When do I use a pooled proportion versus unpooled?

Use unpooled proportions (each sample's own p) for a confidence interval, since you make no assumption about equality. Pool only for the two-proportion significance test, where the null hypothesis assumes the two proportions are equal.

How do I use the interval to decide if two proportions differ?

Check whether the interval contains 0. If it does, there's no convincing evidence of a difference. If it's entirely positive or entirely negative, there is convincing evidence that the proportions differ, and the sign tells you which is larger based on your subtraction order.

What's the difference between interpreting the interval and interpreting the confidence level?

Interpreting the interval means stating that you're C% confident the interval captures the true difference p_1 - p_2 in context. Interpreting the confidence level means explaining that, in repeated sampling, about C% of such intervals would capture the true difference. The exam asks for these separately.

Does the order of subtraction matter?

Yes. p_1 - p_2 is not the same as p_2 - p_1; the signs flip. Clearly define which group is 1 and which is 2, then interpret the sign accordingly. A positive interval means group 1's proportion is larger.

AP-STATS-6.8-6.9

U6.8 Two-Proportion z-Interval

Learn to build and interpret a two-proportion z-confidence interval for p1 - p2: conditions, formula, calculations, and how to judge whether two proportions truly differ.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on U6.8 Two-Proportion z-Interval, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

When you want to know whether two groups differ in the rate of some outcome — say, the proportion of teens who exercise daily in two schools — a single proportion isn't enough. You need to estimate the difference between two population proportions,

p_1 - p_2

. The two-proportion z-interval gives you a plausible range for that difference, complete with a margin of error.

In this lesson you'll verify the Random, Independence (10%), and Large Counts conditions in both samples, plug sample proportions into the interval formula, and — most importantly — interpret the result. A key skill the exam rewards is using whether the interval contains 0 to decide if there's convincing evidence of a real difference.

The Formula and What Each Piece Means

A two-proportion z-interval estimates

p_1 - p_2

using the two sample proportions

\hat{p}_1

and

\hat{p}_2

. The general structure is the familiar

\text{point estimate} \pm \text{critical value} \times \text{standard error}.

The point estimate is the observed difference

\hat{p}_1 - \hat{p}_2

. The standard error combines the variability from both samples:

SE = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}.

The full interval is

(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}.

The critical value

z^*

comes from the standard normal distribution:

1.645

for 90% confidence,

1.96

for 95%, and

2.576

for 99%.

Notice a crucial difference from the two-proportion test: in a confidence interval you do not pool the sample proportions. Pooling is only used in the significance test (Topic 6.10), where the null hypothesis assumes the proportions are equal. Here we make no such assumption, so each sample keeps its own

\hat{p}

in the standard error. Mixing these up is one of the most common point-losing errors on the exam.

Checking Conditions in Both Samples

Before computing anything, you must verify three conditions — and because there are two groups, most conditions are checked twice.

Condition	What to check	Two-sample version
Random	Data from random samples or random assignment	Both groups must qualify
Independence (10%)	Sample < 10% of population when sampling without replacement	$n_1 \le 0.10 N_1$ and $n_2 \le 0.10 N_2$ ; the two samples must also be independent of each other
Large Counts	At least 10 successes and 10 failures	$n_1\hat{p}_1$ , $n_1(1-\hat{p}_1)$ , $n_2\hat{p}_2$ , $n_2(1-\hat{p}_2)$ all $\ge 10$

For Large Counts, show all four counts explicitly using the observed data — graders want to see numbers, not just a claim. For example, write "

34 \ge 10

66 \ge 10

28 \ge 10

72 \ge 10

," then conclude the condition is met.

The independence-between-groups requirement is easy to overlook. If the two samples come from experimental groups formed by random assignment, or from two separately drawn random samples, they're independent. If the same subjects are measured twice (paired data), a two-proportion interval is not appropriate.

