Sampling Distribution of a Proportion (1 of 4)

Assume that 0.80 of all third grade students can pass a test of physical fitness. A random sample of 20 students is chosen: 13 passed and 7 failed. The parameter π is used to designate the proportion of subjects in the population that pass (.80 in this case) and the statistic p is used to designate the proportion who pass in a sample (13/20 = .65 in this case). The sample size (N) in this example is 20. If repeated samples of size N where taken from the population and the proportion passing (p) were determined for each sample, a distribution of values of p would be formed. If the sampling went on forever, the distribution would be the sampling distribution of a proportion. The sampling distribution of a proportion is equal to the binomial distribution. The mean and standard deviation of the binomial distribution are:

μ = π

and .

For the present example, N = 20, π = 0.80, the mean of the sampling distribution of p (μ) is .8 and the standard error of p (σp) is 0.089. The shape of the binomial distribution depends on both N and π. With large values of N and values of π in the neighborhood of .5, the sampling distribution is very close to a normal distribution.

Click here for an interactive demonstration of the normal approximation to the binomial.