Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- To compare two groups, estimate p1 − p2 with the statistic p̂1 − p̂2.
- p̂1 − p̂2 changes from one pair of samples to the next. Its distribution over all possible pairs of samples is its sampling distribution.
- State the order of subtraction (for example, “town A − town B”) and keep it throughout.
- Interpret every value for the two named populations, the variable and the sample sizes.
Key relationships
| Quantity | Formula or rule |
|---|---|
| Mean | μ(p̂1 − p̂2) = p1 − p2 |
| Standard deviation | σ(p̂1 − p̂2) = √[p1(1 − p1)/n1 + p2(1 − p2)/n2] |
| Size of the standard deviation | larger than the standard deviation of either p̂ alone; smaller than their sum |
| Both sample sizes × 4 | standard deviation ÷ 2 |
| Swap the order of subtraction | mean changes sign; standard deviation unchanged |
| z-score | z = [(p̂1 − p̂2) − (p1 − p2)] / σ(p̂1 − p̂2) |
Conditions (in context, with numbers)
- Randomization: two independent random samples, or treatments randomly assigned in an experiment.
- 10%: when sampling without replacement, n1 ≤ 0.10N1 and n2 ≤ 0.10N2. Not needed for an experiment.
- Large counts: n1p1, n1(1 − p1), n2p2 and n2(1 − p2) are all at least 10. This gives an approximately normal shape.
The mean and standard deviation formulas need only independence. The normal shape needs large counts.
Mistakes to avoid
- Adding or subtracting standard deviations instead of adding variances.
- Subtracting the variances because the proportions are subtracted.
- Checking only two counts instead of four.
- Checking the 10% condition for an experiment, or forgetting it for samples.
- Using one combined sample size n1 + n2 in the formula.
- No order of subtraction, so the sign of the mean is unclear.
- “A small probability proves the claim is false.” It only gives reason to doubt it.
Quick self-check
- p1 = 0.5, n1 = 100, p2 = 0.4, n2 = 150. Find the mean and standard deviation of p̂1 − p̂2. (Mean 0.1; standard deviation √(0.0025 + 0.0016) = √0.0041 ≈ 0.0640)
- An experiment randomly assigns volunteers to two treatments. Which conditions do you check? (Randomization, by random assignment, and large counts. Not the 10% condition.)
- Both sample sizes are multiplied by 4. What happens to the standard deviation of p̂1 − p̂2? (It is divided by √4 = 2, so it halves.)
Next: practice questions.