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How do you use a confidence interval for a difference of two proportions to justify a claim?

Topic 6.9 Justifying a Claim Based on a Confidence Interval for a Difference of Population Proportions: use a two-sample proportion interval to judge whether a difference exists and to evaluate claims about the size and direction of that difference.

A focused answer to AP Statistics Topic 6.9, on using a two-sample proportion confidence interval to judge whether two proportions differ and to assess claims about the size and direction of the difference, with worked justifications.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this topic is asking
The zero-check is the heart of it
Judging claims about the size of the difference
Confidence level, sample size, and decisiveness
Try this

What this topic is asking

The College Board (Topic 6.9) wants you to use a two-sample proportion interval to justify a claim: judge whether the two proportions differ (does the interval contain $0$ ?), and evaluate claims about the size and direction of the difference, accounting for confidence level and sample size.

The zero-check is the heart of it

This replaces the one-sample "is $p_0$ inside?" question with "is $0$ inside?" because the relevant null claim for two proportions is equality. The sign of the interval, when it excludes $0$ , tells you the direction of the difference, so always state the subtraction order ("group $1$ minus group $2$ ") to read the sign correctly.

Judging claims about the size of the difference

A directional or magnitude claim, "the new method raises the proportion by at least $0.05$ ," must be assessed against the entire interval, not the point estimate. If even part of the interval falls short of $0.05$ , then a smaller difference remains plausible and the claim is not established, even when the observed difference exceeds $0.05$ . Conversely, if the whole interval lies above $0.05$ , the claim is supported. As with one proportion, the interval (the set of plausible differences), not the single observed difference, carries the justification. This distinction, between "is there a difference at all" and "is the difference at least this big," is a frequent free-response discriminator.

Confidence level, sample size, and decisiveness

A higher confidence level widens the interval, which can pull a once-positive interval back across $0$ and so weaken a difference conclusion; a lower level narrows it and is more decisive (but correct less often). A larger sample in either group narrows the interval by shrinking the unpooled standard error, sharpening any conclusion about a difference and its size. These are the same trade-offs as the one-sample case, applied to a difference, and questions routinely ask you to predict the direction of the change and explain why.

Justifying a difference claim

A retailer compares two checkout designs. A $95\%$ confidence interval for $p_A - p_B$ , the difference in the proportion of customers who complete a purchase (design A minus design B), is $(0.01,\ 0.09)$ . (a) Is there convincing evidence the designs differ? (b) A manager claims design A raises completion by at least $5$ percentage points; assess this. (c) Predict the effect of a $99\%$ interval.

step 1 Zero-check (part a)

The interval $(0.01, 0.09)$ lies entirely above $0$ , so $0$ is not a plausible difference. There is convincing evidence that the two designs differ in completion proportion, with design A higher than design B (the interval is positive and the subtraction is A minus B).

step 2 Assess the magnitude claim (part b)

"At least $5$ percentage points" means $p_A - p_B \ge 0.05$ . The interval includes plausible differences below $0.05$ (for example $0.02$ ), so a difference smaller than $5$ points is still plausible. The interval does not establish that design A raises completion by at least $5$ points.

step 3 Effect of higher confidence (part c)

A $99\%$ interval uses a larger $z^{*}$ ( $2.576$ vs $1.96$ ), so it is wider, centered on the same difference of about $0.05$ . It might become roughly $(-0.005, 0.105)$ , which would contain $0$ and so no longer give convincing evidence of any difference; the conclusion becomes less decisive.

step 4 Summary

There is evidence of a difference (A higher) at $95\%$ , but not enough to confirm a gap of at least $5$ points, and raising confidence to $99\%$ could erase even the basic difference conclusion.

Try this

Q1. A $95\%$ interval for $p_1 - p_2$ is $(-0.08, -0.01)$ . What does it say? [1 point]

Cue. Entirely below $0$ , so there is convincing evidence $p_1 < p_2$ (proportion in group $1$ is lower).

Q2. Why must a magnitude claim be checked against the whole interval, not the observed difference? [1 point]

Cue. The interval is the set of plausible differences; if any plausible value falls short of the claimed size, the claim is not established even if the point estimate meets it.

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2017 (style)1 marksSection I (multiple choice). A

95\%

confidence interval for

p_1 - p_2

(-0.03,\ 0.11)

. Based on this interval, there is (A) convincing evidence

p_1 > p_2

(B) convincing evidence

p_1 < p_2

p_1

and

p_2

(D) proof the proportions are equal

Show worked answer →

The correct answer is (C).

The interval $(-0.03, 0.11)$ contains $0$ , so $0$ (no difference) is a plausible value of $p_1 - p_2$ . Therefore the data give no convincing evidence that the two proportions differ.

(A) and (B) would require the interval to lie entirely above or below $0$ . (D) overstates: an interval never proves equality, it only fails to rule it out.

AP 2021 (style)4 marksSection II (free response). A

90\%

confidence interval for

p_{\text{new}} - p_{\text{old}}

, the difference in the proportion of users who renew under a new versus old plan, is

(0.02,\ 0.10)

. (a) Justify in context whether there is convincing evidence the new plan has a higher renewal proportion. (b) A manager claims the new plan increases the renewal proportion by at least

5

percentage points. Justify whether the interval supports this. (c) Explain how a

99\%

interval would change the strength of the conclusion.

Show worked answer →

A 4-point justification question on a difference.

(a) (2 points) The entire interval $(0.02, 0.10)$ lies above $0$ , so $0$ is not a plausible difference; there is convincing evidence that the new plan has a higher renewal proportion than the old plan.
(b) (1 point) "At least $5$ percentage points" means $p_{\text{new}} - p_{\text{old}} \ge 0.05$ . The interval $(0.02, 0.10)$ includes values below $0.05$ (such as $0.03$ ), so a difference under $5$ points is still plausible; the interval does not establish that the increase is at least $5$ points.
(c) (1 point) A $99\%$ interval is wider; it might extend below $0$ , which would weaken or remove the evidence of any difference at all, making the conclusion less decisive.

Markers reward the zero-check for part (a), reading the whole interval against $0.05$ for part (b), and linking higher confidence to a wider, less decisive interval.

Related dot points

Sources & how we know this

AP Statistics Course and Exam Description — College Board (2020)