728 CHAPTER 15 Holistic Statistics 0.002 shows that among 1000 randomizations, fewer than 2 yielded sample means at least as extreme as 98.2°F.) This shows that if the population mean is really 98.6°F, it is highly unlikely that we would ever get a sample with a mean of 98.2°F, but we really did get a sample with a mean of 98.2°F. The reasonable explanation for that result is that it could not have come from a population with a mean of 98.6°F, so we have strong evidence against 98.6°F being the population mean. Statdisk Randomization Minitab Randomization Bootstrap Resampling: Using the bootstrap resampling method results in the 95% confidence interval of 98.07°F 6 m 6 98.31°F, which is very close to the 95% confidence interval obtained by using the t distribution. (Due to the randomness used in the bootstrap method, the confidence interval could vary somewhat.) Nonparametric test using the Wilcoxon signed-ranks test with a 0.05 significance level: Test statistic is z = -5.67 and the critical values are z = {1.96, so conclude that there is sufficient evidence to warrant rejection of the claim that the median body temperature is equal to 98.6°F. Note that this nonparametric test is not greatly affected by the presence of outliers. Nonparametric test using the sign test for matched pairs with a 0.05 significance level: Test statistic is z = -4.61 and the critical values are z = {1.96, so conclude that there is sufficient evidence to warrant rejection of the claim that body temperatures have a median of 98.6°F. Simulation: For the simulation, use a technology to generate samples of size 106 (as in the sample), a normal distribution (as in the sample), an assumed mean of 98.6°F, and a standard deviation of 0.6228965°F (as in the sample). Listed below in ascending order are the means from 50 samples generated with the assumption that the population mean is 98.6°F. See that the actual sample mean of 98.2°F is not even close to any of the simulated means. We now reason as follows: 1. If the population mean is really 98.6°F, the sample mean of 98.2°F is very unlikely. 2. Because the sample mean of 98.2°F actually did occur, either the sample is very unusual, or the assumed mean of 98.6°F is wrong. 3. Given that none of the simulated samples has a mean close to the sample mean of 98.2°F, it appears that the better explanation is that the assumed population mean of 98.6°F is wrong. 98.5 98.5 98.5 98.5 98.5 98.5 98.5 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.6 98.7 98.7 98.7 98.7 98.7 98.8
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