Sample Proportion
Consider an infinite (or very large) population, where each observation has a probability p of being a success, and a probability (1-p) of being a failure. Let the set of independent and identically distributed random variables X1, X2, ..., Xn represent the observations from a sample of size n, where
Xi = 1 if the ith observation is a success
= 0 if the ith observation is a failure
Then the random variables X1, X2, ..., Xn are Bernoulli variables with parameter p, and the sum (X1 + X2 + ... + Xn) is a binomially distributed random variable with parameters p and n. Letting X = (X1 + X2 + ... + Xn), and utilizing the normal approximation to the binomial distribution,
NowÂ
Therefore
As n ® ¥ , the binomial random variable X approximates a normal random variable with m = np and s2 = np(1-p), while the random variable (X/n) tends towards a normal random variable with m = p and s2 = p(1-p) / n. The random variable (X/n) is also called the sample proportion, defined as the ratio of the number of successes in a sample to the sample size.
