One-sample Proportion z-test

Tests whether a population proportion equals a hypothesized value p₀. With x successes in n trials, p̂ = x ÷ n and z = (p̂ − p₀) ÷ √(p₀(1−p₀)/n).

Inputs

• Successes x, trials n
• Null proportion p₀
• Alternative: p < p₀, p > p₀, or p ≠ p₀
• Significance level α

Assumptions

• Independent Bernoulli trials
• Normal approximation: expected successes np₀ and failures n(1−p₀) at least ~10

Example

Suppose x=52, n=100, p₀=0.50. Then p̂=0.52 and SE=√(0.5×0.5/100)=0.05. z=(0.52−0.50)/0.05=0.40. Two-tailed p-value≈2×(1−Φ(0.40))≈0.689.

Related Calculators

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