8-5 Resampling: Using Technology for Hypothesis Testing 429 confidence interval might be somewhat different. The confidence interval of 0.495 6 p 6 0.548 shows that it is likely that the true value of the population proportion p can be anywhere between 0.495 6 p 6 0.548, so it could be less than or equal to 0.5 and we can’t conclude that p 7 0.5. We can’t support the conclusion that most Internet users utilize two-factor authentication to protect their online data. Randomization Modify the original sample data to have the same value of the proportion claimed in the null hypothesis. (Use a column of 0s and 1s, where the proportion of 1s is the proportion claimed in the null hypothesis.) Next, resample with replacement to determine the likelihood of getting a sample proportion at least as extreme as the one obtained. Example: In this example, we use technology to modify the original sample data 1n = 926; x = 4822 to have a sample proportion of 0.5 (as defined by the null hypothesis) and then generate 1000 resampled data sets using this modified data set. A typical result is that among these 1000 resampled data sets, there are 114 sample proportions that are at least 0.52, so there appears to be about a 0.114 chance of getting a sample proportion of 0.52 or greater. It appears that the sample proportion of 0.52 can easily occur with a population proportion of 0.5. That is, the sample proportion of 0.52 does not appear to be significantly high, so there is not sufficient evidence to support the claim that p 7 0.5. There is not sufficient sample evidence to support the claim that most Internet users utilize two-factor authentication to protect their online data. YOUR TURN. Do Exercise 5 “Cursed Movie.” Testing a Claim About a Mean Section 8-3 presents methods for testing a claim made about a population mean. Example 3 illustrates the use of resampling for testing such claims. Adult Sleep: Resampling Methods EXAMPLE 3 The first example in Section 8-3 included the following sample data consisting of hours slept in one night for randomly selected adults. That example specified that a 0.05 significance level be used to test the claim that the mean amount of sleep for adults is less than 7 hours. This claim can be tested by using the resampling methods of bootstrapping and randomization. 4 8 4 4 8 6 9 7 7 10 7 8 Bootstrapping The confidence interval obtained from the bootstrap resampling method can be used to determine the likely values of the population mean m, and that can be used to form a conclusion about the claim being tested. Example: Using technology for bootstrapping to find the 90% confidence interval limits for this example, we get a confidence interval of approximately 5.9 hours 6 m 6 7.8 hours, which is quite close to the confidence interval of 5.8 hours 6 m 6 7.9 hours obtained using the methods of Section 8-3. The confidence interval shows that there is not sufficient evidence to support the conclusion that the population mean m is less than 7 hours. Randomization Modify the sample data to have the same mean claimed in the null hypothesis, then resample with replacement to determine the likelihood of getting a sample mean at least as extreme as the one obtained. continued
RkJQdWJsaXNoZXIy NjM5ODQ=