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Chapter 7: Introduction to
Sampling Distributions
Section 2: The Central Limit
Theorem
Theorem 7.1

Let x be a random variable with a normal
distribution whose mean is  and
standard deviation is  . Let x be the
sample mean corresponding to random
samples of size n taken from the x
distribution.
1. The x distribution is a normal distribution.
2. The mean of the x distribution is  .
3. The standard deviation of the x
distribution is  n .
To find the z value:
z=
x
 n
Example

Let x represent the length of a single
trout taken at random from the pond.
This is normal distribution with a mean of
10.2 inches and standard deviation of 1.4
inches.
a.) What is the probability that a single trout
taken at random from the pond is
between 8 and 12 inches long?
b.) What is the probability that the mean
length of five trout taken at random is
between 8 and 12 inches?
Central Limit Theorem
 If x possesses any distribution with mean 
and standard deviation , then the sample
mean x based on a random sample of size n
will approach a normal distribution when n is
sufficiently large (n > 30).

 Suppose we know that the x distribution
has a mean  = 30 and a standard
deviation  = 8, but we have no
information as to whether or not the x
distribution is normal. If we draw samples
of size 35 from the x distribution and x
represents the sample mean, what can
you say about the x distribution?
 Suppose x has a normal distribution with
mean  = 20 and  = 5. If we draw
random samples of size 10 from the x
distribution, and x represents the sample
mean, what can you say about the x
distribution?
 Suppose you did not know that x had a
normal distribution. Would you be justified
in saying that the x distribution is
approximately normal if the sample size
was n = 8?
1. Suppose we have no information as to
whether or not the x distribution is normal.
If we draw samples of n = 5 from the x
distribution and x represents the sample
mean, will the x distribution be
approximately normal?
Explain why or why not.
2. Suppose x has a normal distribution
with mean  = 12 and  = 3. If we draw
random samples of size 10 from the x
distribution, and x represents the sample
mean, what can you say about the x
distribution?
3. Suppose you do not know if x has a
normal distribution. If we draw random
samples of size 40, will the x distribution
be approximately normal?
Explain why or why not.