1Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

MARIO F. TRIOLAMARIO F. TRIOLA EIGHTHEIGHTH

EDITIONEDITION

ELEMENTARY STATISTICSSection 5-5 The Central Limit Theorem

2Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Sampling Distribution of the meanis a probability distribution of sample means, with all samples

having the same sample size n.

Definition

3Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Central Limit Theorem

1. The random variable x has a distribution (which may or may not be normal) with mean µ and

standard deviation .

2. Samples all of the same size n are randomly

selected from the population of x values.

Given:

4Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Central Limit Theorem

1. The distribution of sample x will, as the sample size increases, approach a normal distribution.

2. The mean of the sample means µx will be the population mean µ.

3. The standard deviation of the sample means

xwill approach

n

Conclusions:

5Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

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).

6Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Notationthe mean of the sample means

the standard deviation of sample mean

(often called standard error of the mean)

µx = µ

x = n

7Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

0

10

20

0 1 2 3 4 5 6 7 8 9

Distribution of 200 digits

Distribution of 200 digits from Social Security Numbers

(Last 4 digits from 50 students)F

req

uen

cy

Figure 5-19

8Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Table 5-2

1

5

9

5

9

4

7

9

5

7

8

3

8

1

3

2

7

1

3

6

3

8

2

3

6

1

5

3

4

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

2

2

5

0

2

7

8

5

3

7

7

3

4

4

4

5

1

3

6

7

3

7

3

3

8

3

7

6

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

SSN digits x

9Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

0

10

0 1 2 3 4 5 6 7 8 9

Distribution of 50 Sample Means for 50 Students

Fre

qu

ency

Figure 5-20

5

15

10Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

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

normal distribution.

11Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Example: Given the population of women has normally distributed weights with a mean of 143 lb and a standard deviation of 29 lb, a.) if one woman is randomly selected, find the probability

that her weight is greater than 150 lb.b.) if 36 different women are randomly selected, find the

probability that their mean weight is greater than 150 lb.

12Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Example: Given the population of women has normally distributed weights with a mean of 143 lb and a standard deviation of 29 lb, a.) if one woman is randomly selected, find the probability

that her weight is greater than 150 lb.

= 143 150

= 29

0 0.24

0.0948

Normalcdf(0.24,9999) = 0.4052

z = 150-143 = 0.24 29

P(x>150) = 0.405

13Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Example: Given the population of women has normally distributed weights with a mean of 143 lb and a standard deviation of 29 lb, b.) if 36 different women are randomly selected, find the probability that their mean weight is greater than 150 lb.

14Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Example: Given the population of women has normally distributed weights with a mean of 143 lb and a standard deviation of 29 lb, b.) if 36 different women are randomly selected, find the probability that their mean weight is greater than 150 lb.

x = 143 150

x= 4.83333

0 1.45

0.4265

Normalcdf(1.45, 9999) = 0.0735

z = 150-143 = 1.45 29

36

P(x>150) =0.0735

15Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Example: Given the population of women has normally distributed weights with a mean of 143 lb and a standard deviation of 29 lb,

16Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Example: Given the population of women has normally distributed weights with a mean of 143 lb and a standard deviation of 29 lb,

a.) if one woman is randomly selected, find the probability    that her weight is greater than 150 lb.

P(x > 150) = 0.4052

17Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Example: Given the population of women has normally distributed weights with a mean of 143 lb and a standard deviation of 29 lb,

a.) if one woman is randomly selected, find the probability    that her weight is greater than 150 lb.

P(x > 150) = 0.4052b.) if 36 different women are randomly selected, their mean

   weight is greater than 150 lb.

P(x> 150) = 0.0735

18Chapter 5. Section 5-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Example: Given the population of women has normally distributed weights with a mean of 143 lb and a standard deviation of 29 lb,

a.) if one woman is randomly selected, find the probability    that her weight is greater than 150 lb.

P(x > 150) = 0.4052

b.) if 36 different women are randomly selected, their mean    weight is greater than 150 lb.

P(x > 150) = 0.0735

It is much easier for an individual to deviate from the mean than it is for a group of 36 to deviate from the mean.