Section 9.5 Summary

1. The test statistic used for the $Z$ test for the proportion is

   ${\displaystyle Z_{\text{data}}=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0·q_0}{n}}}}$

   where $\hat{p}$ is the observed sample proportion of successes, $p_0$ is the value of $p$ hypothesized in $H_0$, $q_0=1-p_0$, and $n$ is the sample size. $Z_{\text{data}}$ represents the number of standard deviations $\left({\sigma }_{\hat{p}}\right)$ the sample proportion $\hat{p}$ lies above or below the hypothesized proportion $p_0$. Extreme values of $\hat{p}$ will be associated with extreme values of $Z_{\text{data}}$. The $Z$ test for the proportion may be performed using either the $p$-value method or the critical-value method. For the critical-value method, we compare the values of $Z_{\text{data}}$ and $Z_{\text{crit}}$. If $Z_{\text{data}}$ falls in the critical region, we reject $H_0$.

2. For the $p$-value method, we reject $H_0$ if the $p\text{-value}\le \alpha$.
3. We can use a single $100\left(1-\alpha \right)\%$ confidence interval for $p$ to help us perform any number of two-tailed hypothesis tests about $p$ with level of significance $\alpha$.