for a single proportion $p$

If we wish to test a $H_0:p=p_0$ with a z test or a chi test

• $\chi$ is more general than the z test
  • z compares one proportion to our hypothesized one
  • $\chi$ takes all proportions of our sample and compares it to what we would get if they all had the expected proportion we want ($H_0$)
    Or find a confidence interval for the Sample Proportion $p$

Confidence Intervals

• $\hat{p}\pm z_{\alpha /2}SE(\hat{p})$ are the endpoints of a $(1-\alpha )100\mathrm{\%}$ confidence interval
• to test $H_0:p=p_0$
  • $Z=\frac{\hat{p}-p_0}{S{E}_0(\hat{p})}$
  • $S{E}_0(\hat{p})=\sqrt{\frac{p_0(1-p_0)}{n}}$
    • SE is an estimate of the standard deviation

for two proportions $p$

If we wish to

• test null hypothesis $H_0:p_1=p_2$ with a z test or a chi test
• find a confidence interval for $p_1-p_2$
• find relative risk $\frac{p_1}{p_2}$
• find odds ratio$\frac{p_1/(1-p_1)}{p_2/(1-p_2)}$

Confidence Intervals

• $\widehat{p_1}-\widehat{p_2}\pm z_{\alpha /2}SE(\widehat{p_1}-\widehat{p_2})$ are the endpoints of a $(1-\alpha )100\mathrm{\%}$ confidence interval
• to test $H_0:p=p_0$
  • $Z=\frac{\hat{p}-p_0}{S{E}_0(\hat{p})}$
  • $S{E}_0(\hat{p})=\sqrt{\frac{p_0(1-p_0)}{n}}$
• 95% confidence interval that the true proportion is contained in $\widehat{p_1}-\widehat{p_2}\pm z_{\alpha /2}SE(\widehat{p_1}-\widehat{p_2})$