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  • https://stats.libretexts.org/Workbench/Introduction_to_Statistical_Methods_(Yuba_College)/10%3A_Hypothesis_Tests_and_Confidence_Intervals_for_Two_Populations/10.03%3A_Two_Proportion_Z-Test_and_Confidence_Interval
    \(\left(\hat{p}_{1}-\hat{p}_{2}\right)-z_{\alpha / 2} \sqrt{\left(\frac{\hat{p}_{1} \hat{q}_{1}}{n_{1}}+\frac{\hat{p}_{2} \hat{q}_{2}}{n_{2}}\right)}<p_{1}-p_{2}<\left(\hat{p}_{1}-\hat{p}_{2}\right)+z...\left(\hat{p}_{1}-\hat{p}_{2}\right)-z_{\alpha / 2} \sqrt{\left(\frac{\hat{p}_{1} \hat{q}_{1}}{n_{1}}+\frac{\hat{p}_{2} \hat{q}_{2}}{n_{2}}\right)}<p_{1}-p_{2}<\left(\hat{p}_{1}-\hat{p}_{2}\right)+z_{\alpha / 2} \sqrt{\left(\frac{\hat{p}_{1} \hat{q}_{1}}{n_{1}}+\frac{\hat{p}_{2} \hat{q}_{2}}{n_{2}}\right)}
  • https://stats.libretexts.org/Workbench/Statistics_for_Behavioral_Science_Majors/08%3A_Hypothesis_Tests_and_Confidence_Intervals_for_Two_Populations/8.01%3A_Two_Proportion_Z-Test_and_Confidence_Interval
    Where \(\hat{p}=\frac{\left(x_{1}+x_{2}\right)}{\left(n_{1}+n_{2}\right)}=\frac{\left(\hat{p}_{1} \cdot n_{1}+\hat{p}_{2} \cdot n_{2}\right)}{\left(n_{1}+n_{2}\right)}, \quad \hat{q}=1-\hat{p}, \quad ...Where \hat{p}=\frac{\left(x_{1}+x_{2}\right)}{\left(n_{1}+n_{2}\right)}=\frac{\left(\hat{p}_{1} \cdot n_{1}+\hat{p}_{2} \cdot n_{2}\right)}{\left(n_{1}+n_{2}\right)}, \quad \hat{q}=1-\hat{p}, \quad \hat{p}_{1}=\frac{x_{1}}{n_{1}}, \hat{p}_{2}=\frac{x_{2}}{n_{2}}. Now substitute the numbers into the interval estimate: \left(\hat{p}_{1}-\hat{p}_{2}\right) \pm z_{\frac{\alpha}{2}} \sqrt{\left(\frac{\hat{p}_{1} \hat{q}_{1}}{n_{1}}+\frac{\hat{p}_{2} \hat{q}_{2}}{n_{2}}\right)}

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