AP Stats Home
Carrying Out a Test for the Difference of Two Population Proportions
This note page details the practical steps involved in executing a hypothesis test for comparing two population proportions, often following the setup discussed in Setting Up a Test for the Difference of Two Population Proportions. The goal is to determine if there’s a statistically significant difference between two population proportions ( $ p_1 $ and $ p_2 $ ) based on sample data.
1. State Hypotheses
Before carrying out the test, the null and alternative hypotheses must be clearly stated. This was covered in Setting Up a Test for the Difference of Two Population Proportions.
- Null Hypothesis ( $ H_0 $ ): $ p_1 = p_2 $ (or $ p_1 - p_2 = 0 $ ). This assumes no difference between the population proportions.
- Alternative Hypothesis ( $ H_a $ ):
- $ p_1 \neq p_2 $ (Two-sided test)
- $ p_1 < p_2 $ (One-sided, left-tailed test)
- $ p_1 > p_2 $ (One-sided, right-tailed test)
2. Check Conditions
Before performing any calculations, it is crucial to verify that the necessary conditions for inference are met. These ensure the validity of the sampling distribution and the resulting p-value.
| Condition Name | Description |
|---|---|
| Random | The data should come from two independent random samples or two groups in a randomized experiment. Without random assignment/sampling, generalization to the population is problematic. |
| Independent | The 10% Condition: When sampling without replacement, both sample sizes ( $ n_1 $ and $ n_2 $ ) must be less than 10% of their respective population sizes ( $ N_1 $ and $ N_2 $ ). This ensures the independence of observations within each sample. |
| Normal (Large Counts) | The sampling distribution of $ \hat{p}_1 - \hat{p}_2 $ is approximately normal if the number of successes and failures in both samples are sufficiently large. Specifically: |
| $ n_1 \hat{p}_1 \ge 10 $ , $ n_1 (1 - \hat{p}_1) \ge 10 $ | |
| $ n_2 \hat{p}_2 \ge 10 $ , $ n_2 (1 - \hat{p}_2) \ge 10 $ |
Note: For the Large Counts condition for a hypothesis test, we use the pooled proportion ( $ \hat{p}_c $ ) to calculate the expected counts, assuming $ H_0 $ is true. This means we use $ \hat{p}_c $ instead of $ \hat{p}_1 $ and $ \hat{p}_2 $ when checking the Normal condition.
3. Calculate the Test Statistic (z-score)
If the conditions are met, we can calculate the test statistic. For two population proportions, we use a z-statistic. Since we assume $ H_0: p_1 = p_2 $ is true, we pool the sample data to estimate the common population proportion.
Pooled Proportion ( $ \hat{p}_c $ )
The pooled proportion is calculated as: $$ \hat{p}_c = \frac{X_1 + X_2}{n_1 + n_2} $$ where $ X_1 $ and $ X_2 $ are the number of successes in sample 1 and sample 2, respectively, and $ n_1 $ and $ n_2 $ are the respective sample sizes.
Standard Error
The standard error for the difference in sample proportions, using the pooled proportion under $ H_0 $ , is: $$ SE_{\hat{p}_1 - \hat{p}_2} = \sqrt{\frac{\hat{p}_c(1 - \hat{p}_c)}{n_1} + \frac{\hat{p}_c(1 - \hat{p}_c)}{n_2}} $$
z-Test Statistic
The z-test statistic is then calculated as: $$ z = \frac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sqrt{\frac{\hat{p}_c(1 - \hat{p}_c)}{n_1} + \frac{\hat{p}_c(1 - \hat{p}_c)}{n_2}}} $$ Under $ H_0 $ , we assume $ p_1 - p_2 = 0 $ , simplifying the formula to: $$ z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\frac{\hat{p}_c(1 - \hat{p}_c)}{n_1} + \frac{\hat{p}_c(1 - \hat{p}_c)}{n_2}}} $$
4. Calculate the p-value
The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true. This probability is found using the standard normal distribution (Z-distribution). Interpreting p-Values provides more details on its meaning.
- For $ H_a: p_1 \neq p_2 $ (Two-sided): $ P(|Z| \ge |z|) = 2 \cdot P(Z \ge |z|) $
- For $ H_a: p_1 < p_2 $ (Left-sided): $ P(Z \le z) $
- For $ H_a: p_1 > p_2 $ (Right-sided): $ P(Z \ge z) $
You can use a z-table or statistical software/calculator to find this probability.
5. Make a Decision and Conclusion
Finally, compare the p-value to the chosen significance level ( $ \alpha $ ) (e.g., $ \alpha = 0.05 $ ).
- If p-value $ \le \alpha $ : Reject the null hypothesis ( $ H_0 $ ). There is convincing evidence to support the alternative hypothesis ( $ H_a $ ).
- If p-value $ > \alpha $ : Fail to reject the null hypothesis ( $ H_0 $ ). There is not convincing evidence to support the alternative hypothesis ( $ H_a $ ).
State your conclusion in the context of the problem, referring back to the original claim about the two population proportions. Remember to mention the significance level used. Concluding a Test for a Population Proportion offers general guidance on forming conclusions. Be mindful of Potential Errors When Performing Tests.