"A second method for finding a bootstrap confidence interval is called the Bootstrap t-Method. This method requires estimating the standard error of the estimate (such as the standard error of the mean) from the bootstrap sample. For any given set of estimates of a parameter, the standard error of the estimate is found by determining the sample standard deviation of the B bootstrap estimates. For example, in Example 1, we found 2000 estimates of the sample mean mpg. The standard deviation of these 2000 sample means is found to be 0.580. The standard deviation of the 16 observations is 2.38, so


s / sqrt(n) = 2.38 / sqrt(16) = 0.595,


is close to the standard error of the estimate found from the bootstrap samples. The estimate of the standard error from Figure 28 in Example 2 is 0.560. We can use an estimate of the standard error (SE_est) along with critical values from Student's t-distribution (Table VII) to construct confidence intervals as follows:


statistic ± t_(alpha/2) * SE_est


For example, in Example 1 we know x̄ = 28.1 and n = 16, so for a 95% confidence interval, t_0.025 = 2.131 using 15 degrees of freedom. Using the standard error estimate from Example 1, we find the lower bound of the confidence interval to be 28.1 – 2.131(0.580) = 26.86 mpg and the upper bound to be 28.1 + 2.131(0.580) = 29.34.


pH of Rain Revisited See Problem 7. Use the Bootstrap t-Method to find a 95% confidence for the mean pH of rainwater in Tucker County, West Virginia."

Verified step by step guidance

1. Identify the sample mean \( \bar{x} \) and the sample size \( n \) from the given data. These are essential for constructing the confidence interval.
2. Obtain the bootstrap estimates of the parameter (in this case, the mean pH) by resampling the original data with replacement multiple times (e.g., \( B = 2000 \) bootstrap samples). Calculate the mean for each bootstrap sample.
3. Calculate the standard error estimate \( SE_{est} \) by finding the sample standard deviation of the \( B \) bootstrap means. This standard deviation represents the variability of the bootstrap estimates.
4. Determine the critical value \( t_{\alpha/2} \) from the Student's t-distribution table using \( n - 1 \) degrees of freedom and the desired confidence level (for 95%, \( \alpha = 0.05 \), so use \( t_{0.025} \)).
5. Construct the confidence interval using the formula: \[ \bar{x} \pm t_{\alpha/2} \times SE_{est} \] This gives the lower and upper bounds of the 95% confidence interval for the mean pH.

Key Concepts

Bootstrap Sampling and Bootstrap Estimates

Bootstrap sampling involves repeatedly drawing samples with replacement from the original data to create many 'bootstrap samples.' Each sample produces an estimate of the parameter of interest, such as the mean. The collection of these bootstrap estimates approximates the sampling distribution, allowing us to assess variability without relying on strict parametric assumptions.

Standard Error Estimation from Bootstrap Samples

The standard error measures the variability of an estimate across samples. In the bootstrap context, it is estimated by calculating the standard deviation of the bootstrap estimates. This provides an empirical measure of uncertainty for the parameter estimate, which can be compared to traditional formulas like s/√n to validate the bootstrap approach.

Bootstrap t-Method for Confidence Intervals

The Bootstrap t-Method constructs confidence intervals using the bootstrap estimate of the standard error and critical values from the Student's t-distribution. The interval is calculated as the estimate ± t_(alpha/2) times the bootstrap standard error, accounting for sample size and variability. This method improves interval accuracy, especially for small samples or unknown distributions.