Complete Four-Step Two-Proportion Interval Solution

A Complete Interval Is a Connected Argument

A two-proportion confidence interval is more than a pair of calculated endpoints. A complete response identifies the population difference being estimated, explains why the interval procedure is appropriate, shows the calculation, and interprets the result in context. In Common Errors With Two-Proportion Intervals, you learned to keep the group order consistent and to use the unpooled standard error. Here, those ideas become one connected four-step solution.

The four steps are State, Plan, Do, and Conclude. In State, define the population proportions and the difference of interest. In Plan, name the procedure and check its conditions. In Do, calculate the interval. In Conclude, interpret the range of plausible population differences, preserving the group order and the context.

Formula: For a confidence level with critical value $z^*$, a two-proportion interval for $p_1-p_2$ is $$ (\hat{p}_1-\hat{p}_2)\mathbin{\pm}z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} +\frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}. $$ The standard error is unpooled: each group’s contribution uses that group’s sample proportion.

The Four Steps

State.
Define $p_1$ and $p_2$ as the true proportions for the two named groups, with the same success criterion. State that the parameter of interest is $p_1-p_2$, and specify the confidence level.

Plan.
Name the two-proportion $z$-interval. Explain why the design supports inference, address independence between groups and the 10% condition when sampling without replacement, and check Large Counts separately for both groups.

Do.
Calculate each sample proportion, the difference $\hat{p}_1-\hat{p}_2$, the unpooled standard error, and the interval using the appropriate $z^*$.

Conclude.
Interpret the interval as plausible values for the true population difference in the stated order and context. If relevant, describe what the signs of the endpoints indicate.

For an independent random-sample study, explain how the samples were selected and why the groups are independent. When sampling without replacement, check that each sample is no more than 10% of its population. For a randomized experiment, random assignment supports the comparison; the groups must still consist of distinct, unpaired units. The 10% condition concerns sampling without replacement and is not a substitute for explaining the experiment’s random assignment.

For Large Counts in a two-proportion interval, each group needs at least 10 observed successes and at least 10 observed failures. If Group $i$ has $x_i$ successes among $n_i$ observations, check $x_i$ and $n_i-x_i$. As covered in Large Counts for Each Group in Two-Proportion Intervals, check all four counts, not just the total number of observations.

Worked Example: A Randomized Reminder Experiment

Setting: In a fictional experiment, 160 volunteers are randomly assigned to receive text reminders before an appointment, and 150 different volunteers are assigned to receive email reminders. Of those assigned to texts, 112 attend; of those assigned to email, 90 attend. Find and interpret a 95% confidence interval for the difference in attendance proportions, text minus email.

State: Let $p_1$ be the true proportion of volunteers like those in this experiment who would attend after receiving a text reminder, and let $p_2$ be the corresponding proportion after an email reminder. The parameter is $p_1-p_2$, text minus email. We will construct a 95% confidence interval.

Plan: Use a two-proportion $z$-interval. Volunteers were randomly assigned to the two reminder methods, supporting a chance-based comparison. The groups contain different, unpaired volunteers, so the observations in one group are not paired with observations in the other. The outcomes are binary: each volunteer either attended or did not attend. Because this is a randomized experiment rather than sampling without replacement from a population, the 10% condition is not the relevant design condition. For Large Counts, the text group has 112 successes and $160-112=48$ failures; the email group has 90 successes and $150-90=60$ failures. All four counts are at least 10, so the Large Counts condition is met.

Do: The sample proportions are $\hat{p}_1=112/160=0.70$ and $\hat{p}_2=90/150=0.60$. The estimated difference is $0.70-0.60=0.10$. For a 95% interval, use $z^*=1.96$. The unpooled standard error is:

\begin{aligned} SE_{\hat{p}_1-\hat{p}_2} &=\sqrt{\frac{0.70(0.30)}{160}+\frac{0.60(0.40)}{150}}\\ &=\sqrt{0.0013125+0.0016000}\\ &=\sqrt{0.0029125}\approx0.053968. \end{aligned}

The margin of error is $1.96(0.053968)\approx0.105776$. Therefore:

0.10\mathbin{\pm}0.105776 =(-0.005776,\ 0.205776) \approx(-0.0058,\ 0.2058).

