###### Conditions for the sampling distribution of \(\hat{p}\) being nearly normal

The sampling distribution for \(\hat{p}\text{,}\) taken from a sample of size \(n\) from a population with a true proportion \(p\text{,}\) is nearly normal when

the sample observations are independent and

we expected to see at least 10 successes and 10 failures in our sample, i.e. \(np\geq10\) and \(n(1-p)\geq10\text{.}\) This is called the success-failure condition.

If these conditions are met, then the sampling distribution of \(\hat{p}\) is nearly normal with mean \(\mu_{\hat{p}}=p\) and standard deviation \(\sigma_{\hat{p}} = \sqrt{\frac{\ p(1-p)\ }{n}}\text{.}\)