# Carrying Out a Test for the Difference Between Two Population Proportions

> AP Statistics · AP Stats 2024-2026
> Source: https://www.owlsprep.com/study/ap-statistics-u12-carrying-out-a-test-for/

This module walks through the full 4-step AP-aligned workflow for a two-proportion z-test, including condition checks, test statistic calculation, p-value interpretation, and context-rich conclusions.

**Prerequisites:** [One-proportion z-test for a single population proportion](https://www.owlsprep.com/study/ap-statistics-u12-one-proportion-z-test/); [Sampling distribution for the difference in two proportions](https://www.owlsprep.com/study/ap-statistics-u11-sampling-distribution-two-proportions/)

## Learning objectives

- State null and alternative hypotheses for a two-proportion z-test, explicitly defining population parameters in context
- Verify all three required validity conditions for a two-proportion z-test per AP scoring standards
- Calculate the pooled test statistic and corresponding p-value for one-tailed and two-tailed tests
- Draw a statistically correct, context-rich conclusion that links p-values to pre-specified significance levels

## Step 1: State Hypotheses and Define Parameters

The first step of any AP Statistics hypothesis test requires you to explicitly define your population parameters before writing null and alternative hypotheses to avoid losing easy rubric points.

**Null Hypothesis for Two-Proportion Test** — The claim that there is no difference between the two population proportions of successes, or $p_1 = p_2$

*Notation:* $H_0: p_1 - p_2 = 0$

*Example:* $H_0: p_m - p_f = 0$, where $p_m$ is the proportion of male high school students who have tried vaping, and $p_f$ is the proportion of female high school students who have tried vaping.

**Exam command terms**

AP exam questions use specific command terms that signal required content:

- **State a pair of hypotheses** — You must explicitly define $p_1$ and $p_2$ in context; generic notation without context will not earn points

- **Identify the parameters of interest** — You must name both populations and the specific success outcome for each proportion

**Worked example:** A researcher wants to test if the proportion of teens who own a smartphone is different between rural and urban areas. State the correct hypotheses and define parameters.

1. First, define parameters clearly in context:
2. $$p_1 = \text{True proportion of urban teens who own a smartphone}$$
3. $$p_2 = \text{True proportion of rural teens who own a smartphone}$$
4. Write null hypothesis of no difference:
5. $$H_0: p_1 = p_2 \quad \text{or} \quad H_0: p_1 - p_2 = 0$$
6. Write two-sided alternative hypothesis for 'different':
7. $$H_a: p_1 \neq p_2 \quad \text{or} \quad H_a: p_1 - p_2 \neq 0$$

> **Exam tip:** You will lose the full hypotheses point on the AP rubric if you use sample statistics $\hat{p}_1, \hat{p}_2$ instead of population parameters $p_1, p_2$.

## Step 2: Verify Conditions for Inference

All three conditions must be checked and explicitly justified to earn full points on the AP exam, even if they are implied in the problem prompt.

- **Random**: Data comes from two independent random samples, or groups are assigned randomly in a controlled experiment
- **Independence**: Observations within each group are independent, and each sample is no larger than 10% of its respective population
- **Normality**: The sampling distribution of $\hat{p}_1 - \hat{p}_2$ is approximately normal, verified by pooled successes and pooled failures all ≥ 10

> **warning**
>
> Do NOT use unpooled counts to check the normality condition for a two-proportion z-test. Use the pooled total successes and pooled total failures instead.

**Worked example:** For a sample of 200 urban teens with 172 smartphone owners, and 180 rural teens with 135 smartphone owners, verify the normality condition for the test.

1. First calculate pooled total successes: 172 + 135 = 307
2. Calculate pooled total failures: (200 - 172) + (180 - 135) = 28 + 45 = 73
3. Check that all values are greater than 10: 307 ≥ 10, 73 ≥ 10. The normality condition is satisfied.

**Check your understanding**

Which of the following scenarios violates the normality condition for a two-proportion z-test?

1. Sample 1 has 8 successes out of 50, sample 2 has 7 successes out of 50

   *Why:* Pooled successes = 15, pooled failures = 85, both ≥ 10, condition holds.

2. Sample 1 has 3 successes out of 40, sample 2 has 4 successes out of 40

   *Why:* Pooled successes = 7 < 10, normality condition fails.

3. Sample 1 has 12 successes out of 30, sample 2 has 11 successes out of 30

   *Why:* Pooled successes = 23, pooled failures = 37, both ≥ 10, condition holds.

