7-4 Bootstrapping: Using Technology for Estimates 361 2. Bootstrap Sample For the sample data given in Exercise 1, what is a bootstrap sample? 3. How Many? The examples in this section all involved no more than 20 bootstrap samples. How many should be used in real applications? 4. Mean Assume that we want to use the sample data given in Exercise 1 with the bootstrap method to estimate the population mean. The mean of the values in Exercise 1 is 54.3 seconds, and the mean of all of the tobacco times in Data Set 20 “Alcohol and Tobacco in Movies” from Appendix B is 57.4 seconds. If we use 1000 bootstrap samples and find the corresponding 1000 means, do we expect that those 1000 means will target 54.3 seconds or 57.4 seconds? What does that result suggest about the bootstrap method in this case? In Exercises 5–8, use the relatively small number of given bootstrap samples to construct the confidence interval. 5. Online Buying In a Consumer Reports Research Center survey, women were asked if they purchase books online, and responses included these: no, yes, no, no. Letting “yes” = 1 and letting “no” = 0, here are ten bootstrap samples for those responses: 50, 0, 0, 06, 51, 0, 1, 06, 51, 0, 1, 06, 50, 0, 0, 06, 50, 0, 0, 06, 50, 1, 0, 06, 50, 0, 0, 06, 50, 0, 0, 06, 50, 1, 0, 06, 51, 1, 0, 06. Using only the ten given bootstrap samples, construct a 90% confidence interval estimate of the proportion of women who said that they purchase books online. 6. Seating Choice In a 3M Privacy Filters poll, respondents were asked to identify their favorite seat when they fly, and the results include these responses: window, window, other, other. Letting “window” = 1 and letting “other” = 0, those four responses can be represented as {1, 1, 0, 0}. Here are ten bootstrap samples for those responses: 50, 0, 0, 06, 50, 1, 0, 06, 50, 1, 0, 16, 50, 0, 1, 06, 51, 1, 1, 06, 50, 1, 1, 06, 51, 0, 0, 16, 50, 1, 1, 16, 51, 0, 1, 06, 51, 0, 0, 16. Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the proportion of respondents who indicated their favorite seat is “window.” 7. Freshman 15 Here is a sample of amounts of weight change (kg) of college students in their freshman year (from Data Set 13 “Freshman 15” in Appendix B): 11, 3, 0, -2, where -2 represents a loss of 2 kg and positive values represent weight gained. Here are ten bootstrap samples: 511, 11, 11, 06, 511, -2, 0, 116, 511, -2, 3, 06, 53, -2, 0, 116, 50, 0, 0, 36, 53, -2, 3, -26, 511, 3, -2, 06, 5-2, 3, -2, 36, 5-2, 0, -2, 36, 53, 11, 11, 116. a. Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the mean weight change for the population. b. Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the standard deviation of the weight changes for the population. 8. Cell Phone Radiation Here is a sample of measured radiation emissions (cW/kg) for cell phones (based on data from the Environmental Working Group): 38, 55, 86, 145. Here are ten bootstrap samples: 538, 145, 55, 866, 586, 38, 145, 1456, 5145, 86, 55, 556, 555, 55, 55, 1456, 586, 86, 55, 556, 538, 38, 86, 866, 5145, 38, 86, 556, 555, 86, 86, 866, 5145, 86, 55, 866, 538, 145, 86, 556. a. Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the population mean. b. Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the population standard deviation. In Exercises 9–24, use technology to create the large number of bootstrap samples. 9. Freshman 15 Repeat Exercise 7 “Freshman 15” using a confidence level of 90% for parts (a) and (b) and using 1000 bootstrap samples instead of the 10 that were given in Exercise 7. 10. Cell Phone Radiation Repeat Exercise 8 “Cell Phone Radiation” using a confidence level of 90% for parts (a) and (b) and using 1000 bootstrap samples instead of the 10 that were given in Exercise 8.
RkJQdWJsaXNoZXIy NjM5ODQ=