A two-sample z-test decides whether an observed difference between two sample proportions is too big to blame on chance. Enter the data, pick Hₐ and α, and see the test statistic, the shaded p-value, and a full four-step write-up.
Enter the data, choose the alternative and significance level, and follow all four steps: State, Plan, Do, Conclude.
z = (statistic − null value) ÷ standard error. It counts how many null standard errors the observed difference p̂₁ − p̂₂ is from 0.
The p-value is the probability of getting a difference at least as extreme as the one observed, in the direction of Hₐ, if H₀ is true. For ≠, double the one-tail area.
If p-value < α, reject H₀: the result is statistically significant. If p-value ≥ α, fail to reject H₀. Never "accept" H₀.
Say whether there is convincing evidence for Hₐ, in the words of the problem. Mention the p-value and α.
With random assignment, a significant result supports causation. With random sampling, the result generalizes to the populations. Without either, be careful.
With very large samples, tiny differences can be statistically significant. Statistical significance is not the same as practical importance.
From test statistic to conclusion.
| Hₐ | p-value | Calculator |
|---|---|---|
| p₁ − p₂ > 0 | P(Z ≥ z) | normalcdf(z, 1E99, 0, 1) |
| p₁ − p₂ < 0 | P(Z ≤ z) | normalcdf(−1E99, z, 0, 1) |
| p₁ − p₂ ≠ 0 | 2 · P(Z ≥ |z|) | 2 · normalcdf(|z|, 1E99, 0, 1) |
| p-value < α | Reject H₀ | Convincing evidence for Hₐ |
| p-value ≥ α | Fail to reject H₀ | Not convincing evidence for Hₐ |
| All steps | 2-PropZTest | Still write the formula, z, and p-value |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 3.13. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 3 terminology guide.