410 CHAPTER 8 Hypothesis Testing sample but the second condition is violated, there are alternative methods that could be used, including these three alternative methods: ■ Bootstrap Resampling Use the confidence interval method of testing hypotheses, but obtain the confidence interval using bootstrap resampling, introduced in Section 7-4. See Section 8-5 for more detail about using this resampling method. ■ Randomization This method involves resampling (with replacement) after the sample data have been modified to reflect the value of the population parameter that is assumed in the null hypothesis. See Section 8-5 for more details about using this resampling method. ■ Sign Test See Section 13-2. ■ Wilcoxon Signed-Ranks Test See Section 13-3. Two-Tailed Test The preceding examples are based on the same left-tailed test. The following example illustrates a hypothesis test of a claim about a mean using a two-tailed test. Is the Mean Body Temperature Really 98.6°F? EXAMPLE 5 Data Set 5 “Body Temperatures” in Appendix B includes measured body temperatures with these statistics for 12 AM on day 2: n = 106, x = 98.20°F, s = 0.62°F. (This is the author’s favorite data set.) Use a 0.05 significance level to test the common belief that the population mean is 98.6°F. SOLUTION REQUIREMENT CHECK (1) With the study design used, we can treat the sample as a simple random sample. (2) The second requirement is that “the population is normally distributed or n 7 30.” The sample size is n = 106, so the second requirement is satisfied and there is no need to investigate the normality of the data. Both requirements are satisfied. Here are the steps that follow the procedure summarized in Figure 8-1. Step 1: The claim that “the population mean is 98.6°F” becomes m = 98.6°F when expressed in symbolic form. Step 2: The alternative (in symbolic form) to the original claim is m ≠ 98.6°F. Step 3: Because the statement m ≠ 98.6°F does not contain the condition of equality, it becomes the alternative hypothesis H1. The null hypothesis H0 is the statement that m = 98.6°F. H0: m = 98.6°F 1null hypothesis and original claim2 H1: m ≠ 98.6°F 1alternative hypothesis2 Step 4: As specified in the statement of the problem, the significance level is a = 0.05. Step 5: Because the claim is made about the population meanm, the sample statistic most relevant to this test is the sample mean x. We use the t distribution because the relevant sample statistic is x and the requirements for using the t distribution are satisfied. Step 6: The sample statistics are used to calculate the test statistic as follows, but technologies use unrounded values to provide the test statistic of t = -6.61. t = x - mx s2 n = 98.20 - 98.6 0.62 2 106 = -6.64 sa b Human Lie Detectors Researchers tested 13,000 people for their ability to determine when someone is lying. They found 31 people with exceptional skills at identifying lies. These human lie detectors had accuracy rates around 90%. They also found that federal officers and sheriffs were quite good at detecting lies, with accuracy rates around 80%. Psychology Professor Maureen O’Sullivan questioned those who were adept at identifying lies, and she said that “all of them pay attention to nonverbal cues and the nuances of word usages and apply them differently to different people. They could tell you eight things about someone after watching a two-second tape. It’s scary, the things these people notice.” Methods of statistics can be used to distinguish between people unable to detect lying and those with that ability. R t p t t w is
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