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Chapter 18 Sampling Distribution Models
Recall the binomial experiment:
1. There are a fixed number of trials, n.
2. The n trials are all independent.
3. Each trial has one of two outcomes: “success” or “failure”.
4. The probability of success, p, is the same for each trial.
Under just such conditions, the number of successes, X, follows a binomial distribution, X~bin(n,
p).
Also recall that if n is sufficiently large, we can approximate the binomial distribution by the
normal distribution. In this case X~approximately normal(np,
npq ).
If we divide the number of successes, X, by the total number of trials, n, then we get the
proportion of successes in the sample designated as
X
pˆ 
n
and (following from the
distribution above)
p ~approximately normal(p,
pq
).
n
The sampling distribution model for the proportion,
standard deviation
pq
n
p̂ , is a normal model with mean p and
when
1. The sample proportions are independent of one another. (If sampling is done without
replacement, then the sample should be no larger than 10% of the population.)
2. The sample size, n, is large. (Make sure the expected number of successes and failures is at
least 10.)
Central Limit Theorem
Draw an SRS of size n from any population with mean  and finite standard deviation . When n
is large, the sampling distribution of the sample mean
Ie.
x is approximately N(,
x is approximately normal.

n
)
This is a very important concept in statistics. It implies that no matter what the distribution of the
population looks like (skewed left, uniform, clustered, whatever), the possibilities for the sample
mean must
1. follow a normal curve with
2. the same mean as the mean of the population and

3. a standard deviation of
n
when the sample size is large.
Whenever we estimate the standard deviation of a sampling distribution we call it a standard
error. (We do this a lot. It will be impossible to know the standard deviation of the population,,
or the percentage of successes in the population, p, so we estimate them.)
Standard error of a sample proportion
SE p 
pq

.
n
pq

n
is called the standard error of
p . That is,
Standard error of a sample mean
s
.
SEx 
n
s
is called the standard error of x . That is,
n