SECTION 7.3 Hypothesis Testing for the Mean (s Unknown) 385 Using a P-Value with a t-Test In Exercises 27–30, (a) identify the claim and state H0 and Ha, (b) use technology to find the P-value, (c) decide whether to reject or fail to reject the null hypothesis, and (d) interpret the decision in the context of the original claim. Assume the population is normally distributed. 27. Quarter Mile Times A consumer group claims that the mean minimum time it takes for a sedan to travel a quarter mile is greater than 15.3 seconds. A random sample of 22 sedans has a mean minimum time to travel a quarter mile of 15.8 seconds and a standard deviation of 2.36 seconds. At a = 0.10, do you have enough evidence to support the consumer group’s claim? (Adapted from Zero to 60 Times) 28. Dive Duration An oceanographer claims that the mean dive duration of a North Atlantic right whale is 11.5 minutes. A random sample of 34 dive durations has a mean of 12.2 minutes and a standard deviation of 2.2 minutes. Is there enough evidence to reject the claim at a = 0.10? (Source: Marine Ecology Progress Series) 29. Faculty Classroom Hours The dean of a university estimates that the mean number of classroom hours per week for full-time faculty is 11.0. As a member of the student council, you want to test this claim. A random sample of the number of classroom hours for eight full-time faculty for one week is shown in the table at the left. At a = 0.01, can you reject the dean’s claim? 30. Class Size You receive a brochure from a large university. The brochure indicates that the mean class size for full-time faculty is fewer than 32 students. You want to test this claim. You randomly select 18 classes taught by full-time faculty and determine the class size of each. The results are shown in the table at the left. At a = 0.05, can you support the university’s claim? Extending Concepts Deciding on a Distribution In Exercises 31 and 32, decide whether you should use the standard normal sampling distribution or a t-sampling distribution to perform the hypothesis test. Justify your decision. Then use the distribution to test the claim. Write a short paragraph about the results of the test and what you can conclude about the claim. 31. Gas Mileage A car company claims that the mean gas mileage for its luxury sedan is at least 23 miles per gallon. You believe the claim is incorrect and find that a random sample of 5 cars has a mean gas mileage of 22 miles per gallon and a standard deviation of 4 miles per gallon. At a = 0.05, test the company’s claim. Assume the population is normally distributed. 32. Tuition and Fees An education publication claims that the mean in-state tuition and fees at public four-year institutions by state is more than $10,500 per year. A random sample of 30 states has a mean in-state tuition and fees at public four-year institutions of $10,931 per year. Assume the population standard deviation is $2380. At a = 0.01, test the publication’s claim. (Adapted from College Board) 33. Writing You are testing a claim and incorrectly use the standard normal sampling distribution instead of the t@sampling distribution, mistaking the sample standard deviation for the population standard deviation. Does this make it more or less likely to reject the null hypothesis? Is this result the same no matter whether the test is left-tailed, right-tailed, or two-tailed? Explain your reasoning. Classroom hours 11.8 8.6 12.6 7.9 6.4 10.4 13.6 9.1 TABLE FOR EXERCISE 29 Class sizes 35 28 29 33 32 40 26 25 29 28 30 36 33 29 27 30 28 25 TABLE FOR EXERCISE 30
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