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Central Limit Theorem
Section 5-5
M A R I O F. T R I O L A
Copyright © 1998, Triola, Elementary Statistics
Copyright © 1998, Triola, Elementary Statistics
Addison
Wesley Longman
Addison Wesley Longman
1
As the sample size increases the
distribution of sample means will
approach a normal distribution.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
2
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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
3
Table 5-1
x
SSN digits
1
5
9
5
9
4
7
9
5
7
8
3
8
1
3
2
7
1
3
83
2
6
2
2
5
0
2
7
8
5
7
7
3
4
4
4
5
1
3
6
6
3
8
2
3
6
1
5
3
42
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
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Frequency
Distribution of 50 Sample Means
for Students
15
10
5
0
0
1
2
3
4
5
6
7
8
9
Distribution of 50 Sample Means
Figure 5-20
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
5
Normal, Uniform, and Skewed
Distributions
Figure 5-24
Normal
Uniform
Skewed
Original
population
Sample means
(n=5)
Sample means
( n = 10 )
Sample means
( n = 30 )
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Normal, Uniform, and Skewed
Distributions
Figure 5-24
Normal
Uniform
Original
population
Sample means
(n=5)
Sample means
( n = 10 )
Sample means
( n = 30 )
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Normal, Uniform, and Skewed
Distributions
Figure 5-24
Normal
Uniform
Skewed
Original
population
Sample means
(n=5)
Sample means
( n = 10 )
Sample means
( n = 30 )
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
8
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 s.
2. Samples of size n are randomly selected from this
population.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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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 s.
2. Samples of size n are randomly selected from this
population.
Conclusions:
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
10
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 s.
2. Samples of size n are randomly selected from this
population.
Conclusions:
1. The distribution of sample means x will, as the sample
size increases, approach a normal distribution.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
11
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 s.
2. Samples of size n are randomly selected from this
population.
Conclusions:
1. The distribution of sample means x will, as the sample
size increases, approach a normal distribution.
2. The mean of the sample means will be the population
mean µ.
s
—
3. The standard deviation of the sample means will be
n
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
12
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.
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
13
Notation
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Addison Wesley Longman
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Notation
the mean of the sample means
µx = µ
the standard deviation of sample mean
sx =
s
n
(often called standard error of the mean)
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
15
Notation
the mean of the sample means
µx = µ
the standard deviation of sample means
sx =
s
n
(often called standard error of the mean)
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
16
Notation
the mean of the sample means
µx = µ
the standard deviation of sample mean
sx =
s
n
(often called standard error of the mean)
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
17
Sampling Without
Replacement
If n > 0.05 N
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Sampling Without
Replacement
If n > 0.05 N
s
sx = n
N–n
N–1
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Sampling Without
Replacement
If n > 0.05 N
s
sx = n
N–n
N–1
finite population
correction factor
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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