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Bootstrap Confidence Interval Width for Eval Accuracy

Problem Statement

Report the normal-approximation 95% confidence interval for an agent's eval accuracy, so a small benchmark's noise is visible.

Background

For n graded items with k correct, the sample accuracy is p = k/n. The Wald 95% interval is *p +/- 1.96 * sqrt(p(1-p)/n)**, clamped to [0, 1]. Report the lower bound, upper bound, and width.

Your Task

def acc_ci(k, n):

Return a dict {"low": float, "high": float, "width": float}, each rounded to 4 decimals.

Input Format

k (int correct), n (int total, >= 1).

Output Format

A dict of three floats.

Sample

print(acc_ci(80, 100))

Output:

{'low': 0.7216, 'high': 0.8784, 'width': 0.1568}

Example:

Input:

print(acc_ci(80, 100))

Output:

{'low': 0.7216, 'high': 0.8784, 'width': 0.1568}

Reasoning:

Calculate the sample accuracy $p$ by dividing the number of correct items by the total items: $p = 80 / 100 = 0.8$ .
Compute the standard error of the proportion using the formula $\sqrt{\frac{p(1-p)}{n}}$ : $\sqrt{\frac{0.8 \times 0.2}{100}} = \sqrt{0.0016} = 0.04$ .
Determine the margin of error for a 95% confidence level by multiplying the standard error by the Z-score $1.96$ : $1.96 \times 0.04 = 0.0784$ .
Calculate the raw lower and upper bounds by subtracting and adding the margin of error to the sample accuracy: $\text{low} = 0.8 - 0.0784 = 0.7216$ and $\text{high} = 0.8 + 0.0784 = 0.8784$ .
Compute the width of the interval as the difference between the upper and lower bounds: $0.8784 - 0.7216 = 0.1568$ .
The final output is {'low': 0.7216, 'high': 0.8784, 'width': 0.1568}

Constraints:

p = k/n; margin = 1.96*sqrt(p*(1-p)/n).
Clamp low/high to [0,1]; width = high - low.
Round each to 4 decimals.

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1. Background Knowledge

The Wald confidence interval is the most common normal-approximation interval for a population proportion. When you observe $k$ successes out of $n$ independent Bernoulli trials, the sample proportion is $\hat{p} = k/n$ . By the Central Limit Theorem, for large $n$ , the sampling distribution of $\hat{p}$ is approximately normal with mean $p$ and standard error $\sqrt{\hat{p}(1-\hat{p})/n}$ . A 95% confidence interval is then constructed as:

\hat{p} \pm z_{0.975} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

where $z_{0.975} \approx 1.96$ is the 97.5th percentile of the standard normal distribution. This interval captures the range within which we are 95% confident the true accuracy lies.

In the context of AI agent evaluation, a small benchmark (e.g., 100 items) has substantial statistical noise. Reporting only a point estimate like "80% accuracy" hides the uncertainty. The width of the confidence interval ( $\text{high} - \text{low}$ ) quantifies this noise: a wide interval means the benchmark is too small to distinguish between, say, 70% and 90% true accuracy. This is critical for production decisions about whether to trust an eval score or need more test items.

Note that the Wald interval can produce bounds outside $[0, 1]$ when $\hat{p}$ is near 0 or 1, so the result must be clamped to the valid probability range.

2. Algorithm Approach

This is a direct formula evaluation problem — no iterative or search-based algorithm is needed. The approach is:

Compute the sample proportion $\hat{p} = k/n$ .
Compute the standard error (SE) of the proportion.
Compute the margin of error as $1.96 \times \text{SE}$ .
Derive lower and upper bounds, clamping to $[0, 1]$ .
Compute the width as $\text{high} - \text{low}$ .
Round all three values to 4 decimal places and return them in a dictionary.

3. Step-by-Step Strategy

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solution.py

Python 3.13

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