Slide 1What is bootstrapping?Why is this useful?Why is this useful?Why is this useful?Why is this useful?Why is this useful?How would we obtain these by using “brute force”?How would we obtain these by using “brute force”?How would we obtain these by using “brute force”?How would we obtain these by using “brute force”?How would we obtain these by using “brute force”?How would we obtain these by using “brute force”?How would we obtain these by using “brute force”?How would we obtain these by using “brute force”?How would we obtain these by using “brute force”?How would we obtain these by using “brute force”?How would we obtain these by using “brute force”?Why doesn’t anyone do this?Bootstrapping 101Bootstrapping paradigmBootstrapping paradigmBootstrapping paradigmBootstrapping paradigmBootstrapping paradigmBootstrapping paradigmBootstrapping paradigmBootstrapping paradigmBootstrapping paradigmBootstrapping paradigmBootstrapping paradigmGetting bootstrap samples in RGetting SE and sampling distributions using bootstrappingExample code in RExercise (INSR p. 195)The bigger pictureDoes bootstrapping always work?Introduction to “bootstrapping”STA333 Lecture 20What is bootstrapping?•Bootstrapping is a method of resampling your observed data so as to estimate the STANDARD ERROR (SE) and the SAMPLING DISTRIBUTION of some statistic you wish to use.•Recall what these are:–Standard error (SE) of a statistic: a measure of the variability that the statistic has due to random sampling. In short, it is the standard deviation of the statistic.–Sampling distribution of a statistic: the distribution of all possible values that the statistic could take on from all possible samples of a given fixed size n. •Why are we interested in determining these?2Why is this useful?•Think about the necessary ingredients to build a 95% confidence interval for a mean using the traditional approach:3Why is this useful?•Think about the necessary ingredients to build a 95% confidence interval for a mean using the traditional approach:4Statistic used to estimate the population parameter of interest (“point estimate”)Why is this useful?•Think about the necessary ingredients to build a 95% confidence interval for a mean using the traditional approach:5Statistic used to estimate the population parameter of interest (“point estimate”)SE of the statisticWhy is this useful?•Think about the necessary ingredients to build a 95% confidence interval for a mean using the traditional approach:6Statistic used to estimate the population parameter of interest (“point estimate”)A multiplier that determines the # of SEs needed to achieve the required “capture probability” from the statistic’s sampling distributionSE of the statisticWhy is this useful?•Think about the necessary ingredients to build a 95% confidence interval for a mean using the traditional approach:7Statistic used to estimate the population parameter of interest (“point estimate”)A multiplier that determines the # of SEs needed to achieve the required “capture probability” from the statistic’s sampling distributionSE of the statisticThese two together constitute the margin of error of the statistic. Calculating the margin of error requires us to know [1] the SE of the statistic, and [2] the sampling distribution of the statistic.How would we obtain these by using “brute force”?•We’d need to draw all possible samples of a given size n from the population, find the statistic from each sample, and then observe their distribution:8Population True mean = µ (unknown)How would we obtain these by using “brute force”?•We’d need to draw all possible samples of a given size n from the population, find the statistic from each sample, and then observe their distribution:9Population True mean = µ (unknown)SRS of size n5.4 4.35.2 5.74.9 5.3How would we obtain these by using “brute force”?•We’d need to draw all possible samples of a given size n from the population, find the statistic from each sample, and then observe their distribution:10Population True mean = µ (unknown)SRS of size n5.4 4.35.2 5.74.9 5.3How would we obtain these by using “brute force”?•We’d need to draw all possible samples of a given size n from the population, find the statistic from each sample, and then observe their distribution:11Population True mean = µ (unknown)SRS of size n5.4 4.35.2 5.74.9 5.33.3 5.15.0 5.15.8 4.5SRS of size nHow would we obtain these by using “brute force”?•We’d need to draw all possible samples of a given size n from the population, find the statistic from each sample, and then observe their distribution:12Population True mean = µ (unknown)SRS of size n5.4 4.35.2 5.74.9 5.33.3 5.15.0 5.15.8 4.5SRS of size nHow would we obtain these by using “brute force”?•We’d need to draw all possible samples of a given size n from the population, find the statistic from each sample, and then observe their distribution:13Population True mean = µ (unknown)SRS of size n5.4 4.35.2 5.74.9 5.33.3 5.15.0 5.15.8 4.5SRS of size n⁞⁞How would we obtain these by using “brute force”?•We’d need to draw all possible samples of a given size n from the population, find the statistic from each sample, and then observe their distribution:14Population True mean = µ (unknown)SRS of size n5.4 4.35.2 5.74.9 5.33.3 5.15.0 5.15.8 4.5SRS of size nSRS of size n4.4 4.63.9 5.14.4 4.3⁞⁞How would we obtain these by using “brute force”?•We’d need to draw all possible samples of a given size n from the population, find the statistic from each sample, and then observe their distribution:15Population True mean = µ (unknown)SRS of size n5.4 4.35.2 5.74.9 5.33.3 5.15.0 5.15.8 4.5SRS of size nSRS of size n4.4 4.63.9 5.14.4 4.3⁞⁞How would we obtain these by using “brute force”?•We’d need to draw all possible samples of a given size n from the population, find the statistic from each sample, and then observe their distribution:16Population True mean = µ (unknown)SRS of size n5.4 4.35.2 5.74.9 5.33.3 5.15.0 5.15.8 4.5SRS of size nSRS of size n4.4 4.63.9 5.14.4 4.3⁞⁞Standard deviation of all these is the SE of the statisticHow would we obtain these by using “brute force”?•We’d need