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)
2
p-value
0.0455
Critical value
±1.96
Sample mean (x̄)
52
Standard error
1
Rejection region
Reject H₀ if |z| > 1.96
Distribution
Standard normal (z)

p = 0.0455 vs α = 0.05

Rejection regionDensity curve of the Standard normal (z) distribution under H₀ with the rejection region (total area α = 0.05) shaded in both tails beyond ±1.96. The observed statistic z = 2 lands inside the rejection region, so the decision is: Reject H₀.Standard normal (z)‎-1.961.96z = 2Reject H₀α = 0.05 · p = 0.0455

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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.