Exact Binomial Test Calculator
Evaluate success counts against a hypothesised probability using the exact binomial test. This tool reports two-sided and tail-specific p-values, mid-p adjustments, and Clopper-Pearson confidence intervals, making it ideal for small samples and extreme proportions.
1. Provide Test Inputs
Small sample friendlyModel snapshot
Test \(H_0: p = p_0\) against your chosen alternative. The exact p-value sums binomial probabilities \(\binom{n}{k} p_0^k (1 - p_0)^{n-k}\) over outcomes as or more extreme than the observed count.
\[ P(X = k) = \binom{n}{k} p_0^k (1 - p_0)^{n - k} \]
Two-sided p-values follow the “probability ordering” rule used by R's binom.test.
Exact Test Output
Results readyKey Values
- Alternative
- Exact p-value
- Two-sided p
- Mid-p (two-sided)
- P(X = x)
- Alpha threshold
Estimates
- Observed proportion
- Difference (phat - p0)
- Clopper-Pearson CI
Decision
Step-by-step Workflow
Tail conventions
One-sided tests sum probabilities in the specified tail. Two-sided values follow the probability ordering rule: include any outcome whose binomial probability is no larger than the observed \(P(X = x)\), ensuring exact coverage.
Distribution Summary
| k successes | Probability | Cumulative ≤ k | Cumulative ≥ k |
|---|
Formula Reference
Binomial Probability
\\[ P(X = k) = \binom{n}{k} p_0^{k} (1 - p_0)^{n-k} \\]
The exact test sums these probabilities across outcomes as or more extreme than the observed count.
Two-sided & Mid-p
\\[ p_{\text{two-sided}} = \sum_{P(k) \le P(x)} P(k), \quad p_{\text{mid}} = p_{\text{two-sided}} - \tfrac{1}{2} P(x) \\]
The probability ordering rule mirrors the behaviour of R's binom.test.
One-sided Tails
\\[ p_{\text{right}} = \sum_{k=x}^{n} \binom{n}{k} p_0^{k} (1 - p_0)^{n-k}, \quad p_{\text{left}} = \sum_{k=0}^{x} \binom{n}{k} p_0^{k} (1 - p_0)^{n-k} \\]
Select the right tail when testing \(p > p_0\) and the left tail when testing \(p < p_0\).
Clopper-Pearson Interval
\\[ \text{Lower} = B^{-1}\!\left(\tfrac{\alpha}{2}; x, n - x + 1\right), \quad \text{Upper} = B^{-1}\!\left(1 - \tfrac{\alpha}{2}; x + 1, n - x\right) \\]
Here \(B^{-1}\) denotes the inverse beta CDF evaluated at the specified quantile.
Step-by-Step Guide
- Count the number of observed successes \(x\) out of \(n\) identical and independent trials.
- Specify the null proportion \(p_0\) and the alternative direction (two-sided, right, or left).
- Sum binomial probabilities under \(p_0\) across the appropriate tail to obtain the exact p-value.
- Compare the p-value with the chosen alpha level to decide whether to reject \(H_0\).
- Construct the Clopper-Pearson confidence interval to report the plausible range for the true proportion.
References
- Clopper, C. J., & Pearson, E. S. (1934). The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika, 26(4), 404-413.
- Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
- Mehta, C. R., & Hilton, J. F. (1993). Exact power of conditional tests in 2x2 tables. Journal of the American Statistical Association, 88(421), 10-20.