390 CHAPTER 8 Hypothesis Testing Type I and Type II Errors. In Exercises 25–28, provide statements that identify the type I error and the type II error that correspond to the given claim. (Although conclusions are usually expressed in verbal form, the answers here can be expressed with statements that include symbolic expressions such as p = 0.1.) 25. The proportion of people who write with their left hand is equal to 0.1. 26. The proportion of people who know that water boils at a lower temperature at high altitudes is equal to 0.34. 27. The proportion of drivers who make angry gestures is greater than 0.25. 28. The proportion of job interviews that result from online applications is less than 1>2. 29.Interpreting Power Chantix (varenicline) tablets are used as an aid to help people stop smoking. In a clinical trial, 129 subjects were treated with Chantix twice a day for 12 weeks, and 16 subjects experienced abdominal pain (based on data from Pfizer, Inc.). If someone claims that more than 8% of Chantix users experience abdominal pain, that claim is supported with a hypothesis test conducted with a 0.05 significance level. Using 0.18 as an alternative value of p, the power of the test is 0.96. Interpret this value of the power of the test. 30. Calculating Power Consider a hypothesis test of the claim that the Ericsson method of gender selection is effective in increasing the likelihood of having a baby girl, so that the claim is p 7 0.5. Assume that a significance level of a = 0.05 is used, and the sample is a simple random sample of size n = 64. a. Assuming that the true population proportion is 0.65, find the power of the test, which is the probability of rejecting the null hypothesis when it is false. (Hint: With a 0.05 significance level, the critical value is z = 1.645, so any test statistic in the right tail of the accompanying top graph is in the rejection region where the claim is supported. Find the sample proportion pn in the top graph, and use it to find the power shown in the bottom graph.) b. Explain why the green-shaded region of the bottom graph represents the power of the test. 31.Finding Sample Size to Achieve Power Researchers plan to conduct a test of a gender selection method. They plan to use the alternative hypothesis of H1: p 7 0.5 and a significance level of a = 0.05. Find the sample size required to achieve at least 80% power in detecting an increase in p from 0.50 to 0.55. (This is a very difficult exercise. Hint: See Exercise 30.) 32.“At Least” and “At Most” Repeat Exercise 5 after changing the claim to this: “At most 10% of homes have only a landline telephone and no wireless phone.” What do you conclude about this new claim of “at most 10%” if the null hypothesis is rejected? What do you conclude about this new claim of “at most 10%” if we fail to reject the null hypothesis? 8-1 Beyond the Basics b z 5 1.645 a5 0.05 p5 0.5 p5 0.65 Power Key Concept This section describes a complete hypothesis testing procedure for testing a claim made about a population proportion p. We illustrate hypothesis testing with these three methods: (1) the P-value method, (2) the critical value method, (3) the 8-2 Testing a Claim About a Proportion
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