492 CHAPTER 9 Inferences from Two Samples Step 3: Find the difference between the means for each randomization sample, then sort these means in ascending order. Example: The difference between the male/female means for each of the 1000 randomization samples is calculated and sorted. Step 4: Find the number of randomization sample mean differences (found in Step 3) that are “at least as extreme” as d = −4.0 found in Step 1. To determine the number of differences that are “at least as extreme” as -4.0, find the number of differences less than or equal to -4 and greater than or equal to 4. Example: Among 1000 resamplings, there are 89 with sample mean differences of -4 or lower, and there are 90 sample differences greater than or equal to 4. Step 5: Add the two values from Step 4 and divide by the number of randomizations (from Step 2) to get the estimated P-value. Example: P-Value is estimated to be 189 + 902>1000, or 0.179. INTERPRETATION Using a significance level of 0.05, we fail to reject the null hypotheses of H0: m1 = m2 because the estimated P-value of 0.179 is greater than the significance level of 0.05. Using the randomization method, we find that there is not sufficient evidence to warrant rejection of the claim that the mean income of males is different from the mean income of females. YOUR TURN. Do Exercise 9 “Are Weights Changing Over Time?” Two Proportions Section 9-1 presents methods for making inferences about two population proportions. The following example illustrates how resampling can be used to construct confidence intervals and test hypotheses. The first example in Section 9-1 included the following sample data. That example specified a 0.05 significance level for testing the claim that there is no difference in success rates between the two smoking cessation treatment groups. Proportion of Success (not smoking after 52 weeks) E-Cigarette Group: pn 1 = 79>438 = 0.180 Nicotine Replacement Group: pn 2 = 44>446 = 0.099 The above two samples have this difference between the proportions: pn 1 - p n 2 = 0.180 - 0.099 = 0.081 Bootstrapping The procedure for bootstrap resampling with two samples involves creating a bootstrap sample for the first sample by sampling with replacement, then doing the same for the second sample. Next, find the difference between the two unsorted bootstrap sample means. Repeat many times to get a large list of differences, then sort those differences and find P2.5 and P97.5, which are the confidence interval limits of a 95% confidence interval estimate of the difference between the two population proportions. Example: Using the bootstrap procedure with the sample proportions of 79>438 and 44>446, a typical result is this 95% confidence interval: 0.0363 6 p1 - p2 6 0.123. CP EXAMPLE 2 Resampling to Test a Claim About Two Proportions
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