Sampling Proportions From Finite Populations

A population of size $N$ contains $K$ members with a particular attribute. The population proportion is

$$p = \dfrac{K}{N}.$$

We draw a random sample of $n$ items and let $X$ be the count of sampled items having the attribute. The sample proportion is the random variable

$$\hat{p} = \dfrac{X}{n}.$$

When the sample is drawn with replacement, each draw is an independent Bernoulli($p$) trial, so $X$ is a sum of $n$ independent Bernoullis:

$$X \sim \mathrm{Binomial}(n, p), \qquad E[X] = np.$$

Dividing by the constant $n$,

$$E[\hat{p}] = E\!\left[\dfrac{X}{n}\right] = \dfrac{E[X]}{n} = \dfrac{np}{n} = p.$$

The sample proportion is unbiased for $p$: on average, $\hat{p}$ equals the population proportion.