University of California, Los Angeles
Department of Statistics
Statistics 100B
Instructor: Nicolas Christou
The Central Limit Theorem
Suppose that a sample of size
n
is selected from a population that has mean
μ
and standard
deviation
σ
. Let
X
1
,X
2
,
···
,X
n
be the
n
observations that are independent and identically
distributed (i.i.d.). Deﬁne now the sample mean and the total of these
n
observations as
follows:
¯
X
=
∑
n
i
=1
X
i
n
T
=
n
X
i
=1
X
i
The
central limit theorem
states that the sample mean
¯
X
follows approximately the normal
distribution with mean
μ
and standard deviation
σ
√
n
, where
μ
and
σ
are the mean and stan
dard deviation of the population from where the sample was selected. The sample size
n
has
to be large (usually
n
≥
30) if the population from where the sample is taken is nonnormal.
If the population follows the normal distribution then the sample size
n
can be either small
or large.
To summarize:
¯
X
∼
N
(
μ,
σ
√
n
).
To transform
¯
X
into
z
we use:
z
=
¯
x

μ
σ
√
n
Example: Let
X
be a random variable with
μ
= 10 and
σ
= 4. A sample of size 100 is taken
from this population. Find the probability that the sample mean of these 100 observations is
less than 9. We write
P
(
¯
X <
9) =
P
(
z <
9

10
4
√
100
) =
P
(
z <

2
.
5) = 0
.
0062 (from the standard
normal probabilities table).
Similarly the central limit theorem states that sum
T
follows approximately the normal
distribution,
T
∼
N
(
nμ,
√
nσ
), where
μ
and
σ
are the mean and standard deviation of the
population from where the sample was selected.
To transform
T
into
z
we use:
z
=
T

nμ
√
nσ
Example: Let
X
be a random variable with
μ
= 10 and
σ
= 4. A sample of size 100 is
taken from this population. Find the probability that the sum of these 100 observations is
less than 900. We write
P
(
T <
900) =
P
(
z <
900

100(10)
√
100(4)
) =
P
(
z <

2
.
5) = 0
.
0062 (from
the standard normal probabilities table).
Below you can ﬁnd some applications of the central limit theorem.
1
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View Full DocumentEXAMPLE 1
A large freight elevator can transport a maximum of 9800 pounds. Suppose a load of cargo containing 49 boxes must be
transported via the elevator. Experience has shown that the weight of boxes of this type of cargo follows a distribution with
mean
μ
= 205 pounds and standard deviation
σ
= 15 pounds. Based on this information, what is the probability that all 49
boxes can be safely loaded onto the freight elevator and transported?
EXAMPLE 2
From past experience, it is known that the number of tickets purchased by a student standing in line at the ticket window
for the football match of
UCLA
against
USC
follows a distribution that has mean
μ
= 2
.
4 and standard deviation
σ
= 2
.
0.
Suppose that few hours before the start of one of these matches there are 100 eager students standing in line to purchase tickets.
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 Winter '10
 Christou
 Central Limit Theorem, Normal Distribution, Standard Deviation, Variance, σ

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