Problem Statement

Given a dataset of n real numbers, estimate the 95% confidence interval for the population mean using the bootstrap method.

Algorithm:

1. Draw b bootstrap samples, each of size n, sampling with replacement from the original data.
2. Compute the mean of each bootstrap sample.
3. The lower bound is the 2.5th percentile and the upper bound is the 97.5th percentile of the bootstrap means.

Use numpy.random.seed(seed) before sampling.

Function signature:

def bootstrap_ci(X: list, b: int, seed: int) -> tuple:
    # returns (lower, upper)

Input Format

Line 1: n b seed — number of data points, number of bootstrap samples, random seed
Lines 2..n+1: one float per line — the data values

Output Format

One line: lower upper — the 2.5th and 97.5th percentile of bootstrap means (10 significant figures)

Example

Input:

5 1000 42
1.0
2.0
3.0
4.0
5.0

Output:

1.8 4.2

Notes

• Use numpy.random.choice with replace=True to draw bootstrap samples.
• Use numpy.percentile for the 2.5 and 97.5 percentiles.
• The confidence interval captures the variability of the sample mean estimator.