How would you use the bootstrap method to estimate the confidence interval of a sample mean?

The bootstrap method is a resampling technique used to estimate the confidence interval of a parameter such as the sample mean. It involves taking multiple resamples with replacement from the original sample, computing the desired statistic on each resample, and then creating a distribution of these simulated statistics to determine the confidence interval. The main idea behind bootstrapping is that the sample data is a good representation of the population, so resampling from the sample approximates the sampling distribution of the statistic.

Here’s a step-by-step process to use the bootstrap method to estimate the confidence interval of a sample mean:

1. Start with a sample data set X₁, X₂, …, X_n, with n observations.

2. Generate B bootstrap samples, by resampling with replacement from the original sample data set. Each bootstrap sample should have the same size n as the original sample. For example, let X₁^*, X₂^*, …, X_n^* be a bootstrap sample, where X_i^* is an observation chosen randomly with replacement from the original sample.

3. Compute the sample mean for each bootstrap sample. Let X̄_b^* represent the sample mean of the b-th bootstrap sample, where b = 1, 2, …, B.

4. Arrange the bootstrap means in ascending order, X̄₍₁₎^* ≤ X̄₍₂₎^* ≤ … ≤ X̄_(B)^*.

5. Determine the lower and upper bounds of the (1 − α)100% confidence interval by selecting the appropriate percentiles from the ordered list of the bootstrap means. The lower bound corresponds to the $\frac{\alpha}{2}$-th quantile, and the upper bound corresponds to the $1-\frac{\alpha}{2}$-th quantile.

For example, for a 95% confidence interval, we will choose the lower bound at 2.5th percentile (α/2 = 0.025) and the upper bound at 97.5th percentile (1 − α/2 = 0.975) of the bootstrap means.

6. The bootstrap (1 − α)100% confidence interval is given by:

[X̄_{(⌈Bα/2⌉)}^*, X̄_{(⌈B(1 − α/2)⌉)}^*]

where ⌈ ⋅ ⌉ denotes the ceiling function.

In summary, the bootstrap method involves generating multiple resamples with replacement from the original sample, computing the sample mean for each resample, and then determining the lower and upper bounds of the confidence interval based on the appropriate percentiles of the bootstrap means.