384 CHAPTER 8 Hypothesis Testing Resampling Methods (See Section 8-5) In addition to the P-value method, critical value method, and the use of confidence intervals, another approach for testing claims about population parameters is to use resampling methods that involve the use of technology to “resample” the original sample data many times. Section 8-5 is focused on these resampling methods: ■ Bootstrap The bootstrap resampling method was introduced in Section 7-4, and it is used to construct a confidence interval that could be used to estimate a population parameter. The confidence interval can then be used to address the claim being tested. ■ Randomization The randomization method involves resampling after the sample data have been modified to reflect the value of the population parameter that is assumed in the null hypothesis. The resamples help us to determine whether a sample statistic is consistent with the claimed value of a parameter or whether it contradicts the claimed value of a parameter. PART 2 Type I and Type II Errors When testing a null hypothesis, we arrive at a conclusion of rejecting it or failing to reject it. Our conclusions are sometimes correct and sometimes wrong (even if we apply all procedures correctly). With a hypothesis test, it is not simply a matter of being right or wrong. Different types of errors can have dramatically different consequences, and that is why we distinguish between type I errors and type II errors. For example, consider the dramatic difference between these two errors: ■ Conclude that a new medicine is effective when in reality it is not effective. ■ Conclude that a new medicine is not effective when in reality it is effective. Table 8-3 includes two different types of errors and we distinguish between them by calling them type I and type II errors, as described here: ■ Type I error: The mistake of rejecting the null hypothesis when it is actually true. The symbol a (alpha) is used to represent the probability of a type I error. A = P1type I error2 = P1rejecting H0 whenH0 is true2 ■ Type II error: The mistake of failing to reject the null hypothesis when it is actually false. The symbol b (beta) is used to represent the probability of a type II error. B = P1type II error2 = P1failing to reject H0 whenH0 is false2 TABLE 8-3 Type I and Type II Errors True State of Nature Null hypothesis is true Null hypothesis is false Preliminary Conclusion Reject H0 Type I error: Reject a true H0. P(type I error) = a Correct decision ✓ Fail to reject H0 Correct decision ✓ Type II error: Fail to reject a false H0. P (type II error) = b R c cl u fo Sham Surgery as a Placebo Sham surgery is used as a control in clinical trials of surgical interventions (similar to use of a placebo in drug trials). With sham or fake surgery, patients undergo the same pre- and post-surgery procedures as real surgery. They fast, they are given anesthesia, and incisions are made to mimic real surgery, but the actual surgery is not performed. Surprisingly, one study of 53 trials showed that sham surgery worked as well as real surgery in about half of the cases. Meniscus surgery had results that were no better than those from sham surgery. This shows that some surgeries are often not as effective as believed, and there can be some real benefit from sham surgery. Because of ethical considerations, sham surgeries are conducted only with the patient’s approval. S a c o (
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