## Step 3: Calculate Test Statistic and P-Value

Once conditions are verified, you calculate the pooled standard error, z test statistic, and corresponding p-value based on your pre-defined alternative hypothesis.

$$z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}_{pooled}(1-\hat{p}_{pooled}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}}$$

**Pooled Sample Proportion** — Combined proportion of successes across both samples, used to estimate the common population proportion assumed under the null hypothesis

*Notation:* $\hat{p}_{pooled} = \frac{x_1 + x_2}{n_1 + n_2}$

**Worked example:** Using the urban/rural teen smartphone data: n1=200, x1=172, n2=180, x2=135, calculate the z test statistic.

1. Calculate individual sample proportions:
2. $$\hat{p}_1 = 172/200 = 0.86, \quad \hat{p}_2 = 135/180 = 0.75$$
3. Calculate pooled proportion:
4. $$\hat{p}_{pooled} = (172 + 135)/(200 + 180) = 307/380 \approx 0.8079$$
5. Compute standard error:
6. $$SE = \sqrt{0.8079 * 0.1921 * (1/200 + 1/180)} \approx 0.0393$$
7. Compute z statistic:
8. $$z = (0.86 - 0.75)/0.0393 \approx 2.799$$

> **tip**
>
> For a two-sided test with z=2.8, the p-value is 2 * P(Z > 2.8) ≈ 0.005, which is statistically significant at the standard α=0.05 level.

## Step 4: Draw a Conclusion in Context

Your conclusion must explicitly link the p-value to the significance level, reference the hypotheses, and state your result in the full context of the problem to earn all rubric points.

1. Compare your calculated p-value to the pre-stated significance level α (almost always 0.05 for AP exams)
2. State whether you reject or fail to reject the null hypothesis, referencing the p-value comparison
3. Write a final contextual conclusion that directly addresses the original research question

**Worked example:** Given p-value ≈ 0.005 for the two-sided smartphone ownership test, α=0.05, write the full AP-aligned conclusion.

1. Compare p-value to α: 0.005 < 0.05
2. Decision: Reject the null hypothesis
3. Contextual conclusion: There is convincing statistical evidence that the true proportion of teens who own smartphones is different between urban and rural populations.

## Common pitfalls

- **Wrong:** Using unpooled standard error for a two-proportion significance test
  - Why it fails: The null hypothesis assumes $p_1 = p_2$, so pooling is required to get the correct standard error, and AP graders will deduct full points for this mistake
  - Correct: Always use the pooled proportion to calculate standard error for two-proportion z-tests, never use individual sample proportions
- **Wrong:** Defining parameters using the collected sample groups instead of the full population
  - Why it fails: Hypotheses make claims about population values, not the data you collected, so this is a fundamental conceptual error
  - Correct: Explicitly state that $p_1$ and $p_2$ refer to the true, unknown proportions for the entire population of interest, not just your sample
- **Wrong:** Stating 'we accept the null hypothesis' when p-value > α
  - Why it fails: You do not have evidence to prove the null is true, only that you do not have enough evidence to reject it
  - Correct: Always use the phrase 'fail to reject the null hypothesis' for non-significant results
- **Wrong:** Checking the normality condition using individual sample counts instead of pooled counts
  - Why it fails: This can lead you to incorrectly think the normality condition fails when it actually passes, or vice versa
  - Correct: Sum successes across both groups, sum failures across both groups, confirm both totals are ≥ 10
- **Wrong:** Writing a conclusion that does not reference the context of the original problem
  - Why it fails: AP Stats rubrics require conclusions to be tied to the specific scenario, generic conclusions get zero points for that step
  - Correct: Name the populations and the specific success outcome in your final conclusion, do not use generic terms like 'the two proportions'

## Cheatsheet

| Test Step | Required Action for Full AP Points |
| --- | --- |
| 1. Hypotheses | Define $p_1, p_2$ in full context, write $H_0: p_1-p_2=0$ and correct one/two-sided $H_a$ |
| 2. Conditions | Explicitly check Random, Independence (10% condition), Normality (pooled successes/failures ≥10) |
| 3. Calculations | Compute pooled p, pooled SE, z statistic, one or two-tailed p-value |
| 4. Conclusion | Compare p-value to α, state reject/fail to reject $H_0$, write full contextual statement |

## What's next

Mastery of two-proportion z-tests is one of the highest-weighted skills on the AP Statistics exam, appearing in nearly every recent FRQ section focused on inference. This skill directly builds on your prior work with one-proportion tests and sampling distributions for difference in proportions, and it is the required prerequisite for learning chi-square tests for homogeneity later in Unit 6. You will see this scenario tested both as a full 4-step FRQ and as multiple choice questions that ask you to identify correct conditions, calculate p-values, or spot flawed conclusions. Make sure to practice writing full, context-rich conclusions to avoid losing easy rubric points on exam day.

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