7-4 Bootstrapping: Using Technology for Estimates 359 Step 1: In Table 7-4, we created 20 bootstrap samples (with replacement!) from the original sample of 0, 2, 3, 7. (Here we use only 20 bootstrap samples so we have a manageable example that doesn’t occupy many pages of text, but we usually want at least 1000 bootstrap samples.) Step 2: Because we want a confidence interval estimate of the population mean m, we want the sample mean x for each of the 20 bootstrap samples, and those sample means are shown in the column to the right of the bootstrap samples. Step 3: The column of data shown farthest to the right is a list of the 20 sample means arranged in order (“sorted”) from lowest to highest. Step 4: Because we want a confidence level of 90%, we want to find the percentiles P5 and P95. Recall that P5 separates the lowest 5% of values, and P95 separates the top 5% of values. Using the methods from Section 3-3 for finding percentiles, we use the sorted list of bootstrap sample means to find that P5 = 1.75 and P95 = 4.875. The 90% confidence interval estimate of the population mean is 1.75 6 m 6 4.875, where the values are in thousands of dollars. YOUR TURN. Do Exercise 7 “Freshman 15.” TABLE 7-4 Bootstrap Samples for m Bootstrap Sample x Sorted x 3 3 0 2 2.00 1.75 P5 = 1.75 0 3 2 2 1.75 1.75 7 0 2 7 4.00 1.75 3 2 7 3 3.75 2.00 0 0 7 2 2.25 2.00 7 0 0 3 2.50 2.25 3 0 3 2 2.00 2.50 3 7 3 7 5.00 2.50 0 3 2 2 1.75 2.50 0 3 7 0 2.50 2.75 90% Confidence Interval: 0 7 2 2 2.75 3.00 1.75 6 m 6 4.875 7 2 2 3 3.50 3.25 7 2 3 7 4.75 3.25 2 7 2 7 4.50 3.50 0 7 2 3 3.00 3.75 7 3 7 2 4.75 4.00 3 7 0 3 3.25 4.50 0 0 3 7 2.50 4.75 3 3 7 0 3.25 4.75 P95 = 4.875 2 0 2 3 1.75 5.00 Standard Deviations In Section 7-3 we noted that when constructing confidence interval estimates of population standard deviations or variances, there is a requirement that the sample must be from a population with normally distributed values. Even if the sample is large, this normality requirement is much stricter than the normality requirement used for estimating population means. Consequently, the bootstrap method becomes more important for confidence interval estimates of s or s 2. EXAMPLE 4 Incomes: Bootstrap CI for Standard Deviation Use these same incomes (thousands of dollars) from Example 3: 0, 2, 3, 7. Use the bootstrap resampling procedure to construct a 90% confidence interval estimate of the population standard deviation s, the standard deviation of the annual incomes of the population of the author’s statistics students. SOLUTION REQUIREMENT CHECK The same requirement check used in Example 3 applies here. The same basic procedure used in Example 3 is used here. Example 3 already includes 20 bootstrap samples, so here we find the standard deviation of each bootstrap sample, and then we sort them to get this sorted list of sample standard deviations: 1.26 1.26 1.26 1.41 1.41 2.22 2.31 2.38 2.63 2.63 2.87 2.87 2.89 2.94 2.99 3.30 3.32 3.32 3.32 3.56 The 90% confidence interval limits are found from this sorted list of standard deviations by finding P5 and P95. Using the methods from Section 3-3, we get P5 = 1.26 and P95 = 3.44. The 90% confidence interval estimate of the population standard deviation s is 1.26 6 s 6 3.44, where the values are in thousands of dollars. YOUR TURN. Do part (b) of Exercise 8 “Cell Phone Radiation.”
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