Constructing Bootstrap Confidence Intervals Section 3.3 Constructing Bootstrap Confidence Intervals

Review Suppose you each want to estimate a population parameter. You each collect data, calculate your sample statistic and standard error, and compute a 95% confidence interval. About how many of these 112 intervals (there are 112 students in the class) will contain the true population parameter? 112 106 100 95 Adjust for your class size. 95% of 112 is 0.95(112)  106

Bootstrap Sample Your original sample has data values 18, 19, 19, 20, 21 Is the following a possible bootstrap sample? 18, 19, 20, 21, 22 Yes No 22 is not a value from the original sample

Bootstrap Sample Your original sample has data values 18, 19, 19, 20, 21 Is the following a possible bootstrap sample? 18, 19, 20, 21 Yes No Bootstrap samples must be the same size as the original sample

Bootstrap Sample Your original sample has data values 18, 19, 19, 20, 21 Is the following a possible bootstrap sample? 18, 18, 19, 20, 21 Yes No Same size, could be gotten by sampling with replacement

Bootstrap Samples You have a sample of size n = 50. You sample with replacement 1000 times to get 1000 bootstrap samples. What is the sample size of each bootstrap sample? 50 1000 Bootstrap samples are the same size as the original sample

Bootstrap Distribution You have a sample of size n = 50. You sample with replacement 1000 times to get 1000 bootstrap samples. How many bootstrap statistics will you have? 50 1000 One bootstrap statistic for each bootstrap sample

Center The sampling distribution is centered around the population parameter The bootstrap distribution is centered around the Luckily, we don’t care about the center… we care about the variability! population parameter sample statistic bootstrap statistic bootstrap parameter

Reese’s Pieces Based on this sample, give a 95% confidence interval for the true proportion of Reese’s Pieces that are orange. (0.47, 0.57) (0.42, 0.62) (0.41, 0.51) (0.36, 0.56) I have no idea 0.52 ± 2 × 0.05

Atlanta Commutes What’s the mean commute time for workers in metropolitan Atlanta? Data: The American Housing Survey (AHS) collected data from Atlanta in 2004

Atlanta Commutes Go into StatKey and simulate bootstrap distribution, or have them do it if they have computers in the classroom. 29.11 ± 2 × 0.915 95% confidence interval for the average commute time for Atlantans: (a) (28.2, 30.0) (b) (27.3, 30.9) (c) 26.6, 31.8 (d) No idea

“Is there solid evidence of global warming?” Does belief in global warming differ by political party? “Is there solid evidence of global warming?” The sample proportion answering “yes” was 79% among Democrats and 38% among Republicans. (exact numbers for each party not given, but assume n=1000 for each group) Give a 95% CI for the difference in proportions. Source: “Wide Partisan Divide Over Global Warming”, Pew Research Center, 10/27/10. http://pewresearch.org/pubs/1780/poll-global-warming-scientists-energy-policies-offshore-drilling-tea-party

Global Warming www.lock5stat.com/statkey 0.41  2(0.02) = (0.37, 0.45) We are 95% sure that the difference in the proportion of Democrats and Republicans who believe in global warming is between 0.37 and 0.45.

Global Warming Based on the data just analyzed, can you conclude with 95% certainty that the proportion of people believing in global warming differs by political party? (a) Yes (b) No Yes. We are 95% confident that the difference is between 0.37 and 0.45, and this interval does not include 0 (no difference)