Z-Test — Statistical Hypothesis Testing

Input — provide raw data OR summary statistics

Raw Data (optional — leave blank to use summary stats below)

Sample Mean (x̄) or computed from data

Known Population σ

Sample Size (n) or from data

Hypothesized Mean (μ₀)

Alternative Hypothesis

Significance Level (α)

Results

z = (x̄ − μ₀) / (σ / √n)

95% CI for μ:

x̄ ± z_(α/2) × (σ / √n)

Note: Use t-test when σ is unknown.

Input

Number of Successes (x)

Sample Size (n)

Hypothesized Proportion (p₀)

Alternative Hypothesis

Significance Level (α)

Results

p̂ = x / n z = (p̂ − p₀) / √[p₀(1−p₀)/n]

Wilson CI for proportion:

CI = (p̂ + z²/2n ± z√[p̂(1−p̂)/n + z²/4n²]) / (1 + z²/n)

Required: n·p₀ ≥ 5 and n·(1−p₀) ≥ 5

Input

Group 1 — Successes (x₁)

Group 1 — Sample Size (n₁)

Group 2 — Successes (x₂)

Group 2 — Sample Size (n₂)

Alternative Hypothesis

Significance Level (α)

Results

p̂_pool = (x₁ + x₂) / (n₁ + n₂) z = (p̂₁ − p̂₂) / √[p̂_pool(1−p̂_pool)(1/n₁ + 1/n₂)]

CI for (p₁ − p₂) uses unpooled SE:

SE = √[p̂₁(1−p̂₁)/n₁ + p̂₂(1−p̂₂)/n₂]