Sometimes, we might like to compare two proportions.

• \(n_i\) is sample size for the \(i\)th group
• \(p_i\) the proportion for the \(i\)th group
• We will examine their difference: \(p_1 - p_2\).
• Similar to the tests we used for a single proportion.

1. Independence within and between groups
   • generally satisfied if the data are from random samples or a randomized experiment
2. We need \(n_1p_1 > 10\) and \(n_1(1-p_1)>10\) and \(n_2p_2 > 10\) and \(n_2(1-p_2)>10\)

If our conditions are satisfied, the standard error is \[\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}\] and we can calculate confidence intervals and perform hypothesis tests on \(p_1 - p_2\).

A \(100(1-\alpha)\%\) confidence interval for \(p_1-p_2\) is

\[(\hat{p_1} - \hat{p_2}) \pm z_{\alpha/2} \times \sqrt{\frac{\hat{p_1}(1-\hat{p_1})}{n_1} + \frac{\hat{p_2}(1-\hat{p_2})}{n_2}}\]

• We are interested in checking whether \(p_1 = p_2\)
• This is null hypothesis of \[H_0: p_1 - p_2 = 0\]
  • So the null value is zero.
• In this case, we use a pooled proportion to estimate \(p\) in the standard error.

\[\hat{p}_{\text{pooled}} = \frac{\text{total number of successes}}{\text{total number of cases}} = \frac{\hat{p_1}n_1 + \hat{p_2}n_2}{n_1 + n_2}\]

\[ \text{Standard Error} = \sqrt{\frac{\hat{p}_{\text{pooled}}(1-\hat{p}_{\text{pooled}})}{n_1} + \frac{\hat{p}_{\text{pooled}}(1-\hat{p}_{\text{pooled}})}{n_2}}\]

• The critical value is \(z_{\alpha/2}\).
• The test statistic is \[z = \frac{\hat{p_1}-\hat{p_2}}{\sqrt{\frac{\hat{p}_{\text{pooled}}(1-\hat{p}_{\text{pooled}})}{n_1} + \frac{\hat{p}_{\text{pooled}}(1-\hat{p}_{\text{pooled}})}{n_2}}}\]
• The p-value is \[2P(Z > |z|)\] where \(z\) is the test statistic.

1. State the null and alternative hypotheses.
2. Determine the significance level \(\alpha\). Check assumptions, \(n_1p_1 > 10\) and \(n_1(1-p_1)>10\) and \(n_2p_2 > 10\) and \(n_2(1-p_2)>10\).
3. Compute the value of the test statistic.
4. Determine the critical value or p-value.
5. For the critical value approach: If the test statistic is in the rejection region, reject the null hypothesis. For the p-value approach: If \(\text{p-value} < \alpha\), reject the null hypothesis. Otherwise, do not reject.
6. Interpret results.