1

Outline

• Central Limit Theorem• Sampling Distribution of Mean

LESSON 13: SAMPLING DISTRIBUTION

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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 will resemble a normal distribution.X

3

Sample Size and Mean

0

0.02

0.04

0.06

0.08

0.1

1 5 9 13 17 21 25 29 33 37 41 45 49

Class Number

Rel

ativ

e F

req

uen

cy

Distribution of random numbers

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Sample Size and Mean

0

0.02

0.04

0.06

0.08

0.1

1 5 9 13 17 21 25 29 33 37 41 45 49

Class Number

Rel

ativ

e F

req

uen

cy

Distribution of means of n random numbers, n=4

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Sample Size and Mean

0

0.02

0.04

0.06

0.08

0.1

1 5 9 13 17 21 25 29 33 37 41 45 49

Class Number

Rel

ativ

e F

req

uen

cy

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.

• Where,

= Population mean

= Population standard deviation

= Sample size

= Mean of the sample means

= Standard deviation of the sample means

nnXX

,,2

2

n

X

X

7

• Summary: For any general distribution with mean and standard deviation – The distribution of mean of a sample of size can be

approximated by a normal distribution with

• 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 mean and standard deviation follow the above rules.

n

nσ

μ

X

deviation, standard

mean,

SAMPLING DISTRIBUTION OF THE SAMPLE MEAN

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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?

SAMPLING DISTRIBUTION OF THE SAMPLE MEAN

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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)

SAMPLING DISTRIBUTION OF THE SAMPLE MEAN

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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)

SAMPLING DISTRIBUTION OF THE SAMPLE MEAN

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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.

SAMPLING DISTRIBUTION OF THE SAMPLE MEAN

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• For a small, finite population N, the formula for the standard deviation of sampling mean is corrected as follows:

1

N

nN

nX

CORRECTION FOR SMALL SAMPLE SIZE

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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