Download LESSON 13: SAMPLING DISTRIBUTION CENTRAL LIMIT THEOREM

LESSON 13: SAMPLING DISTRIBUTION
Outline
• Central Limit Theorem
• Sampling Distribution of 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
Relative Frequency
0.1
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
1
0
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
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2
Sample Size and Mean
Relative Frequency
0.1
0.08
0.06
0.04
0.02
49
45
41
37
33
29
25
21
17
13
9
5
1
0
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.
µX = µ,
σ2
σ =
,
n
2
X
σ =
σ
n
• Where,
µ = Population mean
σ = Population standard deviation
n = Sample size
µ X = Mean of the sample means
σ X = Standard deviation of the sample means
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SAMPLING DISTRIBUTION OF THE SAMPLE MEAN
• Summary: For any general distribution with mean µ and
standard deviation σ
– The distribution of mean of a sample of size n can be
approximated by a normal distribution with
mean, µ
standard deviation, s X =
σ
n
• Exercise: Generate 1000 random numbers uniformly
distributed between 0 and 1. Consider 200 samples of size
5 each. Compute the sample means. Check if the
histogram of sample means is normally distributed and
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mean and standard deviation follow the above rules.
SAMPLING DISTRIBUTION OF THE SAMPLE MEAN
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, µ=120 cm and standard deviation, σ=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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SAMPLING DISTRIBUTION OF THE SAMPLE MEAN
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, µ=120 cm and standard deviation, σ=5 cm. Find
the probability that one randomly selected unit has a
length greater than 123 cm.
f(x)
σ
µ
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SAMPLING DISTRIBUTION OF THE SAMPLE MEAN
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, µ=120 cm and standard deviation, σ=5 cm. Find
the probability that, if four units are randomly selected,
their mean length exceeds 123 cm.
f(x)
σ
µ
10
5
SAMPLING DISTRIBUTION OF THE SAMPLE MEAN
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, µ=120 cm and standard deviation, σ=5 cm. Find
the probability that, if four units are randomly selected, all
four have lengths that exceed 123 cm.
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CORRECTION FOR SMALL SAMPLE SIZE
• For a small, finite population N, the formula for the standard
deviation of sampling mean is corrected as follows:
σX =
σ
n
N −n
N −1
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READING AND EXERCISES
Lesson 13
Reading:
Sections 8-1, 8-2, 8-3, pp. 260-276
Exercises:
9-3,9-4, 9-8, 9-17, 9-19
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