## proportional confidence intervals

We will measure a single stastistic from a large population, and call it the point estimate. This is usually denoted as \(\hat{p}\).

Given a proportion \(\hat{p}\) (“95% of sample), the range which would possibly contain it as part of its \(2\sigma\) range is the \(95\%\) confidence interval.

Therefore, given a \(\hat{p}\) the plausible interval for its confidence is:

\begin{equation} \hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \end{equation}

where, \(n\) is the sample size, \(\hat{p}\) is the point estimate, and \(z*=1.96\) is the critical value, the z-score denoting \(95\%\) confidence (or any other desired confidence level).

## conditions for proportional confidence interval

There are the conditions that make a proportional confidence interval work

- distribution is normal
- \(n\hat{p}\) and \(n(1-\hat{p})\) are both \(>10\)
- we are sampling with replacement, or otherwise sampling \(<10\%\) of population (otherwise, we need to apply a finite population correction

## value confidence intervals

The expression is:

\begin{equation} \bar{x} \pm t^* \frac{s}{\sqrt{n}} \end{equation}

where \(t*\) is the \(t\) score of the desired power level with the correct degrees of freedom; \(s\) the sample standard deviation, \(n\) the sample size, and \(\har{x}\) the mean.