A calculator’s 2-PropZInt procedure gives the same interval when the successes, sample sizes, and confidence level are entered in the stated group order.

Conclude: We are 95% confident that the true difference in attendance proportions, text minus email, is between about $-0.0058$ and $0.2058$. In context, the interval ranges from a text-reminder attendance proportion about 0.58 percentage points lower to one about 20.58 percentage points higher than the email-reminder proportion. Because zero is in the interval, a zero difference is among the plausible values; the interval does not establish which reminder method has the higher population attendance proportion. Random assignment allows a causal comparison for the experimental units, but these volunteers were not randomly sampled from a broader population.

Using the Same Structure for Random Samples

In an observational study, random sampling can support generalizing to the populations sampled, but it does not establish cause and effect. State the target populations carefully, and make the conclusion match the design. The arithmetic of the interval is the same: calculate the two sample proportions and use the unpooled standard error.

Worked Example: Comparing Two Neighborhood Samples

Setting: In a fictional community survey, separate random samples include 240 residents who live near a large park and 200 residents who live farther away. In the first sample, 84 residents report walking for recreation at least once a week; in the second, 50 do. Assume the near-park population has at least 2,400 residents and the farther-away population has at least 2,000. Find a 95% confidence interval for the difference, near park minus farther away.

State: Let $p_1$ be the true proportion of residents living near the park who walk for recreation at least weekly, and $p_2$ the corresponding proportion of residents living farther away. The parameter is $p_1-p_2$, near park minus farther away, at 95% confidence.

Plan: Use a two-proportion $z$-interval. The two groups were selected using separate random samples, supporting the Random condition and generalization to the respective populations. Residents in one sample are not paired with residents in the other, supporting independence between groups. Each sample is at most 10% of its population: $240\leq0.10(2{,}400)=240$ and $200\leq0.10(2{,}000)=200$. For Large Counts, the near-park sample has 84 successes and $240-84=156$ failures; the farther-away sample has 50 successes and $200-50=150$ failures. All four counts are at least 10.

Do: The sample proportions are $\hat{p}_1=84/240=0.35$ and $\hat{p}_2=50/200=0.25$, so the point estimate is $0.35-0.25=0.10$. Using $z^*=1.96$, the unpooled standard error is:

\begin{aligned} SE_{\hat{p}_1-\hat{p}_2} &=\sqrt{\frac{0.35(0.65)}{240}+\frac{0.25(0.75)}{200}}\\ &=\sqrt{0.0009479+0.0009375}\\ &=\sqrt{0.0018854}\approx0.043421. \end{aligned}

The margin of error is $1.96(0.043421)\approx0.085106$. The interval is:

0.10\mathbin{\pm}0.085106 =(0.014894,\ 0.185106) \approx(0.0149,\ 0.1851).

Conclude: We are 95% confident that the true difference in weekly recreational walking proportions, near park minus farther away, is between about 0.0149 and 0.1851. In context, the proportion near the park is estimated to be about 1.49 to 18.51 percentage points higher. Because the interval is entirely above zero, it supports a higher proportion among residents near the park. This observational comparison does not show that living near a park causes more walking.

When the Interval Includes Zero

An interval that includes zero calls for a careful conclusion. It does not say the proportions are equal, and it does not mean the sample proportions were equal. It says zero is one plausible value for the population difference at the stated confidence level; other values within the interval are plausible too. The interval’s width and endpoints communicate uncertainty and possible direction.

Worked Example: A 99% Interval That Includes Zero

Setting: In a fictional survey, separate random samples include 250 customers who used a mobile checkout option and 200 customers who used a staffed checkout. Of those in the mobile group, 150 report being satisfied; in the staffed-checkout group, 100 report being satisfied. Assume each sampled group is no more than 10% of its respective population. Find a 99% confidence interval for mobile minus staffed checkout.

