annual costs are the same.

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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 two competing hypotheses: the null hypothesis (H0), which represents no effect or no difference, and the alternative hypothesis (Ha), which indicates the presence of an effect or difference. In this case, the null hypothesis would state that the mean costs of pet food for dogs and cats are equal.
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Step 1: Write Hypotheses

Critical Value and Rejection Region

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the significance level (α), which in this case is 0.10. The rejection region is the range of values for the test statistic that would lead to rejecting H0. Understanding where this region lies is crucial for making a correct decision regarding the null hypothesis.
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Critical Values: t-Distribution

Standardized Test Statistic (t)

The standardized test statistic, often denoted as t, is calculated to determine how far the sample mean difference is from the hypothesized mean difference under the null hypothesis, measured in terms of standard error. It allows for comparison against the critical value to decide whether to reject or fail to reject the null hypothesis. In this scenario, it will help assess whether the difference in mean costs for dogs and cats is statistically significant.
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Step 2: Calculate Test Statistic
Related Practice
Textbook Question

Testing a Difference Other Than Zero Sometimes a researcher is interested in testing a difference in means other than zero. In Exercises 27 and 28, you will test the difference between two means using a null hypothesis of Ho: μ1-μ2=k, Ho: μ1-μ2>=k or Ho: μ1-μ2<=k . The standardized test statistic is still

Software Engineer Salaries Is the difference between the mean annual salaries of entry level software engineers in Santa Clara, California, and Greenwich, Connecticut, more than \$4000? To decide, you select a random sample of entry level software engineers from each city. The results of each survey are shown in the figure at the left. Assume the population standard deviations are σ1=\$14,060 and σ2=\$13,050 . At α=0.05, what should you conclude? (Adapted from Salary.com)

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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.

In Exercises 29 and 30, construct the indicated confidence interval for μ1-μ2 .


Architect Salaries Construct a 99% confidence interval for the difference between the mean annual salaries of entry level architects in Denver, Colorado, and Lincoln, Nebraska, using the data from Exercise 28.

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Textbook Question

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=0.05, Assume (σ1)^2≠(σ2)^2

Sample statistics:

x̅1=2410, s1=175, n1=13 and x̅2=2305, s2=52, n2=10

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Textbook Question

Testing a Difference Other Than Zero Sometimes a researcher is interested in testing a difference in means other than zero. In Exercises 27 and 28, you will test the difference between two means using a null hypothesis of Ho: μ1-μ2=k, Ho: μ1-μ2>=k or Ho: μ1-μ2<=k . The standardized test statistic is still

Architect Salaries Is the difference between the mean annual salaries of entry level architects in Denver, Colorado, and Lincoln, Nebraska, equal to \$9000? To decide, you select a random sample of entry level architects from each city. The results of each survey are shown in the figure. Assume the population standard deviations are σ1=\$6560 and σ2=\$6100 . At α=0.01 what should you conclude? (Adapted from Salary.com)

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Textbook Question

In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim about the difference between two population proportions and at the level of significance . Assume the samples are random and independent.


Claim: p1≠p2, α=0.01


Sample statistics: x1=35, n1=70, and x2=36, n2=60

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Textbook Question

Parks and Mental Health In Exercises 13–18, use the figure, which shows the percentages from a survey of two hundred 18- to 24-year-olds in the United States who say that various park and recreation activities have a positive impact on their mental health. (Adapted from National Recreation and Park Association)



Exercising and Taking Classes At α=0.01, can you reject the claim that the proportion of 18- to 24-year-olds who benefit mentally from exercising in parks is greater than or equal to the proportion who benefit mentally from taking classes in parks?

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