Hypothesis Testing Calculator — One-Sample Z-Test

Formula

How this is calculated

A one-sample z-test checks whether a sample mean x̄ is compatible with a hypothesised population mean μ₀, given a known (or assumed) population standard deviation σ. The test statistic z = (x̄ − μ₀) / (σ / √n) expresses the observed difference in units of the standard error SE = σ / √n. Under the null hypothesis H₀: μ = μ₀, the statistic follows a standard normal distribution, so the p-value is the probability of observing a z this extreme or more extreme purely by chance.

For a two-tailed test (H₁: μ ≠ μ₀) the p-value equals 2 × P(Z > |z|). For a right-tailed test (H₁: μ > μ₀) it is P(Z > z); for left-tailed (H₁: μ < μ₀) it is P(Z < z). If the p-value is below the significance level α, reject H₀ and call the result statistically significant at that level. The bell curve highlights the computed z and shades the p-value tail area.

Key assumptions: the sample is drawn randomly and independently; the population standard deviation σ is known (if unknown, use a t-test instead); the Central Limit Theorem applies — generally n ≥ 30 or the population is known to be normal. Statistical significance does not imply practical importance; always consider effect size alongside the p-value.