Interpreting the Interval and Assessing a Difference

A correct interpretation names the confidence level, the parameter, and the context. Template: "We are 95% confident that the interval from ___ to ___ captures the true difference in proportions (

p_1 - p_2

) between [group 1] and [group 2] in [context]."

Remember what "95% confident" means: the method captures the true difference in about 95% of all possible samples. It does not mean there is a 95% probability the true difference lies in this specific interval.

To decide whether the two proportions differ, look at whether the interval contains 0.

Interval location	Conclusion
Contains 0	No convincing evidence the proportions differ
Entirely positive	Convincing evidence $p_1 > p_2$
Entirely negative	Convincing evidence $p_1 < p_2$

Order matters:

p_1 - p_2

means you subtract group 2 from group 1. A positive interval means group 1's proportion is larger. Always state which group you called 1 and which you called 2 so the sign of the difference is interpretable. The exam frequently asks a follow-up like "Does the interval suggest a difference?" — answer by checking for 0 and connecting it to context.

How the Exam Tests This Topic

On the AP exam, two-proportion intervals appear both as standalone free-response questions and as part of larger investigative tasks. A full-credit response almost always requires four steps, often remembered as State, Plan, Do, Conclude.

First, identify the procedure and parameter: a two-proportion z-interval for

p_1 - p_2

. Second, check and verify all conditions with numbers shown. Third, compute the interval (you may use calculator output, but report the interval clearly). Fourth, interpret in context and, if asked, use the interval to answer the research question.

Common mistakes graders penalize include pooling the proportions (wrong for a CI), forgetting to check conditions in both samples, interpreting the interval as a probability statement about the parameter, and reversing the subtraction order without saying so. Another frequent slip is interpreting the confidence level versus interpreting the interval — these are different prompts. "Interpret the interval" asks for the plausible-values statement; "interpret the confidence level" asks about the long-run capture rate of the method.

When a question gives raw counts, convert to

\hat{p}

carefully, and double-check that you used sample sizes

n_1

and

n_2

(not the number of successes) in the standard error denominators.

Key terms

Difference in proportions ( $p_1 - p_2$ ).: The population parameter estimated by a two-proportion interval: the true difference between the proportions of successes in two populations or treatment groups.
Point estimate.: The observed sample difference $\hat{p}_1 - \hat{p}_2$ , which sits at the center of the confidence interval.
Standard error (SE).: An estimate of the variability of $\hat{p}_1 - \hat{p}_2$ , computed as $\sqrt{\hat{p}_1(1-\hat{p}_1)/n_1 + \hat{p}_2(1-\hat{p}_2)/n_2}$ , using unpooled proportions.
Critical value ( $z^*$ ).: The multiplier from the standard normal distribution corresponding to the confidence level: 1.645 (90%), 1.96 (95%), 2.576 (99%).
Margin of error.: The quantity $z^* \times SE$ added and subtracted from the point estimate; it reflects both confidence level and sampling variability.
Large Counts condition.: Requirement that each group has at least 10 successes and 10 failures, so the sampling distribution of the difference is approximately normal.
Independence between samples.: The requirement that the two groups be selected or assigned independently of each other; violated by paired or matched data.
Contains zero.: When a difference interval includes 0, it provides no convincing evidence that the two proportions differ.

Worked example

A researcher takes independent random samples of adults in two cities. In City A, 108 of 300 support a new transit tax. In City B, 84 of 350 support it. Construct and interpret a 95% confidence interval for the difference in the proportion of supporters,

p_A - p_B

First identify the procedure: a two-proportion z-interval for

p_A - p_B

.

Compute the sample proportions. City A:

\hat{p}_A = 108/300 = 0.36

. City B:

\hat{p}_B = 84/350 = 0.24

. The point estimate is

0.36 - 0.24 = 0.12

.

Check conditions. Random: both are stated random samples. Independence: samples are independent of each other, and each city surely has more than

3000

and

3500

adults, so the 10% condition holds. Large Counts: City A has

108 \ge 10

successes and

192 \ge 10

failures; City B has

84 \ge 10

successes and

266 \ge 10

failures. All conditions met.

