A two-sample z-interval estimates how far apart two population proportions are. Enter the data, check the conditions, and follow each step from the standard error to the final interval. Then simulate dozens of intervals to see what "95% confident" really promises.
Enter the successes and sample size for each group and pick a confidence level. The lab checks the conditions and works through every step.
Pretend the true proportions are known. Take many pairs of samples with the sizes above and build an interval from each. The confidence level is the long-run percent of intervals that capture the true difference.
Intervals that miss are drawn in red. Sample sizes and confidence level come from the builder above.
Every confidence interval is point estimate ± (critical value)(standard error). Here the estimate is p̂₁ − p̂₂ and the critical value is z*.
Because p₁ and p₂ are unknown, the standard error plugs in p̂₁ and p̂₂: SE = √( p̂₁(1 − p̂₁)/n₁ + p̂₂(1 − p̂₂)/n₂ ).
Two independent random samples or a randomized experiment; each sample ≤ 10% of its population (for sampling); and at least 10 successes and 10 failures in each group.
"We are 95% confident that the interval from −0.02 to 0.11 captures the true difference p₁ − p₂." Always say which group was subtracted from which, and give context.
95% confidence means that if we took many pairs of samples and built an interval from each, about 95% of those intervals would capture the true difference. It is not a probability about this one interval.
Lower confidence (smaller z*) and larger sample sizes (smaller SE) both shrink the interval. Higher confidence costs precision.
Critical values and the four-step process for a two-sample z-interval for p₁ − p₂.
| Step | What to write | Example |
|---|---|---|
| State | Parameter in words and the confidence level | p₁ − p₂, where p₁ = proportion of all city drivers who wear seatbelts…; 95% confidence |
| Plan | Name the procedure and check conditions | Two-sample z-interval for p₁ − p₂. Random ✓, 10% ✓, 412, 38, 341, 59 all ≥ 10 ✓ |
| Do | Formula, numbers, and the interval | (0.916 − 0.853) ± 1.96(0.0220) = (0.020, 0.106) |
| Conclude | Interpret in context | We are 95% confident the interval from 0.020 to 0.106 captures the true difference… |
| z* values | 90%: 1.645 · 95%: 1.960 · 98%: 2.326 · 99%: 2.576 | invNorm(1 − (1 − C)/2) |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 3.10. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 3 terminology guide.