Hypothesis Testing Calculator (z test, t test, one-proportion test)

Understanding the hypothesis testing calculator — concept and principle

This calculator runs the three most common one-sample tests: a z test for a mean with known σ, a t test for a mean with unknown σ, and a z test for one proportion. It gives the test statistic, the p-value, the critical value, the standard error, the degrees of freedom and the decision at your significance level. Use it for statistics homework and quick checks of survey or experiment results.

t lands beyond the critical value, so p is below 0.05 and H₀ is rejected.

How it is calculated

Mean, σ known: z = (x̄ − μ₀) ÷ (σ/√n)

Mean, σ unknown: t = (x̄ − μ₀) ÷ (s/√n), with n − 1 degrees of freedom

One proportion: z = (p̂ − p₀) ÷ √(p₀(1 − p₀)/n), with p̂ = successes ÷ n

Reject H₀ when p < α (equivalently, when the statistic falls beyond the critical value).

The proportion test needs np₀ and n(1 − p₀) both above 5.
The alternative can be two-sided (≠), less than or greater than.

The principle

A test starts from the assumption that nothing unusual is going on (H₀) and asks how surprising the data would be under it. The test statistic counts how many standard errors the sample result is from the hypothesised value. The p-value is the probability of a result at least that far out if H₀ were true. If that probability is below α, the data are judged inconsistent with H₀. The t distribution replaces z when σ is estimated from the sample, adding a little extra uncertainty.

Worked example

Default inputs: t test, x̄ = 52.3, μ₀ = 50, s = 6, n = 36, α = 0.05, two-sided.

Standard error = 6 ÷ √36 = 1.
t = (52.3 − 50) ÷ 1 = 2.3, with 35 degrees of freedom.
Two-sided critical value at α = 0.05: ±2.03.
p-value = 2 × P(t₃₅ > 2.3) = 0.02753.
Decision: Reject H₀ — significant at this α.

Each point is the calculator re-run with only “Sample size n” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Sample size n	p-value
9	0.2834
18	0.1223
27	0.05699
36	0.02753
45	0.01358
54	0.006797
72	0.001751

How to read the result

The sample mean is 2.3 standard errors above 50, and a difference that large would happen only about 2.8 % of the time if the true mean were 50. That is evidence the mean differs from 50, but it does not say the difference is large or important: 2.3 units may or may not matter in practice. Report the estimate with a confidence interval as well.

Common mistakes

Using z with a sample standard deviation and a small n.
Choosing a one-sided test after seeing the data.
Reading p as the probability that H₀ is true.
Saying "accept H₀" when the result is not significant; the correct phrase is "fail to reject".

Keep in mind: Tests assume random, independent observations and, for small samples, roughly normal data. Statistical significance is not proof of cause or of practical importance.

Frequently asked questions

What does a p-value of 0.0275 mean?

If H₀ were true, a result at least this extreme would occur about 2.75 % of the time.

When do I use a t test instead of a z test?

When the population standard deviation is unknown and estimated from the sample.

What is the significance level α?

The false-alarm risk you accept, commonly 0.05.

Sources: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.2 Confidence Limits for the Mean · OpenStax, Introductory Statistics 2e (2023), Chapter 9 Hypothesis Testing with One Sample

Checked 9 Nov 2026. NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.2 Confidence Limits for the Mean (t = (Ȳ − μ₀)/(s/√N), N − 1 degrees of freedom, two- and one-sided rejection regions); OpenStax, Introductory Statistics 2e, §9.3 (normal and t distributions for means; p′ normal with mean p and SD √(pq/n) when np and nq are above 5).

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