Compute the standard error (unpooled):

SE = \sqrt{\frac{0.36(0.64)}{300} + \frac{0.24(0.76)}{350}} = \sqrt{0.000768 + 0.000521} = \sqrt{0.001289} \approx 0.0359.

With

z^* = 1.96

, the margin of error is

1.96 \times 0.0359 \approx 0.0704

.

The interval is

0.12 \pm 0.070

, or about

(0.050, 0.190)

.

Interpret: We are 95% confident that the true difference in the proportion of supporters (City A minus City B) is between

0.050

and

0.190

. Because the interval is entirely positive and does not contain 0, there is convincing evidence that City A has a higher proportion of supporters than City B.

Practice questions

A 95% two-proportion z-interval for

p_1 - p_2

is calculated to be

(-0.04, 0.11)

. Which conclusion is best supported?

Because the interval contains 0, there is not convincing evidence that the two proportions differ.
Because the interval contains 0, we have proven that the two proportions are equal.
Because the interval is mostly positive, $p_1$ is definitely greater than $p_2$ .
Because the interval is narrow, the sample sizes must have been small.

Answer: Because the interval contains 0, there is not convincing evidence that the two proportions differ.

An interval containing 0 means a difference of zero is a plausible value, so we lack convincing evidence of a real difference. It does not prove equality (a CI never proves a value), and being partly positive does not establish that

p_1 > p_2

. Interval width relates to sample size inversely, and a narrow interval suggests larger, not smaller, samples.

Explain why the two-proportion z-interval uses unpooled sample proportions in the standard error, while the two-proportion z-test may use a pooled proportion.

Answer: A confidence interval makes no assumption that the proportions are equal, so each sample uses its own $\hat{p}$ ; a significance test assumes equality under $H_0$ , justifying a pooled estimate.

The interval's goal is simply to estimate

p_1 - p_2

without presupposing any relationship, so the natural standard error keeps

\hat{p}_1

and

\hat{p}_2

separate. In contrast, the test's null hypothesis

H_0: p_1 = p_2

implies a single common proportion, which is best estimated by combining all successes over all trials. Using the pooled value in an interval, or the unpooled value in the test, would be a procedural error.

A survey found that 45 of 90 students at School X and 60 of 150 students at School Y have a part-time job. Verify the Large Counts condition for a two-proportion z-interval.

Answer: School X: 45 successes and 45 failures; School Y: 60 successes and 90 failures. All four counts are at least 10, so the condition is met.

For School X, successes

= 45

and failures

= 90 - 45 = 45

. For School Y, successes

= 60

and failures

= 150 - 60 = 90

. Since

45 \ge 10

45 \ge 10

60 \ge 10

, and

90 \ge 10

, the sampling distribution of the difference is approximately normal and the Large Counts condition is satisfied. Always show all four counts with numbers to earn full credit.

FAQ

When do I use a pooled proportion versus unpooled?: Use unpooled proportions (each sample's own $\hat{p}$ ) for a confidence interval, since you make no assumption about equality. Pool only for the two-proportion significance test, where the null hypothesis assumes the two proportions are equal.
How do I use the interval to decide if two proportions differ?: Check whether the interval contains 0. If it does, there's no convincing evidence of a difference. If it's entirely positive or entirely negative, there is convincing evidence that the proportions differ, and the sign tells you which is larger based on your subtraction order.
What's the difference between interpreting the interval and interpreting the confidence level?: Interpreting the interval means stating that you're C% confident the interval captures the true difference $p_1 - p_2$ in context. Interpreting the confidence level means explaining that, in repeated sampling, about C% of such intervals would capture the true difference. The exam asks for these separately.
Does the order of subtraction matter?: Yes. $p_1 - p_2$ is not the same as $p_2 - p_1$ ; the signs flip. Clearly define which group is 1 and which is 2, then interpret the sign accordingly. A positive interval means group 1's proportion is larger.

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