State: Let $p_1$ be the true satisfaction proportion among customers like those sampled who use mobile checkout, and let $p_2$ be the corresponding proportion for customers who use staffed checkout. The parameter is $p_1-p_2$, mobile minus staffed checkout, at 99% confidence.

Plan: Use a two-proportion $z$-interval. The samples are separate random samples, and each is at most 10% of its population as stated. The groups are independent and unpaired. For Large Counts, the mobile group has 150 successes and $250-150=100$ failures; the staffed group has 100 successes and $200-100=100$ failures. All four counts are at least 10.

Do: The sample proportions are $\hat{p}_1=150/250=0.60$ and $\hat{p}_2=100/200=0.50$, giving a point estimate of $0.10$. For 99% confidence, use $z^*=2.576$. The standard error is:

\begin{aligned} SE_{\hat{p}_1-\hat{p}_2} &=\sqrt{\frac{0.60(0.40)}{250}+\frac{0.50(0.50)}{200}}\\ &=\sqrt{0.000960+0.001250}\\ &=\sqrt{0.002210}\approx0.047011. \end{aligned}

The margin of error is $2.576(0.047011)\approx0.121099$. Therefore:

0.10\mathbin{\pm}0.121099 =(-0.021099,\ 0.221099) \approx(-0.0211,\ 0.2211).

Conclude: We are 99% confident that the true difference in satisfaction proportions, mobile minus staffed checkout, is between about $-0.0211$ and $0.2211$. The interval includes zero and values on both sides of zero, so it is plausible that either group has a somewhat higher population satisfaction proportion. This interval does not establish that the proportions are equal.

Common Mistakes and AP Exam Tips

Leaving group sizes unclear: State $n_1$ and $n_2$ explicitly, and divide each group’s success count by its own sample size. Do not use the total number of participants as both denominators unless that is actually each group’s size.
Skipping the conditions: A statement such as “the conditions are met” is not a check. Name the study’s random process, address group independence and the 10% condition when relevant, and list all four success and failure counts.
Using a pooled proportion: For this interval, use the two separate sample proportions in the standard error. Pooling is not the interval method for estimating a difference that may be nonzero.
Changing the order in the conclusion: If the parameter is $p_1-p_2$, interpret the result as Group 1 minus Group 2. For negative endpoints, translate the direction without silently reversing the defined difference.
Overstating what the design shows: Random samples can support generalization to the sampled populations; random assignment can support a causal claim about experimental units. An observational association alone does not establish cause and effect.
Claiming equality when zero is included: Say that zero is plausible at the stated confidence level, along with the other values in the interval. Do not say that the population proportions are proven equal.

AP Exam Tip: Write enough detail for a reader to follow the argument: define the parameter and subtraction order, justify the procedure with explicit condition checks, show the unpooled calculation, and interpret both endpoints in context. A correct interval without a contextual conclusion is not a complete solution.

Key Takeaway

A full two-proportion interval response links the study design to the method and the method to a contextual conclusion. Keep the group order fixed, check each condition explicitly, calculate with the separate sample proportions, and explain what the interval’s endpoints—and the presence or absence of zero—mean for the population difference.

Key takeaway: State the population difference, Plan with justified conditions, Do the unpooled interval calculation, and Conclude in context with the same group order.

Check Your Understanding

Use the four-step structure and the distinctions in this tutorial to answer each question.

In a study, 63 of 90 people in Group 1 and 48 of 80 people in Group 2 have the defined outcome. What are the two sample proportions, and what is the point estimate for Group 1 minus Group 2?
For a two-proportion interval, what four counts must be checked for the Large Counts condition?
A survey uses independent random samples of 300 and 250 people. What population-size information would establish the 10% condition for both samples?
An interval for $p_1-p_2$ is $(-0.03,\ 0.16)$. What does the inclusion of zero mean, and what should the conclusion avoid claiming?
Why does random assignment in an experiment support a different kind of conclusion from random sampling in an observational study?