Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- To estimate how far apart two population proportions are, use a two-sample z-interval for a difference between population proportions.
- Define the parameter with the difference (order stated), the response and both populations (or both treatments).
- The interval has the shape point estimate ± margin of error, with point estimate p̂1 − p̂2.
- Interpreting the interval and using it to judge a claim come next, in Topic 3.11.
Key relationships
| Quantity | Formula or rule |
|---|---|
| Point estimate | p̂1 − p̂2, with p̂1 = x1/n1 and p̂2 = x2/n2 |
| Standard error | SE = √[p̂1(1 − p̂1)/n1 + p̂2(1 − p̂2)/n2] |
| Critical value | z* = invNorm((1 + C)/2): 1.645 (90%), 1.960 (95%), 2.576 (99%) |
| Margin of error | MOE = z* × SE |
| Interval | (p̂1 − p̂2) ± MOE |
| From an interval (a, b) | point estimate = (a + b)/2; MOE = (b − a)/2; SE = MOE ÷ z* |
| Reverse the order | interval (a, b) becomes (−b, −a); width unchanged |
Conditions (in context, with numbers)
- Randomization: two independent random samples, or a randomized experiment.
- 10%: n1 ≤ 0.10N1 and n2 ≤ 0.10N2 when sampling without replacement. Not needed for an experiment.
- Normality: n1p̂1, n1(1 − p̂1), n2p̂2 and n2(1 − p̂2) are all at least 10 (observed successes and failures).
Mistakes to avoid
- Pooling the two proportions. That is for tests, not intervals.
- Defining the parameter with the samples or leaving out the order of subtraction.
- Checking two counts, or n ≥ 30, instead of four observed counts.
- Applying the 10% condition to an experiment.
- Adding standard errors instead of adding the variance terms under one root.
- Using invNorm(C) for z*; use (1 + C)/2.
- Rounding p̂1 and p̂2 early, which moves the endpoints.
Quick self-check
- Independent random samples: 60 of 100 and 45 of 100 are successes. Find the 95% interval for p1 − p2. (Point estimate 0.15, SE ≈ 0.0698, MOE ≈ 0.137, interval (0.013, 0.287))
- A 90% interval for p1 − p2 is (0.04, 0.16). Find the point estimate and the margin of error. (0.10 and 0.06)
- One group has 9 failures. Can you use the z-interval? (No: the normality condition needs at least 10 failures in each group.)
Next: practice questions.