The central limit theorem states
that the sampling distribution of the
mean approaches a normal
distribution as N, the sample size, increases. A sample proportion can be
thought of as a mean in the followingway: For each trial, give a "success"
a score of 1 and a "failure" a score of 0. The mean score will be
the proportion of successes. Therefore, the sample proportion is a sample mean.
The central limit theorem implies that the larger the sample size, the closer
the distribution of p will be to a normal distribution.