A one-proportion z-test checks whether a sample proportion is far enough from a claimed population proportion to be surprising. Enter the claimed proportion p0, the number of successes x and the sample size n, and the calculator gives the z-statistic, the p-value and the decision at your significance level — with every step shown.
One-Proportion Z-Test Calculator
Test H0: p = p0 against your sample.
Conditions: random sample, np0 ≥ 10 and n(1 − p0) ≥ 10. Use α = 0.05 unless your course says otherwise.
What Is a One-Proportion Z-Test?
A one-proportion z-test tests a claim about a single population proportion. The null hypothesis H0 says p = p0 (the claimed value); the alternative Ha says p ≠ p0, p > p0 or p < p0. You collect a sample, compute the sample proportion p̂ = x/n, and ask: is p̂ far enough from p0 that chance alone is unlikely to explain it? The z-statistic measures that distance in standard errors.
The One-Proportion Z-Test Formula
The test statistic is z = (p̂ − p0) ÷ √(p0(1 − p0) ÷ n). Note that the standard error uses p0, the claimed proportion, not p̂ — because the test assumes H0 is true. The p-value is the probability of a z-statistic at least as extreme as yours, if H0 were true. For a two-tailed test, double the one-tail area. If the p-value is smaller than α (usually 0.05), reject H0.
Conditions for the Z-Test
Check before you test: (1) the sample is random, (2) np0 ≥ 10 and n(1 − p0) ≥ 10 so the sampling distribution is close to normal, and (3) the sample is less than 10% of the population (the 10% condition) when sampling without replacement. If the counts are too small, use an exact binomial test instead of the z-test.
Worked Example
A coin maker claims a coin is fair (p0 = 0.5). You flip it 100 times and get 60 heads. p̂ = 0.60, SE = √(0.5 × 0.5 ÷ 100) = 0.05, z = (0.60 − 0.50) ÷ 0.05 = 2.00. The two-tailed p-value is about 0.0455. Because 0.0455 < 0.05, you reject H0 at α = 0.05: the evidence suggests the coin is not fair. At α = 0.01, you would fail to reject it.
One-Proportion Z-Test FAQs
When do you use a one-proportion z-test?
When you have one sample and want to test a claim about a single population proportion — for example, whether 60% of voters really support a candidate, or whether a coin is fair.
What does the p-value mean?
The probability of getting a sample proportion at least as far from p0 as yours, if the null hypothesis were true. A small p-value means the data are surprising under H0.
What is the difference between a z-test and a confidence interval?
A z-test gives a yes/no decision about a claimed value; a confidence interval gives a range of plausible values for the true proportion. Both use the same standard error idea.
Related tools: Two-Proportion Z-Test · Proportion Confidence Interval · Sample Proportion · Statistics