Problems

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Bootstrap Mean & Confidence Interval

Probability and Statistics

Medium

Estimate the mean of a 1D dataset using bootstrap resampling. Repeatedly sample with replacement from the array, compute the mean for each resample, and use these bootstrap means to estimate a confidence interval.

Mathematical Definition

Bootstrap Mean Distribution:

\bar{X}^{*}_{b} = \frac{1}{N} \sum_{i=1}^{N} X^{*}_{i}, \qquad b = 1, 2, \ldots, B

where X_i∗ are sampled with replacement from the original data, and B is the number of bootstrap samples.

Confidence Interval:

\left[ Q_{\alpha/2}(\bar{X}^{*}), \; Q_{1-\alpha/2}(\bar{X}^{*}) \right]

where α=1−ci and Q_p pdenotes the p-th quantile.

Function Arguments

x: 1D array-like, shape (N,) - input observations
n_bootstrap: int - number of bootstrap samples
ci: float - confidence level (e.g., 0.95 for 95%)
rng: np.random.Generator or None - random number generator for reproducibility

Loading visualization...

Examples

Input: x = [1.0, 2.0, 3.0, 4.0], n_bootstrap=1000, ci=0.90

Output: boot_means.shape = (1000,), lower ≈ 1.5, upper ≈ 3.5

Input: x = [5.0], n_bootstrap=1000, ci=0.95

Output: boot_means.shape = (1000,), lower = 5.0, upper = 5.0

Hint 1

Use rng.integers() to generate bootstrap indices for each iteration.

Hint 2

For 95% CI, use quantiles at 0.025 and 0.975. For general ci, use alpha = (1 - ci) / 2.

Requirements

Return tuple: (boot_means, lower, upper)
boot_means: NumPy array of shape (n_bootstrap,)
lower, upper: scalars with lower < upper
Use rng.integers() if rng provided, else np.random
Use np.quantile() for confidence bounds

Constraints

1 ≤ len(x) ≤ 10,000
100 ≤ n_bootstrap ≤ 10,000
Use NumPy only
Time limit: 2s

Try Similar Problems

Sample Var Std Mean Median Mode Percentiles T Test One Sample Chi2 Independence

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Bootstrap Mean & Confidence Interval

Probability and Statistics

Medium

Mathematical Definition

Bootstrap Mean Distribution:

\bar{X}^{*}_{b} = \frac{1}{N} \sum_{i=1}^{N} X^{*}_{i}, \qquad b = 1, 2, \ldots, B

where X_i∗ are sampled with replacement from the original data, and B is the number of bootstrap samples.

Confidence Interval:

\left[ Q_{\alpha/2}(\bar{X}^{*}), \; Q_{1-\alpha/2}(\bar{X}^{*}) \right]

where α=1−ci and Q_p pdenotes the p-th quantile.

Function Arguments

x: 1D array-like, shape (N,) - input observations
n_bootstrap: int - number of bootstrap samples
ci: float - confidence level (e.g., 0.95 for 95%)
rng: np.random.Generator or None - random number generator for reproducibility

Loading visualization...

Examples

Input: x = [1.0, 2.0, 3.0, 4.0], n_bootstrap=1000, ci=0.90

Output: boot_means.shape = (1000,), lower ≈ 1.5, upper ≈ 3.5

Input: x = [5.0], n_bootstrap=1000, ci=0.95

Output: boot_means.shape = (1000,), lower = 5.0, upper = 5.0

Hint 1

Use rng.integers() to generate bootstrap indices for each iteration.

Hint 2

For 95% CI, use quantiles at 0.025 and 0.975. For general ci, use alpha = (1 - ci) / 2.

Requirements

Return tuple: (boot_means, lower, upper)
boot_means: NumPy array of shape (n_bootstrap,)
lower, upper: scalars with lower < upper
Use rng.integers() if rng provided, else np.random
Use np.quantile() for confidence bounds

Constraints

1 ≤ len(x) ≤ 10,000
100 ≤ n_bootstrap ≤ 10,000
Use NumPy only
Time limit: 2s

Try Similar Problems

Sample Var Std Mean Median Mode Percentiles T Test One Sample Chi2 Independence

Case 1

Case 2

x =

Accepts: array

n_bootstrap =

Accepts: number

ci =

Accepts: number

seed =

Accepts: number

Mathematical Definition

Bootstrap Mean Distribution:

\bar{X}^{*}_{b} = \frac{1}{N} \sum_{i=1}^{N} X^{*}_{i}, \qquad b = 1, 2, \ldots, B

where X_i∗ are sampled with replacement from the original data, and B is the number of bootstrap samples.

Confidence Interval:

\left[ Q_{\alpha/2}(\bar{X}^{*}), \; Q_{1-\alpha/2}(\bar{X}^{*}) \right]

where α=1−ci and Q_p pdenotes the p-th quantile.

Function Arguments

x: 1D array-like, shape (N,) - input observations
n_bootstrap: int - number of bootstrap samples
ci: float - confidence level (e.g., 0.95 for 95%)
rng: np.random.Generator or None - random number generator for reproducibility