496 CHAPTER 9 Inferences from Two Samples Bootstrap Resampling with Two Standard Deviations or Two Variances The bootstrap method can be applied as follows: (1) Generate separate bootstrap samples from each of the two sets of sample data; (2) obtain the standard deviation from each of the bootstrap samples; (3) Using the two lists of unsorted standard deviations, divide the first list by the second list to obtain a list of ratios of the type s1>s2; (4) sort those ratios; (5) find the percentile values (such as P2.5 and P97.5) from the sorted list of ratios. If the confidence interval includes the ratio of 1 (indicating that s1 = s2), that is evidence that there is not a significant difference between s1 and s2. If the confidence interval does not include 1, that suggests that there is a significant difference between s1 and s2. As of this writing, it is rare to find statistics software that does bootstrapping with two standard deviations or variances, but we can apply the bootstrap method using any software capable of bootstrapping one sample and generating standard deviations or variances. Statdisk, Minitab, and StatCrunch generate such bootstrap samples, and they can be used for the above procedure. Example: Using the above procedure with the samples, a typical result is this 95% confidence interval: 0.67 6 s1>s2 6 2.72. Because the confidence interval does include the ratio of 1, conclude that there is not sufficient evidence to warrant rejection of the claim that the two standard deviations (or variances) are equal. It appears that the variation among weights of male U.S. Army personnel did not change from 1988 to 2012. Randomization with Two Standard Deviations or Two Variances Randomization would first require that the two samples be combined into one big sample, random samples of the same sizes are then drawn without replacement, and the ratio s2 1>s 2 2 is found. That process is repeated many times (such as 1000). We can then find the number of ratios s2 1>s 2 2 that are at least as extreme as the ratio s 2 1>s 2 2 found from the original two samples. We can then determine whether the two variances (or standard deviations) are significantly different. As of this writing, it is rare to find statistics software that does randomization with two standard deviations or variances. Bootstrapping and Randomization – Two Samples Access tech supplements, videos, and data sets at www.TriolaStats.com TECH CENTER Statdisk 1. Click Resampling in the top menu. 2. Select the desired type of bootstrapping or randomization from the dropdown menu. Options include: – Bootstrap Two Proportions – Bootstrap Two Means –Randomization Two Proportions – Randomization Two Means –Randomization Matched Pairs 3. Enter the required inputs which includes the desired number of resamplings. 4. Click Evaluate. StatCrunch 1. Click Applets in the top menu. 2. Select Resampling in the dropdown menu and select the desired randomization from the submenu. Options include: –Randomization test for two proportions –Randomization test for two means 3. Enter the required Sample 1 and Sample 2 data. 4. Click Compute! and the applet window will appear. 5. Click 1000 times for 1000 resamplings. The results will be displayed. StatCrunch does not currently have a function for bootstrapping two proportions or two means. Minitab 1. Click Calc in the top menu. 2. Select Resampling from dropdown menu. 3. Select the desired resampling function from the submenu: – Bootstrapping for 2-Sample Means –Randomization Test for 2-Sample Means 4. Enter the required inputs which includes the desired number of resamplings. 5. Click the Options button. For randomization, select the format of the alternative hypothesis. For bootstrapping, enter the desired confidence level. 6. Click OK twice. TIP: If bootstrapping and/or randomization is desired for two proportions, enter two columns of 0’s and 1’s that represent the two sample proportions, then proceed to use the bootstrapping and randomization functions for two means.
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