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CHAPTER 8
SAMPLING DISTRIBUTIONS
Outline
• Central limit theorem
• Sampling distribution of the sample mean
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CENTRAL LIMIT THEOREM
Central Limit Theorem: If a random sample is drawn from any
population, the sampling distribution of the sample mean is
approximately normal for a sufficiently large sample size.
The larger the sample size, the more closely the sampling
distribution of x will resemble a normal distribution.
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Sample Size and Mean
0.08
0.06
0.04
0.02
Class Number
Distribution of random numbers
49
45
41
37
33
29
25
21
17
13
9
5
0
1
Relative Frequency
0.1
3
Sample Size and Mean
0.08
0.06
0.04
0.02
45
49
41
37
33
29
25
21
17
13
9
5
0
1
Relative Frequency
0.1
Class Number
Distribution of means of n random numbers, n=4
4
Sample Size and Mean
0.08
0.06
0.04
0.02
49
45
41
37
33
29
25
21
17
13
9
5
0
1
Relative Frequency
0.1
Class Number
Distribution of means of n random numbers, n=10
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SAMPLING DISTRIBUTION OF THE SAMPLE MEAN
• If the sample size increases, the variation of the sample
mean decreases.
x  ,  
2
x
2
n
, 

n
• Where,
 = Population mean
 = Population standard deviation
n = Sample size
 x = Mean of the sample means
 x = Standard deviation of the sample means
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CENTRAL LIMIT THEOREM
Example 1: An automatic machine in a manufacturing
process requires an important sub-component. The
lengths of the sub-component are normally distributed with
a mean, =120 cm and standard deviation, =5 cm. What
does the central limit theorem say about the sampling
distribution of the mean if samples of size 4 are drawn
from this population?
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CENTRAL LIMIT THEOREM
Example 2: An automatic machine in a manufacturing
process requires an important sub-component. The
lengths of the sub-component are normally distributed with
a mean, =120 cm and standard deviation, =5 cm. Find
the probability that one randomly selected unit has a
length greater than 123 cm.
f(x)


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CENTRAL LIMIT THEOREM
Example 3: An automatic machine in a manufacturing
process requires an important sub-component. The
lengths of the sub-component are normally distributed with
a mean, =120 cm and standard deviation, =5 cm. Find
the probability that, if four units are randomly selected,
their mean length exceeds 123 cm.
f(x)


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CENTRAL LIMIT THEOREM
Example 4: An automatic machine in a manufacturing
process requires an important sub-component. The
lengths of the sub-component are normally distributed with
a mean, =120 cm and standard deviation, =5 cm. Find
the probability that, if four units are randomly selected, all
four have lengths that exceed 123 cm.
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READING AND EXERCISES
• Reading: pp. 289-298
• Exercises: 8.2, 8.4, 8.6
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