Hypothesis tests for proportions

Math

Definition

A statistical procedure that uses sample data to test a claim about a population proportion. It calculates a z-test statistic comparing the sample proportion to the hypothesized proportion and uses the p-value to decide whether to reject the null hypothesis.

How It Works

  1. State the null hypothesis H₀: p = p₀ and alternative hypothesis Hₐ.
  2. Check conditions: random sample, independence (10% rule), and normality (np₀ ≥ 10 and n(1−p₀) ≥ 10).
  3. Calculate the test statistic: z = (p̂ − p₀) / √(p₀(1−p₀)/n).
  4. Find the p-value from the z-distribution.
  5. Compare p-value to significance level α and make a conclusion.
  6. State the conclusion in context.

Examples

  • Testing whether more than 50% of voters support a candidate
  • Checking if a factory's defect rate exceeds 3%
  • Determining if a new drug's success rate differs from the existing treatment
Key Fact

z = (p̂ − p₀) / √(p₀(1−p₀)/n)

Related Terms

Confidence intervals for proportions Two-proportion z-test Sampling distribution of a proportion Normal distribution

Study This Concept

Practice hypothesis tests for proportions with free review games in these units:

AP Statistics Unit 6: Inference for Proportions