Z-Test Calculator

Perform one-sample Z-test to test hypotheses about population means when the population standard deviation is known. Calculate z-statistics and determine statistical significance.

Mean of your sample data

Hypothesized population mean

Known population standard deviation

Number of observations in sample

Z = (x̄ - μ₀) / (σ/√n), where x̄ = sample mean, μ₀ = hypothesized population mean, σ = population standard deviation, n = sample size
Sample: x̄=105, n=30. Population: μ₀=100, σ=15. Z = (105-100)/(15/√30) = 1.826. Two-tailed α=0.05: Not significant (|1.826| < 1.96)

What is a Z-test and when should I use it?

A Z-test is a statistical hypothesis test used to determine if a sample mean differs significantly from a known population mean when the population standard deviation is known. Use it when: (1) Population σ is known, (2) Sample size n ≥ 30 (or population is normal), (3) Testing one sample mean against a hypothesized value.

What is the difference between Z-test and t-test?

Z-test: Use when population standard deviation (σ) is KNOWN. Uses standard normal distribution (Z-distribution). T-test: Use when population σ is UNKNOWN (estimate from sample). Uses t-distribution. For large samples (n>30), results are similar. T-test is more common in practice since σ is rarely known.

How do I interpret the Z-statistic?

Z-statistic measures how many standard errors the sample mean is from the hypothesized population mean. |Z| > 1.96 (α=0.05, two-tailed) indicates significance. Example: Z=2.5 means sample mean is 2.5 standard errors away, suggesting a real difference (p < 0.05).

What are the null and alternative hypotheses for Z-test?

Two-tailed: H₀: μ = μ₀ vs H₁: μ ≠ μ₀ (mean differs). Left-tailed: H₀: μ = μ₀ vs H₁: μ < μ₀ (mean is less). Right-tailed: H₀: μ = μ₀ vs H₁: μ > μ₀ (mean is greater). Choose based on research question: two-tailed for difference, one-tailed for direction.

What assumptions does the Z-test require?

Requirements: (1) Random sample from population, (2) Population standard deviation (σ) is known, (3) Either: population is normally distributed OR sample size n ≥ 30 (Central Limit Theorem), (4) Independent observations. Violations can lead to incorrect conclusions.