10. Hypothesis Testing for Two Samples

Explain how to perform a two-sample z-test for the difference between two population means using independent samples with and known.

Describe another way you can perform a hypothesis test for the difference between the means of two populations using independent samples with and known that does not use rejection regions.

Independent and Dependent Samples In Exercises 5–8, classify the two samples as independent or dependent and justify your answer.

Sample 1: The commute times of 10 workers when they use their own vehicles

Sample 2: The commute times of the same 10 workers when they use public transportation

In Exercises 11–14, test the claim about the difference between two population means and at the level of significance . Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1<μ2; α=0.05

Population statistics:σ1=75 and σ2=105

Sample Statistics: x̅1=2435, n1=35, x̅2=2432, n2=90

In Exercises 11–14, test the claim about the difference between two population means and at the level of significance . Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1<μ2; α=0.03

Population statistics:σ1=136 and σ2=215

Sample Statistics: x̅1=5004, n1=144, x̅2=4895, n2=156

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.

Braking Distances To compare the dry braking distances from 60 to 0 miles per hour for two makes of automobiles, a safety engineer conducts braking tests for 16 compact SUVs and 11 midsize SUVs. The mean braking distance for the compact SUVs is 131.8 feet. Assume the population standard deviation is 5.5 feet. The mean braking distance for the midsize SUVs is 132.8 feet. Assume the population standard deviation is 6.7 feet. At α=0.10 , can the engineer support the claim that the mean braking distances are different for the two categories of SUVs? (Adapted from Consumer Reports)

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

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.

ACT Mathematics and Science Scores The mean ACT mathematics score for 60 high school students is 20.2. Assume the population standard deviation is 5.7. The mean ACT science score for 75 high school students is 20.6. Assume the population standard deviation is 5.9. At α=0.01, can you reject the claim that ACT mathematics and science scores are equal? (Source: ACT, Inc.)

Getting at the Concept Explain why the null hypothesis Ho: μ1=μ2 is equivalent to the null hypothesis .Ho: μ1-μ2=0

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.

ACT English and Reading Scores The mean ACT English score for 120 high school students is 19.9. Assume the population standard deviation is 7.2. The mean ACT reading score for 150 high school students is 21.2. Assume the population standard deviation is 7.1. At α=0.10, can you support the claim that ACT reading scores are higher than ACT English scores? (Source: ACT, Inc.)

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)

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.

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

Software Engineer Salaries Construct a 95% confidence interval for the difference between the mean annual salaries of entry level software engineers in Santa Clara, California, and Greenwich, CT, using the data from Exercise 27.

What conditions are necessary to use the t-test for testing the difference between two population means?

Explain how to perform a two-sample t-test for the difference between two population means.

In Exercises 5–8, 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=0.05

Population statistics: σ1= 0.30 and σ2= 0.23

Sample statistics: x̅1 = 1.28, n1 = 96, and x̅2= 1.34, n2= 85