AP Statistics Topic 3.11: Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Proportions

01 · Read the interval before judging the claim

The concept in 60 seconds

Suppose a study compares the proportion of students who prefer digital study notes at School A with the corresponding proportion at School B. A justified $95\%$ confidence interval for $p_{1}-p_{2}$ , A minus B, is approximately $(0.0557, 0.2643)$ .

We can use it to answer three questions:

What difference is estimated? A’s population preference proportion is estimated to be $5.57\text{ to }26.43$ percentage points higher than B’s, with $95\%$ confidence.
Is there evidence of a difference? Yes, using this interval approach: $0$ is outside the interval, and every value in the interval is positive.
Is an exact $10$ -percentage-point difference plausible? Yes. The value $0.10$ is inside the interval, so it is compatible with this interval. That does not prove it is the true difference.

Use an interval as evidence, not a guarantee. A confidence interval is calculated from random samples and may or may not capture the true population difference. A conclusion depends on a justified method and the stated confidence level.

Visual guide 1: the parameter is a population difference

Population 1 · School A $p_{1}$ is unknown

Proportion of all School A students who prefer digital notes. The sample estimate is $\hat{p}_{1}=0.62$ .

Population 2 · School B $p_{2}$ is unknown

Proportion of all School B students with the same preference. The sample estimate is $\hat{p}_{2}=0.46$ .

A minus BEstimate: $0.16$

The $95\%$ interval $(0.0557, 0.2643)$ estimates $p_{1}-p_{2}$ , not either school’s individual rate.

Sample data are fictional: $\frac{124}{200}$ at A and $\frac{69}{150}$ at B. State both target populations, the shared success outcome and the subtraction order before interpreting the interval.

Quick check: if

0.10

is inside the interval, have we proved that the true difference is

0.10

No. It is $1$ value compatible with the interval. Other values inside are also compatible; the interval does not identify $1$ exact true difference.

02 · Keep the parameter, signs and units together

Read the difference between proportions

All examples here are fictional teaching scenarios. In the main study, independent SRSs without replacement give $124$ successes out of $200$ at School A and $69$ out of $150$ at School B. Each success means “prefers digital notes.” The populations contain $10{,}000$ students at A and $8{,}000$ at B.

What each quantity describes in the school example.
Quantity	Meaning	Value or status
$p_{1}$ , $p_{2}$	Population preference proportions at Schools A and B	Unknown
$p_{1}-p_{2}$	Fixed population difference, A minus B	Unknown parameter
$\hat{p}_{1}$ , $\hat{p}_{2}$	Observed sample preference proportions	$0.62$ and $0.46$
$\hat{p}_{1}-\hat{p}_{2}$	Observed point estimate	$0.16=16\,\text{percentage points}$
$95\%$ confidence interval	Range estimating the population difference	Approximately $(0.0557, 0.2643)$
$0$ difference	Population proportions are equal	$p_{1}=p_{2}$

\hat p_1-\hat p_2=\frac{124}{200}-\frac{69}{150}=0.16

$95\%$ confidence interval for $p_1-p_2$ : approximately $(0.0557,0.2643)$ .

Percentage-point range: approximately $(5.57,26.43)$ .

The sample difference is $16$ percentage points. The interval estimates the unknown difference between all students’ preference proportions at the $2$ schools. It is not an interval for either school’s own preference rate.

Percentage points are differences, not relative increases

Multiply a proportion difference by $100$ to express it in percentage points. A difference of $0.10$ is $10$ percentage points. A “ $10\%$ relative increase” requires a reference rate and a division; it is a different quantity.

A negative difference is meaningful

Each individual proportion lies between $0$ and $1$ , but $p_{1}-p_{2}$ can be negative. For example, $-0.12$ means group 1’s success proportion is $12$ percentage points lower than group 2’s.

Reversing group order changes the signs

An interval $(L, U)$ for A minus B becomes $(-U, -L)$ for B minus A. The main interval becomes approximately $(-0.2643, -0.0557)$ . It describes the same comparison in the opposite order; confidence level and width do not change.

Quick check: does

(0.0557, 0.2643)

mean

5.57\%\text{ to }26.43\%

of School A students prefer digital notes?

No. Those endpoints describe an A-minus-B population difference, expressed as $5.57\text{ to }26.43$ percentage points. They are not School A’s own population proportion.

03 · Zero means equal population proportions

Use $0$ to assess a difference

p_1-p_2=0\iff p_1=p_2

For a justified interval, check whether it contains $0$ . Then check whether its nonzero values are positive or negative.

Visual guide 2: check the interval against

0

Separate fictional independent-SRS studies, all with justified conditions and $95\%$ unpooled $z$ -intervals. Sample counts: positive: $\frac{124}{200}$ versus $\frac{69}{150}$ ; containing $0$ : $\frac{108}{200}$ versus $\frac{100}{200}$ ; negative: $\frac{52}{100}$ versus $\frac{80}{100}$ . Each study meets its separately stated population-size and observed-count checks in the lesson. All intervals use the same group order and common $-45\text{ to }+35$ percentage-point scale. Endpoints are rounded from full-precision calculations. $0$ in an interval means insufficient evidence of a difference, not proof of equality.

Case 1: the interval is entirely positive

If $L > 0$ , every value is positive. The interval provides convincing evidence that $p_{1} > p_{2}$ , using the confidence level and method stated.

For the school interval $(0.0557, 0.2643)$ , $0$ is excluded. We have evidence that School A’s population preference proportion is higher than School B’s.

