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ELEMENTARY
STATISTICS
Section 5-5
The Central Limit Theorem
EIGHTH
Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman EDITION
MARIO F. TRIOLA
1
Definition
Sampling Distribution of the mean
is a probability distribution of
sample means, with all samples
having the same sample size n.
Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
2
Central Limit Theorem
Given:
1. The random variable x has a distribution (which
may or may not be normal) with mean µ and
standard deviation .
2. Samples all of the same size n are randomly
selected from the population of x values.
Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
3
Central Limit Theorem
Conclusions:
1. The distribution of sample x will, as the
sample size increases, approach a normal
distribution.
2. The mean of the sample means µx will be the
population mean µ.
3. The standard deviation of the sample means
n
x will approach 
Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
4
Practical Rules Commonly Used:
1. For samples of size n larger than 30, the distribution of
the sample means can be approximated reasonably well
by a normal distribution. The approximation gets better
as the sample size n becomes larger.
2. If the original population is itself normally distributed,
then the sample means will be normally distributed for
any sample size n (not just the values of n larger than 30).
Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
5
Notation
the mean of the sample means
µx = µ
the standard deviation of sample mean

x = n
(often called standard error of the mean)
Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
6
Distribution of 200 digits from
Social Security Numbers
Frequency
(Last 4 digits from 50 students)
20
10
0
0
1
2
3
4
5
6
7
8
9
Distribution of 200 digits
Figure 5-19
Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
7
Table 5-2
x
SSN digits
1
5
9
5
9
4
7
9
5
7
2
6
2
2
5
0
2
7
8
5
8
3
8
1
3
2
7
1
3
3
7
7
3
4
4
4
5
1
3
6
6
3
8
2
3
6
1
5
3
4
6
7
3
7
3
3
8
3
7
6
4
6
8
5
5
2
6
4
9
4.75
4.25
8.25
3.25
5.00
3.50
5.25
4.75
5.00
2
6
1
9
5
7
8
6
4
0
7
4.00
5.25
4.25
4.50
4.75
3.75
5.25
3.75
4.50
6.00
Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
8
Frequency
Distribution of 50 Sample Means
for 50 Students
15
10
5
0
0
1
2
3
4
5
6
7
8
9
Figure 5-20
Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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As the sample size increases,
the sampling distribution of
sample means approaches a
normal distribution.
Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
10
Example: Given the population of women has normally
distributed weights with a mean of 143 lb and a standard
deviation of 29 lb,
a.) if one woman is randomly selected, find the probability
that her weight is greater than 150 lb.
b.) if 36 different women are randomly selected, find the
probability that their mean weight is greater than 150 lb.
Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
11
Example: Given the population of women has normally
distributed weights with a mean of 143 lb and a standard
deviation of 29 lb,
a.) if one woman is randomly selected, find the probability
that her weight is greater than 150 lb.
z = 150-143 = 0.24
29
P(x>150) = 0.405
Normalcdf(0.24,9999) = 0.4052
= 29
0.0948
 = 143
0
150
0.24
Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
12
Example: Given the population of women has normally
distributed weights with a mean of 143 lb and a standard
deviation of 29 lb,
b.) if 36 different women are randomly selected, find the
probability that their mean weight is greater than 150 lb.
Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
13
Example: Given the population of women has normally
distributed weights with a mean of 143 lb and a standard
deviation of 29 lb,
b.) if 36 different women are randomly selected, find the
probability that their mean weight is greater than 150 lb.
P(x>150) =0.0735
z = 150-143 = 1.45
29
36
Normalcdf(1.45, 9999) = 0.0735
x = 4.83333
0.4265
x = 143
0
150
1.45
Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
14
Example: Given the population of women has normally
distributed weights with a mean of 143 lb and a standard
deviation of 29 lb,
Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
15
Example: Given the population of women has normally
distributed weights with a mean of 143 lb and a standard
deviation of 29 lb,
a.) if one woman is randomly selected, find the
probability that her weight is greater than 150 lb.
P(x > 150) = 0.4052
Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
16
Example: Given the population of women has normally
distributed weights with a mean of 143 lb and a standard
deviation of 29 lb,
a.) if one woman is randomly selected, find the
probability that her weight is greater than 150 lb.
P(x > 150) = 0.4052
b.) if 36 different women are randomly selected, their
mean weight is greater than 150 lb.
P(x > 150) = 0.0735
Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
17
Example: Given the population of women has normally
distributed weights with a mean of 143 lb and a standard
deviation of 29 lb,
a.) if one woman is randomly selected, find the
probability that her weight is greater than 150 lb.
P(x > 150) = 0.4052
b.) if 36 different women are randomly selected, their
mean weight is greater than 150 lb.
P(x > 150) = 0.0735
It is much easier for an individual to deviate from the
mean than it is for a group of 36 to deviate from the mean.
Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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