Math
Hypothesis tests for proportions.
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 · 6 phases
Step by step.
- State the null hypothesis H₀: p = p₀ and alternative hypothesis Hₐ.
- Check conditions: random sample, independence (10% rule), and normality (np₀ ≥ 10 and n(1−p₀) ≥ 10).
- Calculate the test statistic: z = (p̂ − p₀) / √(p₀(1−p₀)/n).
- Find the p-value from the z-distribution.
- Compare p-value to significance level α and make a conclusion.
- State the conclusion in context.
Examples
Real-world.
- 1 Testing whether more than 50% of voters support a candidate
- 2 Checking if a factory's defect rate exceeds 3%
- 3 Determining if a new drug's success rate differs from the existing treatment
Key Fact
z = (p̂ − p₀) / √(p₀(1−p₀)/n)