Problem Statement

Given a dataset X of n values, perform a bootstrap analysis of the sample mean and return:

1. The bootstrap standard error of the mean (std of bootstrap means, ddof=0)
2. The 2.5th percentile of bootstrap means (lower CI bound)
3. The 97.5th percentile of bootstrap means (upper CI bound)
4. The width of the 95% CI (upper - lower)

Use numpy.random.default_rng(seed) and rng.choice(n, size=n, replace=True) for sampling.

Function signature:

def bootstrap_analysis(X: np.ndarray, b: int, seed: int) -> tuple:
    # returns (se, lo, hi, width)

Input Format

Line 1: n b seed — number of data points, bootstrap resamples, random seed
Line 2: n space-separated floats — the data values

Output Format

Four lines (8 decimal places each):

1. Bootstrap standard error
2. 2.5th percentile (lower bound)
3. 97.5th percentile (upper bound)
4. CI width

Example

Input:

5 200 42
1.0 2.0 3.0 4.0 5.0

Output:

0.59268794
1.80000000
4.00500000
2.20500000

Derivation

The bootstrap estimates the sampling distribution of the mean empirically. With b resamples, each of size n:

• Each resample mean m_i = mean(X[resample_i])
• SE = std({m_i})
• The percentile bootstrap CI directly uses the α/2 and 1-α/2 quantiles of {m_i}

This is valid even when the underlying distribution is unknown, relying only on the data.

Notes

• Use numpy.random.default_rng (new-style RNG), not numpy.random.seed.
• The standard error decreases as n grows; the CI width reflects estimation uncertainty.
• With only 5 points and 200 resamples, variability is high.