Section 9.4 Hypothesis Tests for Proportions 447 samples 100 messages and finds the mean length to be 7.3 characters. Both Juan and Mary obtained the same standard deviation. Will Juan’s P-value be greater than, less than, or the same as Mary’s P-value? Explain. greater 39. Interpret a P-value: A real estate agent believes that the mean size of houses in a certain city is greater than 1500 square feet. He samples 100 houses, and performs a test of H0 : �� = 1500 versus H1: �� > 1500. He obtains a P-value of 0.0002. a. The real estate agent concludes that because the P-value is very small, the mean house size must be much greater than 1500. Is this conclusion justified? No b. Another real estate agent says that because the P-value is very small, we can be fairly certain that the mean size is greater than 1500, but we cannot conclude that it is a lot greater. Is this conclusion justified? Yes 40. Interpret a P-value: The manufacturer of a medication designed to lower blood pressure claims that the mean systolic blood pressure for people taking their medication is less than 135. To test this claim, blood pressure is measured for a sample of 500 people who are taking the medication. The P-value for testing H0 : �� = 135 versus H1: �� < 135 is P = 0.001. a. The manufacturer concludes that because the P-value is very small, we can be fairly certain that the mean pressure is less than 135, but we cannot conclude that it is a lot smaller. Is this conclusion justified? Yes b. Someone else says that because the P-value is very small, we can conclude that the mean pressure is a lot less than 135. Is this conclusion justified? No Extending the Concepts 41. Using z instead of t: When the sample size is large, some people treat the sample standard deviation s as if it were the population standard deviation ��, and use the standard normal distribution rather than the Student’s t distribution, to find a critical value. Assume that a right-tailed test will be made with a sample of size 100 from a normal population, using the �� = 0.05 significance level. a. Find the critical value under the assumption that �� is known. 1.645 b. In fact, �� is unknown. How many degrees of freedom should be used for the Student’s t distribution? 99 c. What is the probability of rejecting H0 when it is true if the critical value in part (a) is used? You will need technology to find the answer. 0.0516 Answers to Check Your Understanding Exercises for Section 9.3 1. a. P-value is between 0.01 and 0.025 Tech: 0.0127 b. P-value is between 0.025 and 0.05 Tech: 0.0485 c. P-value is between 0.005 and 0.01 Tech: 0.0096 d. P-value is between 0.01 and 0.02 Tech: 0.0148 2. a. t = 2.309 b. 40 c. Between 0.01 and 0.025 Tech: 0.0131 d. Weaker; the P-value is larger. 3. 0.0128 4. a. H0: �� = 7, H1: �� < 7 b. 6 c. 5 d. 6.68 e. 0.205 f. −3.823593745 g. 0.00616394 h. Yes 5. 0.0312 6. a. 2.920 b. −2.787, 2.787 c. 1.292 d. −2.160, 2.160 SECTION 9.4 Hypothesis Tests for Proportions Objectives 1. Test a hypothesis about a proportion using the P-value method 2. Test a hypothesis about a proportion using the critical value method Objective 1 Test a hypothesis about a proportion using the P-value method How cool is Facebook? In a recent GenX2Z American College Student Survey, 90% of female college students rated the social network site Facebook as ‘‘cool.’’ The other 10% rated it as ‘‘lame.’’ Assume that the survey was based on a sample of 500 students. A marketing executive at Facebook wants to advertise the site with the slogan ‘‘More than 85% of female college students think Facebook is cool.’’ Before launching the ad campaign, he wants to be confident that the slogan is true. Can he conclude that the proportion of female college students who think Facebook is cool is greater than 0.85? This is an example of a problem that calls for a hypothesis test about a population proportion. There are two categories, ‘‘cool’’ and ‘‘lame.’’ The quantity 0.85 represents the proportion in the ‘‘cool’’ category. To perform the test, we will need some notation, which we summarize as follows.
navidi_monk_elementary_statistics_2e_ch7-9
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