Topic 6.11 Notes – Carrying Out a Test for the Difference of Two Population Proportions

1. What a Two-Proportion z-Test Is

A two-proportion z-test compares two independent population proportions to see if there’s convincing evidence they are different.

• Parameter of interest:

  $$p_1 - p_2$$ where
  $p_1$ = true proportion in Population 1
  $p_2$ = true proportion in Population 2

The null hypothesis usually says there is no difference:

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

This test uses a normal model to approximate the sampling distribution of $\hat{p}_1 - \hat{p}_2$, assuming the null is true.

You use this when:

• The response variable is categorical (success/failure).
• You have two independent groups.
• You want to test a difference, not estimate it (that would be a confidence interval).

2. Hypotheses and Conditions

Writing Hypotheses in Context

Always define parameters in words.

Example structure:

• $p_1$ = true proportion of registered voters in City A who support the policy
• $p_2$ = true proportion of registered voters in City B who support the policy

Then:

$$H_0: p_1 - p_2 = 0$$

$$H_a: p_1 - p_2 \neq 0 \quad \text{(or > 0, or < 0 depending on the claim)}$$

Be clear which group is 1 and which is 2. Order matters for interpretation.

Conditions for Using the Test

You need all of these:

1. Random
   • Random sample or random assignment is stated.
2. Independent Groups
   • The two groups are separate.
   • Not matched pairs.
3. Independence Within Groups
   • 10% condition: each sample is less than 10% of its population (if sampling without replacement).
4. Large Counts Condition

Because we assume $p_1 = p_2$ under $H_0$, we use a pooled proportion when checking counts.

3. Calculating the Test Statistic

Here’s the structure. Everything comes from the general test statistic formula:

$$\text{test statistic} = \frac{\text{statistic} - \text{parameter under } H_0}{\text{standard error}}$$

Step 1: Sample Proportions

$$\hat{p}_1 = \frac{x_1}{n_1}, \quad \hat{p}_2 = \frac{x_2}{n_2}$$

Step 2: Pooled Proportion

Under $H_0$, we assume the proportions are equal, so we combine successes:

$$\hat{p}_c = \frac{x_1 + x_2}{n_1 + n_2}$$

This estimates the common population proportion.

Step 3: Standard Error (Pooled)

$$SE = \sqrt{\hat{p}_c(1-\hat{p}_c)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}$$

Notice this uses pooled data. That’s specific to hypothesis tests.

Step 4: z Test Statistic

$$z = \frac{\hat{p}_1 - \hat{p}_2 - 0}{SE}$$

The “− 0” comes from $H_0: p_1 - p_2 = 0$.

A large positive or negative $z$ means the observed difference is far from what we’d expect if the proportions were equal.

On your calculator, this is 2-PropZTest. You must record:

• the z-value
• the p-value

On an FRQ, forgetting to report z often costs a point.

4. Interpreting the P-Value and Making a Decision

What the P-Value Means

In context, it should sound like this:

The probability of observing a difference in sample proportions at least as extreme as the one found, assuming the true population proportions are equal.

That “assuming the true proportions are equal” part is required for full credit.

Formal Decision Rule

Compare p-value to $\alpha$ (often 0.05):

• If $p \le \alpha$ → Reject $H_0$
• If $p > \alpha$ → Fail to reject $H_0$

Never say “accept $H_0$.”

Writing a Conclusion in Context

Your conclusion must:

1. State the decision.
2. Answer the research question.
3. Refer to population proportions, not sample data.

Example wording:

Since the p-value is less than 0.05, we reject $H_0$. There is convincing statistical evidence that the true proportion of customers who prefer Brand A differs from the true proportion who prefer Brand B.

Notice it says true proportion and names the populations.

5. Common Mistakes

• Using the unpooled standard error for a test. Pooling is required under $H_0$.
• Interpreting the p-value as the probability that $H_0$ is true.
• Forgetting to define $p_1$ and $p_2$.
• Concluding about the sample instead of the populations.
• Using this test for matched pairs data.

A quick mental check before submitting:
Two groups? Categorical outcome? Pooled SE? z and p-value reported? Conclusion about population proportions?