Section 8.3 Summary

1. The sample proportion of successes

   $\hat{p}={\displaystyle \frac{x}{n}}={\displaystyle \frac{\text{number of successes}}{\text{sample size}}}$

   is a point estimate of the population proportion $p$.

2. The $100(1-\alpha )\%$ confidence interval for the population proportion $p$ is given by

   $\hat{p}\pm Z_{\alpha /2}\sqrt{{\displaystyle \frac{\hat{p}\cdot \hat{q}}{n}}}$

   where $\hat{p}$ is the sample proportion of successes $\hat{q}=1-\hat{p}$, $n$ is the sample size, and $Z_{\alpha /2}$ depends on the confidence level. The $Z$ interval for $p$ may be constructed only if both the following conditions apply: $n\cdot \hat{p}\ge 5$ and $n\cdot \hat{q}\ge 5$ (alternatively, $x\ge 5$ and $n-x\ge 5$).

3. Note that the confidence interval for $p$ takes on the form

   $\text{point estimate}\pm \text{margin of error}$

   where $\hat{p}$ is the point estimate of $p$ and $E=Z_{\alpha /2}\sqrt{\hat{p}\cdot \hat{q}/n}$ is the margin of error.

4. Suppose we want to estimate the population proportion $p$ to within a margin of error $E$ with confidence $100(1-\alpha )\%$. If $\hat{p}$ is known, then the required sample size needed is given by

   $n=\hat{p}\cdot \hat{q}{\left({\displaystyle \frac{Z_{\alpha /2}}{E}}\right)}^2$

   If $\hat{p}$ is not known, then the required sample size needed is given by

   $n={\left[{\displaystyle \frac{0.5\cdot Z_{\alpha /2}}{E}}\right]}^2$