Hypothesis Test

z- and t-tests with p-value and rejection region

z = (x̄ − μ₀)/(σ/√n) = (52 − 50)/(10/√100)

Decision

Reject H₀

Statistically significant at α = 0.05: the evidence favors the alternative.

Hypotheses

H₀: μ = 50

H₁: μ ≠ 50

Test statistic (z)

p-value

0.0455

Critical value

±1.96

Sample mean (x̄)

Standard error

Rejection region

Reject H₀ if |z| > 1.96

Distribution

Standard normal (z)

p = 0.0455 vs α = 0.05

About this calculator

The Hypothesis Test Calculator runs a one-sample z-test or t-test for a mean, or a z-test for a proportion, with a two-sided or one-tailed alternative. It reports the test statistic, the exact p-value, the critical value and rejection region, the standard error, and the reject / fail-to-reject decision at your chosen significance level α. A shaded density curve shows where the statistic falls relative to the rejection region, making the decision easy to see. Useful for statistics coursework, A/B testing, and quality control.

How to use the Hypothesis Test calculator

Enter the values for your problem into the input fields.
Read the result — it updates instantly as you type.
Check the formula and the visual explanation to follow how the answer was found.
Copy the page URL to share the exact calculation.

Common examples

z-test: x̄ = 52, μ₀ = 50, σ = 10, n = 100 → z = 2, p ≈ 0.0455 < 0.05 → reject H₀
t-test: x̄ = 105 vs μ₀ = 100, s = 12, n = 16 → t ≈ 1.667 (df 15), p ≈ 0.116 → fail to reject
Proportion: 60 successes in 100 vs p₀ = 0.5, right-tailed → z = 2, p ≈ 0.0228 → reject
Left-tailed z at α = 0.01: x̄ = 495 vs 500, σ = 20, n = 50 → z ≈ −1.768, p ≈ 0.0385 → fail to reject

Frequently asked questions

How does the Hypothesis Test calculator work?

Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.

When would I use the Hypothesis Test calculator?

It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.

How accurate are the results?

Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.

Can I share a specific calculation with someone else?

Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.