Ch. 8 - Hypothesis Testing with Two Samples

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 8 - Hypothesis Testing with Two Samples

Problem 8.4.20

Chapter 8, Problem 8.4.20

Daily Activities In Exercises 19–22, the results of a survey of 200 U.S. randomly selected U.S. men and 300 randomly selected U.S. women are shown in the figure at the left, which displays the percentages engaged in working or socializing and communicating on an average day. (Adapted from U.S. Bureau of Labor Statistics)

Women’s Activities At α=0.01, can you reject the claim that the proportion of women who work is the same as the proportion of women who socialize and communicate on an average day?

Verified step by step guidance

Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁). H₀: The proportion of women who work is equal to the proportion of women who socialize and communicate (p₁ = p₂). H₁: The proportion of women who work is not equal to the proportion of women who socialize and communicate (p₁ ≠ p₂).

Step 2: Identify the sample proportions from the data. From the graph, the proportion of women who work is 33% (p₁ = 0.33), and the proportion of women who socialize and communicate is 37% (p₂ = 0.37).

Step 3: Calculate the test statistic for comparing two proportions. Use the formula for the z-test statistic:

z = (p

₁-p₂)p(1-p)n, where p is the pooled proportion and n is the sample size.

Step 4: Compute the pooled proportion (p) using the formula:

p = x

₁+x₂n₁+n₂, where x₁ and x₂ are the number of successes (working and socializing) and n₁ and n₂ are the sample sizes. For women, n₁ = n₂ = 300.

Step 5: Compare the calculated z-test statistic to the critical value at α = 0.01. If |z| > zₐ/₂ (critical value for a two-tailed test), reject the null hypothesis; otherwise, fail to reject the null hypothesis.

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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 a population 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 context, the null hypothesis would state that the proportion of women who work is equal to the proportion who socialize and communicate.

Proportions

Proportions are a way to express a part of a whole as a fraction of 1. In this question, we are comparing the proportions of women engaged in two different activities: working and socializing. Understanding how to calculate and interpret these proportions is crucial for analyzing the survey results and making statistical inferences.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether to reject the null hypothesis in hypothesis testing. A common significance level is 0.05, but in this case, it is set at 0.01, indicating a stricter criterion for rejecting the null hypothesis. This means that there must be strong evidence against H0 to conclude that the proportions are different.

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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=0.01, Assume (σ1)^2=(σ2)^2

Sample statistics:

x̅1=33.7, s1=3.5 , n1=12 and x̅2=35.5 , s2=2.2 , n2=17

Textbook Question

Constructing Confidence Intervals for μ1-μ2. When the sampling distribution for x̅1-x̅2 is approximated by a t-distribution and the populations have equal variances, you can construct a confidence interval for μ1-μ2, as shown below.

Construct the indicated confidence interval for μ1-μ2 . Assume the populations are approximately normal with equal variances.

Family Doctor

To compare the mean number of days spent waiting to see a family doctor for two large cities, you randomly select several people in each city who have had an appointment with a family doctor. The results are shown at the left. Construct a 90% confidence interval for the difference in mean number of days spent waiting to see a family doctor for the two cities. (Adapted from Merritt Hawkins)

Textbook Question

Testing the Difference Between Two Means In Exercises 15–24, (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.

Bed-in-a-Box To compare customer satisfaction with mattresses that are delivered compressed in a box and traditional mattresses, a researcher randomly selects 30 ratings of mattresses in boxes and 30 ratings of traditional mattresses. The mean rating of mattresses in boxes is 68.7 out of 100. Assume the population standard deviation is 6.6. The mean rating of traditional mattresses is 70.9 out of 100. Assume the population standard deviation is 5.6. At α=0.01, can the researcher support the claim that the mean rating of traditional mattresses is greater than the mean rating of mattresses in a box? (Adapted from Consumer Reports)

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Textbook Question

Home Prices A real estate agency says that the mean home sales price in Casper, Wyoming, is the same as in Cheyenne, Wyoming. The mean home sales price for 35 homes in Casper is \$349,237. Assume the population standard deviation is \$158,005. The mean home sales price for 41 homes in Cheyenne is \$435,244. Assume the population standard deviation is \$154,716. At α=0.01, is there enough evidence to reject the agency’s claim? (Adapted from Realtor.com)

Textbook Question

Repair Costs: Washing Machines You want to buy a washing machine, and a salesperson tells you that the mean repair costs for Model A and Model B are equal. You research the repair costs. The mean repair cost of 24 Model A washing machines is \$208. Assume the population standard deviation is \$18. The mean repair cost of 26 Model B washing machines is \$221. Assume the population standard deviation is \$22. At α=0.01, can you reject the salesperson’s claim?

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)

Taking Classes and Enjoying Nature At α=0.05, can you support the claim that the proportion of 18- to 24-year-olds who benefit mentally from taking classes in parks is less than the proportion who benefit mentally from enjoying nature in parks?