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.
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:
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.
for an interactive demonstration of the normal approximation to the binomial.