Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-1

Developing a Sampling Distribution

Assume there is a population …

Population size N=4

Random variable, X,

is age of individuals

Values of X: 18, 20,

22, 24 (years)

A B C D

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-2

.3

.2

.1

0 18 20 22 24

A B C D

Uniform Distribution

P(x)

x

(continued)

Summary Measures for the Population Distribution:

Developing a Sampling Distribution

214

24222018

N

Xμ i

2.236N

μ)(Xσ

2i

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-3

1st 2nd Observation Obs 18 20 22 24

18 18 19 20 21

20 19 20 21 22

22 20 21 22 23

24 21 22 23 24

Sampling Distribution of All Sample Means

18 19 20 21 22 23 240

.1

.2

.3 P(X)

X

Sample Means Distribution

16 Sample Means

_

Developing a Sampling Distribution

(continued)

(no longer uniform)

_

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-4

Summary Measures of this Sampling Distribution:

Developing aSampling Distribution

(continued)

2116

24191918

N

Xμ i

X

1.5816

21)-(2421)-(1921)-(18

N

)μX(σ

222

2Xi

X

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-5

Comparing the Population Distributionto the Sample Means Distribution

18 19 20 21 22 23 240

.1

.2

.3 P(X)

X 18 20 22 24

A B C D

0

.1

.2

.3

PopulationN = 4

P(X)

X _

1.58σ 21μXX

2.236σ 21μ

Sample Means Distributionn = 2

_

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-6

Sample Mean Sampling Distribution:Standard Error of the Mean

Different samples of the same size from the same population will yield different sample means

A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean:(This assumes that sampling is with replacement or sampling is without replacement from an infinite population)

Note that the standard error of the mean decreases as the sample size increases

n

σσ

X

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-7

Sample Mean Sampling Distribution:If the Population is Normal

If a population is normal with mean μ and

standard deviation σ, the sampling distribution

of is also normally distributed with

and

X

μμX

n

σσ

X

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-8

Z-value for Sampling Distributionof the Mean

Z-value for the sampling distribution of :

where: = sample mean

= population mean

= population standard deviation

n = sample size

Xμσ

n

σμ)X(

σ

)μX(Z

X

X

X

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-9

Normal Population Distribution

Normal Sampling Distribution (has the same mean)

Sampling Distribution Properties

(i.e. is unbiased )xx

x

μμx

μ

xμ

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-10

Sampling Distribution Properties

As n increases,

decreasesLarger sample size

Smaller sample size

x

(continued)

xσ

μ

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-11

Sample Mean Sampling Distribution:If the Population is not Normal

We can apply the Central Limit Theorem:

Even if the population is not normal, …sample means from the population will be

approximately normal as long as the sample size is large enough.

Properties of the sampling distribution:

andμμx n

σσx

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-12

n↑

Central Limit Theorem

As the sample size gets large enough…

the sampling distribution becomes almost normal regardless of shape of population

x

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-13

Population Distribution

Sampling Distribution (becomes normal as n increases)

Central Tendency

Variation

x

x

Larger sample size

Smaller sample size

Sample Mean Sampling Distribution:If the Population is not Normal

(continued)

Sampling distribution properties:

μμx

n

σσx

xμ

μ

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-14

How Large is Large Enough?

For most distributions, n > 30 will give a sampling distribution that is nearly normal

For fairly symmetric distributions, n > 15

For normal population distributions, the sampling distribution of the mean is always normally distributed

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-15

Example

Suppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected.

What is the probability that the sample mean is between 7.8 and 8.2?

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-16

Example

Solution:

Even if the population is not normally distributed, the central limit theorem can be used (n > 30)

… so the sampling distribution of is approximately normal

… with mean = 8

…and standard deviation

(continued)

x

xμ

0.536

3

n

σσx

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-17

Example

Solution (continued):(continued)

0.31080.4)ZP(-0.4

363

8-8.2

nσ

μ- X

363

8-7.8P 8.2) X P(7.8

Z7.8 8.2 -0.4 0.4

Sampling Distribution

Standard Normal Distribution .1554

+.1554

Population Distribution

??

??

?????

??? Sample Standardize

8μ 8μX

0μz xX