Question

Why is a binomial distribution a reasonable approximation of a sampling distribution for a population proportion when the population is large relative to the sample size?

Answer 1

Ans:

When population is very large as compared to sample size,then we can consider it as if we are sampling with replacement,as n is very large,so it will not make much difference if replace or not.

As, the standard deviation of all sample proportions is directly related to the sample size, n as given below.

The standard deviation of all sample proportions(p-hat) is=sqrt(p(1-p)/n)

Since the sample size n appears in the denominator of the square root, the standard deviation does decrease as sample size increases. Finally, the shape of the distribution of p-hat will be approximately normal as long as the sample size n is large enough. The convention is to require both np and n(1 – p) to be at least 10.