9-5 Resampling: Using Technology for Inferences 495 Bootstrapping For the bootstrap resampling procedure with matched pairs, first find the differences d between the matched pairs of data, and then use the same bootstrapping procedure described in Section 7-4 for one mean. Example: The matched pairs of data have these “measured – reported” differences: 2.6, 1.3, 4.8, 0.5, 9.9, 0.3, -8.6, 0.6. A 90% confidence interval is required, so we find the confidence interval limits of P0.05 and P0.95. A typical result is the confidence interval: -1.46 lb 6 md 6 4.08 lb. This confidence interval is reasonably close to the confidence interval obtained in Section 9-3. Because the confidence interval does include 0, it appears that there is no significant difference between measured weights and reported weights. Randomization For the randomization procedure for matched pairs, we are assuming in the null hypothesis that md = 0, so each pair of values can occur in any order. We therefore randomly select the order for each pair of values, then find the difference d, then find the mean d. We repeat that procedure many times, such as 1000, and we proceed to find the number of d values that are at least as extreme as the original value of d found from the original sample values. Example: Using the sample data, we get d = 1.425 lb. We use the randomization procedure to find the likelihood of getting a value d Ú 1.425 (or at least as extreme as the value of 1.425 that was obtained) for this right-tailed test. Here is a typical result: With 1000 resamples, the mean of 1.425 or greater occurs 222 times so its likelihood is about 0.222. This is similar to the P@value = 0.231 found in Section 9-3. This shows that d = 1.425 is not rare and can easily occur by chance, so it appears that there is no significant difference between measured weights and reported weights. TABLE 9-2 Measured and Reported Weights (lb) Subject 1 2 3 4 5 6 7 8 Measured Weight (lb) 152.6 149.3 174.8 119.5 194.9 180.3 215.4 239.6 Reported Weight (lb) 150 148 170 119 185 180 224 239 YOUR TURN. Do Exercise 13 “Measured and Reported Weights.” Two Variances or Standard Deviations Section 9-4 presents methods for making inferences about two population variances (or standard deviations). For our resampling methods involving two variances or standard deviations, we will use these same two stipulations: 1. Let the first sample be the one with the larger standard deviation (or variance). 2. Use the test statistic format of F = s2 1>s 2 2. Resampling With Two Variances or Standard Deviations EXAMPLE 5 Example 1 in Section 9-4 tested the claim that the variation among male U.S. Army personnel weights did not change from the ANSUR I study in 1988 to the ANSUR II study in 2012, and that example used a 0.05 significance level. Here are the data used in that example after the two samples have been switched so that the first sample has the larger standard deviation. ANSUR II 2012 90.8 86.1 101.1 76.9 63.0 98.4 83.5 65.1 111.5 78.0 ANSUR I 1988 63.088.9 71.183.684.276.369.574.4 81.472.085.5111.1 continued
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