Case 2: the interval is entirely negative

If $U < 0$ , every value is negative. The interval provides convincing evidence that $p_{1} < p_{2}$ .

An A-minus-B interval of $(-0.4054, -0.1546)$ , for example, gives evidence that A’s population success proportion is lower than B’s. The negative signs do not invalidate an interval for a difference.

Case 3: the interval contains $0$

If $L\le 0\le U$ , equality is compatible with the interval. There is insufficient evidence from this interval to conclude a difference between the population proportions.

For $(-0.0578, 0.1378)$ , both negative and positive differences are also plausible. The interval does not establish a direction.

“Insufficient evidence of a difference” does not mean “evidence of equality.” An interval containing $0$ also contains other possible differences. It may simply be too imprecise to establish whether a difference exists.

If an endpoint is exactly $0$ , $0$ is included for this decision. If a displayed endpoint is rounded to $0$ , use more precision before deciding whether $0$ is truly included.

Quick check: a sample difference is positive, but its interval is

(-0.03, 0.12)

. Can we conclude the population difference is positive?

No. The interval includes $0$ and negative differences. A positive point estimate alone does not establish that the population difference is positive.

04 · Translate the exact wording into a value or threshold

Assess a specific claim

For an exact claimed difference, check its location

Suppose someone claims that $p_{1}-p_{2}=d_{0}$ . Compare $d_{0}$ with the justified interval:

Inside the interval: the claimed value is compatible with the interval. It is plausible under this confidence-interval approach, but not proved correct.
Outside the interval: the interval provides evidence against that exact claimed value at the stated confidence level. It does not make the claim logically impossible.

Visual guide 3: an exact claim is a value to locate

Exact claimed values compared with the school interval $(0.0557, 0.2643)$ .
Exact claimed difference	Location	What the interval says
$0$ : equal proportions	Outside, below the lower bound	Evidence against $0$ ; the interval is entirely positive.
$0.10$ : $10$ percentage points	Inside	Compatible with this interval, but not proved correct.
$0.20$ : $20$ percentage points	Inside	Compatible with this interval, but not proved correct.
$0.30$ : $30$ percentage points	Outside, above the upper bound	Evidence against this exact value at the stated confidence level.

These are exact-value claims. To assess “at least $10$ percentage points higher,” compare the lower bound with $0.10$ instead of checking whether $0.10$ is inside. Compatibility does not mean certainty or equal likelihood of all interval values.

For the main school interval, $0.10$ and $0.20$ are compatible, while $0$ and $0.30$ are excluded. Compatibility does not assign a probability to a claimed value or make all values inside equally likely.

“Exactly $10$ points” and “at least $10$ points” ask different questions

An exact claim asks whether $0.10$ is compatible. An “at least $10$ percentage points higher” claim asks whether the interval supports the whole difference being at least $0.10$ .

The main interval extends below $0.10$ . It therefore does not establish that A’s population proportion is at least $10$ percentage points higher than B’s, even though an exact $10$ -point difference is compatible.

Use the lower bound for a stronger positive claim

To support $p_1-p_2\gt t$ by the entire interval, require $L\gt t$ .

To support $p_1-p_2\ge t$ by the entire interval, require $L\ge t$ .

For the main interval, the unrounded lower bound is about $0.0556607$ :

More than $5$ percentage points higher: supported by this interval because $0.0556607 > 0.05$ .
At least $10$ percentage points higher: not established because the interval includes values below $0.10$ .
At least $20$ percentage points higher: not established because the interval includes values below $0.20$ .

Similarly, to support a whole difference being below a threshold, compare the upper bound with that threshold. An interval may provide evidence of some difference without establishing that the difference is large enough to matter for a particular decision.

Quick check: an interval is

(0.06, 0.14)

. Does it support “more than

5

points higher” and “more than

10

points higher”?

It supports more than $5$ percentage points higher, because the lower bound $0.06$ exceeds $0.05$ . It does not establish more than $10$ points: values below $0.10$ remain in the interval.

05 · Write the parameter, not a vague numerical range

Write a clear interval interpretation

A strong interpretation names the confidence level, both populations, the success outcome, the group order and the endpoint range.

Visual guide 4: build a contextual interpretation

Confidence + target“We are $95\%$ confident…”

Name the population preference proportion at School A and compare it with School B’s.

Range + direction“… $5.57\text{ to }26.43$ points higher…”

Convert the difference endpoints to percentage points; use “higher” because both are positive.

Outcome + populations“…prefer digital notes…”

Make clear that the statement concerns all students in the $2$ school populations, not just the sampled students.

Connect these ingredients into one sentence. For a range crossing $0$ , describe the lower endpoint as “lower” and the upper endpoint as “higher,” or keep the signed population-difference range.

Main-example sentence: “We are $95\%$ confident that the proportion of all School A students who prefer digital notes is between $5.57$ and $26.43$ percentage points higher than the proportion of all School B students who prefer digital notes.”

When the interval crosses $0$ , retain the signed range

For a justified $95\%$ A-minus-B interval of $(-0.03, 0.12)$ , say:

“We are $95\%$ confident that the population proportion with the stated outcome in group A is between $3$ percentage points lower and $12$ percentage points higher than the corresponding proportion in group B.”

Replace the group names and outcome with the actual context. Do not describe the whole range as “higher,” because the lower endpoint is negative.

Justify a claim with a reason and a conclusion

Use this pattern: interval fact → meaning → contextual conclusion.

Example: “Because every value in the $95\%$ interval $(0.0557, 0.2643)$

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Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Proportions

By the end of this lesson, you should be able to:

The concept in 60 seconds

Read the difference between proportions