Once you have a confidence interval for p₁ − p₂, you can use it to judge claims. Is there a difference? Which group is higher? Is a specific difference plausible? Pick a claim and see whether every plausible value agrees with it.
A confidence interval gives the plausible values of p₁ − p₂. A claim is supported only if every plausible value agrees with it. Pick a study, choose a claim, and see the verdict.
Keep the same sample proportions and change the sample size or the confidence level. A narrower interval can turn "not convincing" into "convincing."
Values inside a confidence interval are plausible for the parameter. Values outside are not plausible at that confidence level.
If 0 is in the interval for p₁ − p₂, no difference is plausible, so there is not convincing evidence the proportions differ. If 0 is outside, there is.
An interval entirely above 0 supports p₁ > p₂. An interval entirely below 0 supports p₁ < p₂. Check the order of subtraction before you decide.
A value inside the interval is plausible, not established. "0 is in the interval" never proves the proportions are equal; it means we cannot rule equality out.
Larger samples give narrower intervals. Higher confidence gives wider intervals. To halve the margin of error, use about four times as many people in each group.
Random assignment lets you conclude cause and effect. Random sampling lets you generalize to the populations. A claim needs the right design, not just the right interval.
Reading a 95% interval for p₁ − p₂ against common claims.
| Interval | Claim | Verdict | Why |
|---|---|---|---|
| (0.03, 0.12) | p₁ ≠ p₂ | Supported | 0 is not in the interval |
| (−0.04, 0.07) | p₁ ≠ p₂ | Not convincing | 0 is plausible |
| (0.03, 0.12) | p₁ − p₂ > 0.10 | Not convincing | Values from 0.03 to 0.10 are also plausible |
| (0.03, 0.12) | p₁ − p₂ = 0.05 | Plausible | 0.05 is inside; other values are too |
| (−0.14, −0.03) | p₁ > p₂ | Contradicted | Every plausible value is negative |
| Any | Wider at 99% than 95% | Always true | Larger z* |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 3.11. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 3 terminology guide.