Tests for one proportion (z-test and exact binomial test)

Tests whether a population proportion equals p₀, by the score z-test (large samples) or the exact binomial test (any size).

z = \frac{\phat - p_0}{\sqrt{p_0(1-p_0)/n}} = \frac{x - np_0}{\sqrt{np_0(1-p_0)}}

Score z-test: the SE uses the null value p₀, not p̂. Its matching interval is the Wilson interval.

$x, n$: Number of successes and trials.
$\phat$: Sample proportion $x/n$ .
$p_0$: Proportion stated by H₀ (0 < p₀ < 1).

p_{\text{two-sided}} = \sum_{k:\ P(X=k) \le P(X=x)} P(X = k),\qquad X \sim \text{Binomial}(n, p_0)

Exact binomial test: the two-sided p-value sums the probabilities of all outcomes no more likely than the observed one (R's rule); one-sided p-values are binomial tails.

$P(X=k)$: Binomial probability $\binom{n}{k}p_0^k(1-p_0)^{n-k}$ .

When to use

A binary outcome (yes/no) from n independent trials and a benchmark proportion: a claimed defect rate, a historical pass rate, 50% for "no preference". Use the exact test for small samples or when $np_0$ or $n(1-p_0)$ is below about 10; the z-test is the usual large-sample choice.

Assumptions

Independent trials with a common probability of success (random sample).
z-test: $np_0 \ge 10$ and $n(1-p_0) \ge 10$ (a common rule of thumb).

How to interpret

The z-test is reported with a Wilson interval and the exact test with a Clopper–Pearson interval (each is the interval that inverts its test). The engine's z-test uses no continuity correction by default.
The exact test is conservative: its actual Type I error rate is at most α, often less.

Common mistakes

Using $\phat$ instead of $p_0$ in the test's SE.
Doubling a one-sided exact p-value: the exact two-sided rule is different (sum of probabilities ≤ the observed).
Applying the z-test to 3 successes out of 15.

Example

14 of 20 customers prefer the new design; $H_0: p = 0.5$ . z-test: $\phat = 0.70$ , $z = (0.70 - 0.5)/\sqrt{0.25/20} = 1.79$ , $p = 0.074$ , Wilson 95% CI 0.481 to 0.855. Exact binomial test: $p = 0.115$ , Clopper–Pearson CI 0.457 to 0.881. Neither gives significant evidence at α = 0.05; 20 customers are too few to pin down the preference.