Two-sample Proportion z-test

Tests whether two population proportions differ. With counts x₁, n₁ and x₂, n₂, the pooled proportion is p̂ = (x₁ + x₂) ÷ (n₁ + n₂). The test statistic is z = (p̂₁ − p̂₂) ÷ SE with SE = √(p̂(1−p̂)(1/n₁ + 1/n₂)).

Inputs

• Successes and trials in group 1: x₁, n₁
• Successes and trials in group 2: x₂, n₂
• Alternative: left, right, or two-tailed
• Significance level α

Assumptions

• Independent samples
• Approximate normality: each group has at least ~10 successes and ~10 failures

Example

Suppose x₁=45, n₁=100 and x₂=35, n₂=100. Then p̂₁=0.45, p̂₂=0.35, pooled p̂=0.40, SE≈√(0.4×0.6×(1/100+1/100))≈0.0693. z≈(0.10)/0.0693≈1.442. For a two-tailed test, p-value≈2×(1−Φ(1.442))≈0.149.

Related Calculators

• One-sample Proportion Test
• P-value Calculator
• Z-score Calculator
• Binomial Test