428 CHAPTER 8 Hypothesis Testing Criteria for “At Least As Extreme” The following criteria for “at least as extreme” are based on means, but proportions or standard deviations could also be addressed using these same criteria. In the following, x is the original sample mean found from the unmodified original sample, and m is the value of the population mean assumed in the null hypothesis. “Values” refers to the sample mean for each resampling. ■ Left-tailed test: Values at least as extreme as the original sample mean x are sample means that are less than or equal to x. ■ Right-tailed test: Values at least as extreme as x are sample means that are greater than or equal to x. ■ Two-tailed test: Find the difference d between x and m (the claimed value of the population parameter used in the null hypothesis), and express d as a positive value. That is, d = x - m . Using that difference d, values at least as extreme as x are those that are . . . less than or equal to 1m - d2 or greater than or equal to 1m + d2 In Example 1 we have a sample with x = 6.0 and we have a two-tailed hypothesis test in which we assume that m = 6.5. For Example 1, d = 6.0 - 6.5 = 0.5. The value of m - d is 6.5 - 0.5 = 6.0; the value of m + d is 6.5 + 0.5 = 7.0. So values that are at least as extreme as x are those that are less than or equal to 6.0 or greater than or equal to 7.0 How Many Randomizations? It would be wise to repeat the randomization at least 1000 times. Professional statisticians commonly resample 10,000 or more times. It is obviously impractical to resample that many times using any manual procedure, so the use of software such as Statdisk is very strongly recommended. Testing a Claim About a Proportion Section 8-2 presents methods for testing a claim made about a population proportion. Example 2 shows how resampling can be used to test such claims. HINT Technologies such as Statdisk will automatically execute the steps listed in the preceding example, so enter and use the original unmodified data. There is no need to manually perform the calculations illustrated in the preceding example. The claim given in the Chapter Problem, “Most Internet users utilize two-factor authentication to protect their online data,” can be tested by using the resampling methods of bootstrapping (introduced in Section 7-4) and randomization. Bootstrapping The confidence interval obtained from the bootstrap resampling method can be used to determine the likely values of the population proportion p, and that can be used to form a conclusion about the claim being tested. Example: Using bootstrap resampling to find the 90% confidence interval limits for the Chapter Problem 1n = 926; x = 4822 yields the result of approximately 0.495 6 p 6 0.548. Because of the random nature of this process, your resulting CP EXAMPLE 2 Resampling to Test a Claim About a Proportion
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