Compute the mean and standard deviation of the sampling distribution
of p

State the relationship between the sampling distribution of p and the
normal distribution

The distribution of p is closely related to the
binomial distribution. The binomial distribution is the distribution
of the total number of successes (favoring Candidate A, for example)
whereas the distribution of p is the distribution of the mean
number of successes. The mean, of course, is the total divided
by the sample size, N.

The binomial distribution has a mean of

μ = Nπ

Dividing by N to adjust for the fact that we
are now dealing with means instead of totals, we find the mean
of the sampling distribution of p is

μp = π

The standard deviation of the binomial distribution
is:

Dividing by N to get the standard error of p,
we find that:

The sampling distribution of p is a discrete rather
than a continuous distribution.

The sampling distribution of p is approximately
normally distributed if N is fairly large and π is not close
to 0 or 1. A rule of thumb is that the approximation is good if
both N π and N(1 - π) are both greater than 10.