494 CHAPTER 9 Inferences from Two Samples Matched Pairs Section 9-3 presents methods for making inferences about matched pairs. The following example illustrates the use of resampling with matched pairs. Example: A 90% confidence interval is required for this left-tailed case, so we create a large list of differences (1000 resamples) and then find the confidence interval limits of P0.05 and P0.95. A typical result is a 90% confidence interval: -74.1 mm 6 m1 - m2 6 0.683 mm. This result is similar to the confidence interval found in Example 2 in Section 9-2. Because the confidence interval does include 0, it appears that there is not a significant difference between the mean height in 1988 and the mean height in 2012. (Note: Repeating this bootstrap procedure several times will easily lead to confidence interval limits that do not include 0, so it could easily appear that there is a significant difference between the mean height in 1988 and the mean height in 2012. The P-value of 0.0546 from the t-test from Example 1 in Section 9-2 also confirms that the sample data are close to the borderline between supporting and failing to support the claim that the mean height of the 1988 population is less than the mean height of the 2012 population. In this case, the sample data do not provide compelling evidence in favor or against that claim.) Randomization As previously described, the randomization procedure involves combining both sets of sample data, then randomly selecting samples without replacement using the same sample sizes as the original samples. Find the difference in sample means, and then repeat many times to determine whether the original difference rarely occurs or commonly occurs. Example: Using the sample data, we get x1 - x2 = 1739.4 - 1777.8 = -38.4 mm. If we use the randomization procedure described in Example 1 with the two sets of sample data to generate 1000 simulated differences, a typical result is that the difference of -38.4 mm or below will occur 51 times (for a proportion of 0.051). This is quite close to the P-value of 0.0546 from Example 1 in Section 9-2. As in Example 1, we conclude that there is not sufficient evidence to support the claim that the mean height of the 1988 male population is less than the mean height of the 2012 male population. (Note: As in the preceding bootstrapping example, the sample data are close to the borderline between supporting and failing to support the claim that the mean height of the 1988 population is less than the mean height of the 2012 population. The sample data do not provide compelling evidence in favor or against that claim.) YOUR TURN. Do Exercise 11 “Regular Coke and Diet Coke.” HINT For the randomization in Example 3, it was found that there were 51 differences of -38.4 mm or below. Instead of getting a result of 51 each time, repeating that randomization process will result in counts typically ranging from about 35 to 65. In such cases, answers in Appendix B include the statement that “results vary.” Know that such results can vary by somewhat substantial amounts. Resampling with Matched Pairs EXAMPLE 4 Examples 1 and 2 in Section 9-3 use the following eight measured and reported weights to test the claim that for males, the measured weights tend to be higher than the reported weights. Use a 0.05 significance level.
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