358 CHAPTER 7 Estimating Parameters and Determining Sample Sizes Step 1: In Table 7-3, we created 20 bootstrap samples from the original sample of 0, 0, 1, 0. Step 2: Because we want a confidence interval estimate of the population proportion p, we want the sample proportion pn for each of the 20 bootstrap samples, and those sample proportions 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 proportions 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 proportions to find that P5 = 0.00 and P95 = 0.75. The 90% confidence interval estimate of the population proportion is 0.00 6 p 6 0.75. INTERPRETATION The confidence interval of 0.00 6 p 6 0.75 is quite wide. After all, every confidence interval for every proportion must fall between 0 and 1, so the 90% confidence interval of 0.00 6 p 6 0.75 doesn’t seem to be helpful, but it is based on only four sample values. YOUR TURN. Do Exercise 5 “Online Buying.” TABLE 7-3 Bootstrap Samples for p Bootstrap Sample pn Sorted pn 1 0 0 1 0.50 0.00 P5 = 0.00 1 0 1 0 0.50 0.00 0 1 1 1 0.75 0.00 0 0 0 0 0.00 0.00 0 1 0 0 0.25 0.25 1 0 0 0 0.25 0.25 0 1 0 1 0.50 0.25 1 0 0 0 0.25 0.25 0 0 0 0 0.00 0.25 0 0 1 1 0.50 0.25 90% Confidence Interval: 0 0 0 1 0.25 0.25 0.00 6 p 6 0.75 0 0 1 0 0.25 0.25 1 1 1 0 0.75 0.50 0 0 0 0 0.00 0.50 0 0 0 0 0.00 0.50 0 1 1 0 0.50 0.50 0 0 1 0 0.25 0.50 1 0 0 0 0.25 0.75 1 1 1 0 0.75 0.75 P95 = 0.75 0 0 0 1 0.25 0.75 HINT: Example 2 uses only 20 bootstrap samples, but effective use of the bootstrap method typically requires the use of software to generate 1000 or more bootstrap samples. Means In Section 7-2 we noted that when constructing a confidence interval estimate of a population mean, there is a requirement that the sample is from a normally distributed population or the sample size is greater than 30. The bootstrap method can be used when this requirement is not satisfied. EXAMPLE 3 Incomes: Bootstrap CI for Mean When the author collected a simple random sample of annual incomes of his statistics students, he obtained these results (in thousands of dollars): 0, 2, 3, 7. Use the bootstrap resampling procedure to construct a 90% confidence interval estimate of the mean annual income of the population of all of the author’s statistics students. SOLUTION REQUIREMENT CHECK The sample is a simple random sample and there is no requirement that the sample must be from a normally distributed population. Because distributions of incomes are typically skewed instead of normal, we should not use the methods of Section 7-2 for finding the confidence interval, but the bootstrap method can be used.
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