Question

Suppose that you are sampling to estimate a proportion. When the sample size is large relative to the population size (say, above 20%), will the binomial distribution’s estimation of the sampling distribution’s variance generally be accurate, too big, or too small?

Answer 1

Binomial distribution’s estimation of the sampling distribution’s variance = np(1-p)

where n is the sample size and p is the sample proportion.

For large sample size relative to the population size, sampling without replacement (as is usually done in surveys and many other situations) from a finite population does not give independent Bernoulli trials. In fact, it follows a hypergeometric distribution.

Variance of hypergeometric distribution is (N - n) / (N - 1) np(1-p)

where N is the population size and n is the sample size.

Let say that n = 0.2N and N - 1 N, then

Variance of hypergeometric distribution is (N - 0.2N) / (N) np(1-p) = 0.8 np(1-p)

So, the binomial distribution’s estimation of the sampling distribution’s variance would not be accurate and will be smaller.