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One Proportion Z-Test Calculator
The one-proportion z-test tests whether the share of a population with some characteristic equals a claimed value p₀, for example whether a pass rate is really 70% or whether a coin is fair. Enter the number of successes, the sample size and p₀, then pick the alternative hypothesis and significance level.
The calculator first checks the normal-approximation conditions np₀ ≥ 10 and n(1 − p₀) ≥ 10, then finds the sample proportion, the standard error √[p₀(1 − p₀)/n] under the null hypothesis, the z statistic, the p-value and the critical value. A Wald confidence interval for p is reported below the decision.
Note that the standard error uses the hypothesised p₀, not the sample proportion, because the test is carried out assuming H₀ is true. For a confidence interval on its own, which uses p̂, see the confidence interval for a proportion. When the conditions fail, an exact binomial test is the safer choice.
Result and step-by-step solution
- State the hypothesesH₀: p = 0.5 H₁: p ≠ 0.5 (two-tailed test)
- Check the conditionsnp₀ = 50, n(1 − p₀) = 50. Both are at least 10, so the normal approximation is reasonable.
- Sample proportionp̂ = x ÷ n = 58 ÷ 100 = 0.58
- Standard error under H₀SE = √[p₀(1 − p₀) ÷ n] = √[0.5 × 0.5 ÷ 100] = 0.05
- Test statisticz = (p̂ − p₀) ÷ SE = (0.58 − 0.5) ÷ 0.05 = 1.6
- p-value2 × P(Z ≥ |1.6|) = 0.1096
- Decisionp = 0.1096 ≥ α = 0.05. critical values ±1.96 (reject when |z| > 1.96).
| 95% Wald confidence interval for p | 0.4833 to 0.6767 |
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Frequently asked questions
- What conditions must hold for a one-proportion z-test?
- The sample should be random and both np₀ and n(1 − p₀) should be at least 10 (some texts say 5) so the sampling distribution of p̂ is approximately normal.
- Why does the standard error use p₀ rather than p̂?
- Because a hypothesis test assumes the null hypothesis is true while computing the statistic. Confidence intervals, which make no such assumption, use p̂.
- How do I test whether a coin is fair?
- Set p₀ = 0.5, enter the number of heads as successes and the number of flips as n, and use a two-tailed test.
- What if my sample is small?
- Use an exact binomial test. The z-test can be badly off when np₀ or n(1 − p₀) is below 10.
Last reviewed: September 25, 2026. Calculations run in your browser and were validated against SciPy.