430 CHAPTER 8 Hypothesis Testing Example: In this example, we use technology to modify the original sample (n = 12, x = 6.83333333 hours) to have a sample mean of 7.0 hours, and then generate 1000 resampled data sets using this modified data set. A typical result is that among 1000 resampled data sets, there are 377 sample means that are at 6.83333333 or lower, so there appears to be about a 0.377 chance of getting a sample mean of 6.83333333 or lower. It appears that the sample mean of 6.83333333 can easily occur with a population mean of 7.0. The sample mean of 6.83333333 does not appear to be significantly low, so there is not sufficient evidence to support the claim that m 6 7 hours. There is not sufficient sample evidence to support the claim that the mean amount of sleep for adults is less than 7 hours. YOUR TURN. Do Exercise 11 “Cell Phone Radiation.” Testing a Claim About a Standard Deviation or Variance Section 8-4 presents methods for testing a claim made about a population standard deviation or variance. Example 4 shows how resampling can be used to test such claims. Minting Quarters: Bootstrap Resampling Method EXAMPLE 4 The first example in Section 8-4 included the following sample data consisting of weights of randomly selected quarters. That example specified that a 0.05 significance level be used to test the claim that these weights are from a population with a standard deviation that is less than 0.062 g. This claim can be tested by using the resampling method of bootstrapping. 5.7424 5.7328 5.7268 5.5938 5.6342 5.6839 5.6651 5.6925 5.6803 5.6245 5.7985 5.7180 5.7299 5.6582 5.7360 5.6546 5.7222 5.6619 5.7041 5.6528 5.6210 5.6613 5.6484 5.6502 Bootstrapping The confidence interval obtained from the bootstrap resampling method can be used to determine the likely values of the population standard deviation or variance, and that can be used to form a conclusion about the claim being tested. Example: Using technology with bootstrapping to obtain the 90% confidence interval using the bootstrap resampling method (from Section 7-4), we get this approximate confidence interval: 0.03374 g 6 s 6 0.05851 g. Because of the randomness used, this confidence interval might differ somewhat. In testing the claim that “the standard deviation is less than 0.062 g,” we see that the assumed value of s = 0.062 g is not contained within the confidence interval and all values of the confidence interval are less than 0.062 g, so we support the claim that the sample weights are from a population with a standard deviation less than 0.062 g. This conclusion is different than the one reached in Example 2 in Section 8-4. Randomization As of this writing, there is no statistical software that does randomization for a standard deviation or variance. See Exercise 13 for an approach used by the author.
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