Two-proportion z-test

Math

Definition

A two-proportion z-test is a hypothesis test used to determine whether there is a significant difference between the proportions of two independent populations. It compares the sample proportions and uses a pooled proportion under the null hypothesis that the two population proportions are equal.

How It Works

  1. State H₀: p₁ = p₂ and the alternative hypothesis.
  2. Calculate the sample proportions p̂₁ and p̂₂ and the pooled proportion p̂c.
  3. Compute the standard error using the pooled proportion.
  4. Calculate the z-statistic: z = (p̂₁ − p̂₂) / SE.
  5. Find the p-value and compare to the significance level to make a conclusion.

Examples

  • Comparing the proportion of students who passed a test at two different schools
  • Testing whether a new drug has a different cure rate than the existing treatment
  • Determining if male and female voters support a candidate at different rates
Key Fact

z = (p̂₁ − p̂₂) / √[p̂c(1 − p̂c)(1/n₁ + 1/n₂)] where p̂c is the pooled proportion

Related Terms

Hypothesis tests for proportions Confidence intervals for proportions Two-sample t-test Sampling distribution of a proportion

Study This Concept

Practice two-proportion z-test with free review games in these units:

AP Statistics Unit 6: Inference for Proportions