BINOMTEST

Overview

The BINOMTEST function performs a binomial test, a statistical hypothesis test used to determine whether the observed proportion of successes in a series of independent trials differs significantly from a hypothesized probability. This test is fundamental in quality control, clinical trials, and any scenario involving binary outcomes (success/failure, yes/no, pass/fail).

The binomial test evaluates the null hypothesis that the probability of success in a Bernoulli experiment equals a specified value p. Given k observed successes out of n trials, the test calculates a p-value representing the probability of observing a result at least as extreme as k, assuming the null hypothesis is true.

This implementation uses the scipy.stats.binomtest function from the SciPy library. The function performs a two-sided test by default, meaning it tests whether the observed proportion differs from p in either direction.

The p-value is computed exactly using the binomial distribution probability mass function:

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

where \binom{n}{k} is the binomial coefficient. For a two-sided test, the p-value sums probabilities of all outcomes as extreme or more extreme than the observed result.

Interpreting results: A small p-value (typically < 0.05) suggests the observed proportion differs significantly from the hypothesized probability, leading to rejection of the null hypothesis. A p-value close to 1 indicates the observed result is consistent with the null hypothesis.

This example function is provided as-is without any representation of accuracy.

Excel Usage

=BINOMTEST(k, n, p)

• k (int, required): Number of successes observed in the experiment.
• n (int, required): Total number of trials in the experiment.
• p (float, optional, default: 0.5): Hypothesized probability of success on each trial.

Returns (list[list]): 2D list [[k, p_value]], or error message string.

Examples

Example 1: Fair coin test (5 heads in 10 flips)

Inputs:

k	n	p
5	10	0.5

Excel formula:

=BINOMTEST(5, 10, 0.5)

Expected output:

Result
5	1

Example 2: Biased coin test (8 heads in 10 flips)

Inputs:

k	n	p
8	10	0.5

Excel formula:

=BINOMTEST(8, 10, 0.5)

Expected output:

Result
8	0.1094

Example 3: Custom probability (7 successes in 12 trials at p=0.6)

Inputs:

k	n	p
7	12	0.6

Excel formula:

=BINOMTEST(7, 12, 0.6)

Expected output:

Result
7	1

Example 4: Default probability (3 successes in 6 trials)

Inputs:

k	n
3	6

Excel formula:

=BINOMTEST(3, 6)

Expected output:

Result
3	1

Python Code

import math
from scipy.stats import binomtest as scipy_binomtest

def binomtest(k, n, p=0.5):
    """
    Perform a binomial test for the probability of success in a Bernoulli experiment.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.binomtest.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        k (int): Number of successes observed in the experiment.
        n (int): Total number of trials in the experiment.
        p (float, optional): Hypothesized probability of success on each trial. Default is 0.5.

    Returns:
        list[list]: 2D list [[k, p_value]], or error message string.
    """
    try:
        k_int = int(k)
        n_int = int(n)
        p_float = float(p)
    except (TypeError, ValueError):
        return "Invalid input: k and n must be integers, p must be a float."

    if n_int <= 0:
        return "Invalid input: n must be a positive integer."
    if k_int < 0:
        return "Invalid input: k must be non-negative."
    if k_int > n_int:
        return "Invalid input: k cannot exceed n."
    if not (0 <= p_float <= 1):
        return "Invalid input: p must be between 0 and 1."

    result = scipy_binomtest(k_int, n_int, p_float)
    pval = float(result.pvalue)

    if not math.isfinite(pval):
        return "Invalid output: p-value is not finite."

    return [[k_int, pval]]

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