The Central Limit Theorem [PDF]

Central Limit Theorem. 1. The distribution of sample means xwill, as the sample size increases, approach a normal distri

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The Central Limit Theorem

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

1

Definition Sampling Distribution of the Mean the probability distribution of sample means, with all samples having the same sample size n. (In general, the sampling distribution of any statistic is the probability distribution of that statistic.) Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

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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 σ . 2. Simple random samples all of the same size n are selected from the population. (The samples are selected so that all possible samples of size n have the same chance of being selected.)

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

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Central Limit Theorem Conclusions:

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

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Central Limit Theorem Conclusions: 1. The distribution of sample means x will, as the sample size increases, approach a normal distribution.

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

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Central Limit Theorem 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.

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

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Central Limit Theorem 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. 3. The standard deviation of the sample means will approach σ/ n . Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

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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).

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

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Notation

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

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Notation the mean of the sample means

µx = µ

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

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Notation the mean of the sample means

µx = µ the standard deviation of sample means

σx = nσ Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

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Notation the mean of the sample means

µx = µ the standard deviation of sample means

σx = nσ (often called standard error of the mean) Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

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x

SSN digits 1

8

6

4

4.75

5

3

3

6

4.25

9

8

8

8

8.25

5

1

2

5

3.25

9

3

3

5

5.00

4 7

2 7

6 1

2 6

3.50 5.25

9

1

5

4

4.75

5

3

3

9

5.00

7

3

4

2

2

7

6

6

4.00

6 2

7 3

7 3

1 9

5.25 4.25

2

4

7

5

4.50

5

4

3

7

4.75

0

4

3

8

3.75

2

5

8

6

5.25

7

1

3

4

3.75

8 5

3 6

7 6

0 7

4.50 6.00

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

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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 Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

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Frequency

Distribution of 50 Sample Means for 50 Students

15 10 5 0 0

1

2

3

4

5

6

7

8

9

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

15

As the sample size increases, the sampling distribution of sample means approaches a normal distribution.

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

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Example: Given the population of men has normally distributed weights with a mean of 172 lb and a standard deviation of 29 lb, a) if one man is randomly selected, find the probability that his weight is greater than 167 lb. b) if 12 different men are randomly selected, find the probability that their mean weight is greater than 167 lb.

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

17

Example: Given the population of men has normally distributed weights with a mean of 172 lb. and a standard deviation of 29 lb, a) if one man is randomly selected, find the probability that his weight is greater than 167 lb.

z = 167 – 172 = –0.17 29

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

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Example: Given the population of men has normally distributed weights with a mean of 172 lb. and a standard deviation of 29 lb, a) if one man is randomly selected, find the probability that his weight is greater than 167 lb. The probability that one man randomly selected has a weight greater than 167 lb. is 0.5675

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

19

Example: Given the population of men has normally distributed weights with a mean of 172 lb. and a standard deviation of 29 lb, b.) if 12 different men are randomly selected, find the probability that their mean weight is greater than 167 lb.

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

20

Example: Given the population of men has normally distributed weights with a mean of 172 lb. and a standard deviation of 29 lb, b.) if 12 different men are randomly selected, find the probability that their mean weight is greater than 167 lb.

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

21

Example: Given the population of men has normally distributed weights with a mean of 172 lb. and a standard deviation of 29 lb, b.) if 12 different men are randomly selected, find the probability that their mean weight is greater than 167 lb.

z = 167 – 172 = –0.60 29 12

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

22

Example: Given the population of men has normally distributed weights with a mean of 172 lb. and a standard deviation of 29 lb, b.) if 12 different men are randomly selected, find the probability that their mean weight is greater than 167 lb.

z = 167 – 172 = –0.60 29 12

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

23

Example: Given the population of men has normally distributed weights with a mean of 172 lb. and a standard deviation of 29 lb, b.) if 12 different men are randomly selected, find the probability that their mean weight is greater than 167 lb. The probability that the mean weight of 12 randomly selected men is greater than 167 lb. is 0.7257 .

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

24

Example: Given the population of men has normally distributed weights with a mean of 172 lb and a standard deviation of 29 lb, a) if one man is randomly selected, find the probability that his weight is greater than 167 lb.

P(x P( x > 167) = 0.5675

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

25

Example: Given the population of men has normally distributed weights with a mean of 172 lb and a standard deviation of 29 lb, a) if one man is randomly selected, find the probability that his weight is greater than 167 lb.

P(x P( x > 167) = 0.5675 b) if 12 different men are randomly selected, their mean weight is greater than 167 lb.

P(x P( x > 167) = 0.7257 Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

26

Example: Given the population of men has normally distributed weights with a mean of 172 lb and a standard deviation of 29 lb, a) if one man is randomly selected, find the probability that his weight is greater than 167 lb.

P(x P( x > 167) = 0.5675 b) if 12 different men are randomly selected, their mean weight is greater than 167 lb.

P(x P( x > 167) = 0.7257 It is much easier for an individual to deviate from the mean than it is for a group of 12 to deviate from the mean. Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

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Sampling Without Replacement

If n > 0.05 N

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

28

Sampling Without Replacement

If n > 0.05 N

σx =

σ n

N-n N-1

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

29

Sampling Without Replacement

If n > 0.05 N

σx =

σ n

N-n N-1

finite population correction factor Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

30

Example: IQ scores are normally distributed and have a mean of 100 and a standard deviation of 15. If 8 people are randomly selected, find the probability that their mean is at least 109.

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

Triola,, Essentials of Statistics, Third Edition. Copyright 2008. Pear son Education, Inc. Triola

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