Topic 3.12 Notes – Setting Up a Test for the Difference Between Two Population Proportions

What a Test for Two Population Proportions Is

This test compares two populations or two independent treatment groups on a binary categorical outcome.

• Binary categorical means there are two outcomes, like support/do not support, passed/did not pass, recovered/did not recover.
• Success means the outcome you care about. It does not have to be good. If the study is about getting sick, then “success” could be “got sick.”

The parameter is

$$p_1 - p_2$$

where $p_1$ and $p_2$ are the true population proportions of success.

The sample statistics are

$$\hat p_1=\frac{x_1}{n_1} \qquad\text{and}\qquad \hat p_2=\frac{x_2}{n_2}$$

where $x$ is the number of successes and $n$ is the sample size.

Use this when:

• the response variable has two categories
• the groups are independent
• the question asks whether one proportion is bigger, smaller, or different

Do not use it for:

• comparing means of quantitative data
• paired or matched data

Be careful with order. If you define the parameter as $p_A - p_B$, then a positive result means A’s proportion is larger. Reverse the order, and the sign and one-sided hypothesis direction reverse too.

Hypotheses for $p_1 - p_2$

The null says no difference:

$$H_0: p_1=p_2 \qquad\text{or}\qquad H_0: p_1-p_2=0$$

The alternative depends on the wording:

• “higher,” “greater,” “more likely”
  $\rightarrow H_a: p_1>p_2$
• “lower,” “less”
  $\rightarrow H_a: p_1<p_2$
• “different,” “not the same,” “has an effect”
  $\rightarrow H_a: p_1\ne p_2$

A common trap is choosing the direction after seeing the sample results. On the AP exam, the direction comes from the research question, not from which sample proportion happened to be bigger.

Also, hypotheses are always about population proportions $p_1, p_2$, never sample proportions $\hat p_1, \hat p_2$.

Why Pooling Happens Under the Null

If $H_0$ says $p_1=p_2$, then both groups are assumed to share one common proportion. So we estimate that common value by pooling the samples:

$$\hat p_c=\frac{x_1+x_2}{n_1+n_2} =\frac{n_1\hat p_1+n_2\hat p_2}{n_1+n_2}$$

This is a weighted average, so the larger sample has more influence.

A two-way display like this is a good way to picture the setup. Each group has its own observed counts, but under the null you combine the successes and totals to estimate one shared proportion.

Two-way table of group counts and outcomes. Group 1 has $\hat p_1 = 0.60$ and Group 2 has $\hat p_2 = 0.40$; pooling uses the Success total over the grand total, $\hat p_c = 84/170 \approx 0.494$, which falls between them.

You use $\hat p_c$ here because the null assumes equal proportions. That pooled proportion is used for the large-counts check in this test. It is not used to describe each group’s observed proportion, and it is not used in a confidence interval for $p_1-p_2$.

Conditions for a Valid Two-Sample $z$-Test

All three conditions must be checked in context.

Randomization and independent groups

You need either:

• two independent random samples, or
• a randomized experiment with independent treatment groups

Random sampling lets you generalize to a population. Random assignment lets you talk about cause and effect. Big sample sizes do not fix bad sampling.

10% condition

This matters for random samples taken without replacement from finite populations:

$$n_1 \le 0.10N_1 \qquad\text{and}\qquad n_2 \le 0.10N_2$$

Check it separately for each group. You do not need this for a randomized experiment.

Large counts condition

Because the null assumes one common proportion, use the pooled proportion:

• $n_1\hat p_c \ge 10$
• $n_1(1-\hat p_c) \ge 10$
• $n_2\hat p_c \ge 10$
• $n_2(1-\hat p_c) \ge 10$

These are expected successes and failures in each group under the null. This is what justifies the normal approximation.

How to Write a Complete Setup and What to Avoid

A good setup sounds like this:

1. Define the parameters in context.
   $p_A$ = true proportion of all voters in District A who support the measure.
   $p_B$ = true proportion of all voters in District B who support the measure.
2. State hypotheses with the correct direction.
3. Name the procedure.
   Two-sample $z$-test for a difference between two population proportions
4. Verify randomization, 10% condition if needed, and large counts.

Common mistakes:

• writing hypotheses with $\hat p$ instead of $p$
• mixing up subtraction order and inequality direction
• forgetting groups must be independent
• checking large counts with separate $\hat p_1$ and $\hat p_2$ instead of $\hat p_c$
• mixing up the 10% condition with the large-counts condition
• trying to calculate the test statistic here. This topic stops at setup.