AP Stats 3.11: Claims from a Two-Prop CI

What topic 3.11 covers

Topic 3.11 is about interpreting and using the interval you built in topic 3.10, not computing a new one. You state what the interval means in context, interpret the confidence level for a difference, and judge a claim about whether the two population proportions differ. It is the two-group counterpart of topic 3.4, and the same care with wording applies.

Interpreting the interval and the level

For a $C\%$ interval $(a, b)$ for $p_1 - p_2$, you are $C\%$ confident that the interval captures the true difference between the two population proportions, described with the response variable and the two populations in context. The confidence level describes the method: in repeated random sampling with the same sample sizes from the same populations, about $C\%$ of intervals built this way capture the true difference. A single interval either does or does not contain $p_1 - p_2$, and you cannot tell which from the data alone.

Using 0 to justify a claim

The value 0 is the reference point because $p_1 - p_2 = 0$ means the two proportions are equal.

• If the interval contains 0, then 0 is a plausible difference, so there is not convincing evidence that the two proportions differ.
• If the interval does not contain 0, then 0 is not plausible, so there is convincing evidence of a difference. The sign of the interval shows which proportion is larger.

Common wording mistakes

Do not say the interval shows "no difference" when it contains 0; the correct statement is that there is not enough evidence to conclude a difference exists. Failing to find a difference is not the same as proving the proportions are equal, just as failing to reject a null is not proving it true. When the interval excludes 0, name the direction of the difference and connect it back to the populations, rather than stopping at "the proportions differ." Matching your conclusion to the exact question keeps the interpretation from drifting into a claim the interval does not support. A tidy way to phrase a significant result is to name the direction of the difference, the range of plausible values, and the two populations, all in one sentence tied to the response variable.