Topic 6.10 Notes – Setting Up a Test for the Difference of Two Population Proportion

What a two‑proportion z‑test is for

We use this test when we’re comparing one categorical variable (success/failure) across two independent groups.

Example situations:

• Do a higher proportion of students at School A pass the AP exam than at School B?
• Is the proportion of voters who support a policy different in two states?

The parameter of interest is:

$$p_1 - p_2$$

where:

• $p_1$ = true population proportion of “success” in group 1
• $p_2$ = true population proportion of “success” in group 2

This is the hypothesis‑test version of the two‑proportion confidence interval. Here, we are testing whether the difference is actually 0.

Writing hypotheses correctly

Before writing anything symbolic, define your parameters in context.

For example:

• $p_1$ = the true proportion of all adults in City A who support the new tax.
• $p_2$ = the true proportion of all adults in City B who support the new tax.

Null hypothesis

For a difference in proportions, the null always represents no difference:

$$H_0: p_1 = p_2$$

or equivalently,

$$H_0: p_1 - p_2 = 0$$

The value under the null is always 0. That’s built into this type of test.

Alternative hypothesis

This depends on the research question.

• Two‑sided: $$H_a: p_1 \ne p_2$$
• One‑sided: $$H_a: p_1 > p_2 \quad \text{or} \quad H_a: p_1 < p_2$$

The wording of the question determines direction. Don’t choose the direction based on what the sample data suggest.

Common mistake on quizzes: writing hypotheses using $\hat{p}_1$ and $\hat{p}_2$. Hypotheses are always about population proportions.

Choosing the correct test

If you are:

• Comparing two proportions
• From two independent groups
• For a single categorical variable

Then the correct method is a:

Two‑sample z‑test for a difference between two population proportions
(On your calculator: 2‑PropZTest.)

This is different from:

• A one‑proportion z‑test (only one group)
• A chi‑square test (used for larger tables or independence)
• A t‑test (used for means, not proportions)

Conditions for inference

We are using a normal (z) model, so we must justify it.

1. Random

• Two independent random samples
• or
• Random assignment in an experiment

If random sampling was used, you can generalize to the population.
If random assignment was used, you can talk about cause and effect.

2. Independence

Two parts:

• The two samples are independent of each other.
• 10% condition (if sampling without replacement):

$$n_1 \le 0.1N_1 \quad \text{and} \quad n_2 \le 0.1N_2$$

Random assignment automatically makes the two groups independent.

3. Normal (Large Counts Condition with pooling)

Here’s the key idea that students forget:

For a two‑proportion test, we assume $H_0$ is true. That means we assume $p_1 = p_2$. So we combine the samples to estimate a common proportion.

Step 1: Compute the pooled proportion

$$\hat{p}_c = \frac{n_1\hat{p}_1 + n_2\hat{p}_2}{n_1 + n_2}$$

This is the combined proportion of successes from both samples.

Step 2: Check large counts using $\hat{p}_c$

All four must be at least 10:

• $n_1 \hat{p}_c$
• $n_1 (1 - \hat{p}_c)$
• $n_2 \hat{p}_c$
• $n_2 (1 - \hat{p}_c)$

If these are satisfied, the sampling distribution of $\hat{p}_1 - \hat{p}_2$ is approximately normal, so using a z‑test is reasonable.

Important distinction:

• Tests → pooled proportion
• Confidence intervals → not pooled

That difference shows up often in AP questions.

What you should be able to do quickly

Given a scenario, you should be able to:

• Identify that the response variable is categorical.
• State and define $p_1$ and $p_2$ clearly in context.
• Write $H_0: p_1 - p_2 = 0$.
• Choose the correct alternative based on wording.
• Name the correct procedure: two‑sample z‑test for a difference in population proportions.
• Check random, independence, and large counts using the pooled proportion.

On free response, missing the pooled check or forgetting to define parameters is an easy way to lose points.