Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- The parameter p₁ − p₂ is a fixed number. The interval changes from sample to sample, so one interval may or may not capture it.
- Interval: “We are C% confident that the interval from a to b captures the difference (population 1 − population 2) in the proportions of all [individuals] who [response].”
- Confidence level: “In repeated random sampling with the same sample sizes from the same populations, about C% of intervals built this way would capture the true difference p₁ − p₂.”
- For an experiment, the populations are the treatments: “the proportion of subjects like these who would [respond] with treatment 1, minus with treatment 2”.
Key relationships
| The interval | Conclusion about p₁ − p₂ |
|---|---|
| Contains 0 | 0 is plausible: not convincing evidence of a difference (not proof of no difference) |
| All values positive | Convincing evidence of a difference, with p₁ > p₂ |
| All values negative | Convincing evidence of a difference, with p₁ < p₂ |
| Claimed difference d inside | d is plausible: no convincing evidence against the claim |
| Claimed difference d outside | Convincing evidence against the claim that p₁ − p₂ = d |
| Order reversed | Signs flip and endpoints swap; same conclusion |
| Higher confidence level | Larger z*, wider interval: a borderline conclusion can change |
Assumptions and conventions
- The interval was built correctly (Topic 3.10): random samples or random assignment, 10% condition for sampling, at least 10 observed successes and failures in each group.
- Endpoints are differences in proportions. In words, use percentage points.
- Random sampling allows generalising to the populations. Only random assignment allows a cause-and-effect conclusion.
Mistakes to avoid
- “95% probability that p₁ − p₂ is in the interval.” Use “95% confident”; the 95% is about the method.
- “Contains 0, so the proportions are equal.” Plausible is not proved.
- No order of subtraction, so the sign has no meaning.
- Talking about the samples instead of the populations or treatments.
- Judging from the point estimate alone instead of the whole interval.
- Claiming cause from independent random samples.
- Choosing the confidence level after seeing the data.
Quick self-check
- A 95% interval for p₁ − p₂ is (0.03, 0.11). Is there convincing evidence of a difference? (Yes: 0 is not in the interval; every value is positive, so p₁ > p₂.)
- A 95% interval is (−0.05, 0.02). What can you conclude? (It contains 0, so there is not convincing evidence of a difference. It does not show the proportions are equal.)
- Rewrite (0.04, 0.10) for p₂ − p₁. ((−0.10, −0.04))
- Using (0.03, 0.11), is a claimed difference of 8 percentage points plausible? (Yes: 0.08 is inside the interval.)
Next: practice questions.