Statistical techniques in business ecohomics chap008

Chapter EightSampling Methods and the Central Limit Theorem GOALS When you have completed this chapter, you will be able to: ONE Explain why a sample is the only feasible way to learn a

Eight

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Irwin © 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.

Chapter Eight

Sampling Methods and the Central Limit Theorem

GOALS

When you have completed this chapter, you will

be able to:

ONE

Explain why a sample is the only feasible way to learn about a

population.

TWO

Describe methods to select a sample.

THREE

Define and construct a sampling distribution of the sample

mean.

FOUR

Explain the central limit theorem Goals

Chapter Eight continued

Sampling Methods and the

Central Limit Theorem

GOALS

When you have completed this chapter, you will

be able to:

FIVE

Use the Central Limit Theorem to find probabilities of selecting possible sample means from a specified population

Goals

Why Sample the Population?

Why sample?

The destructive

nature of

certain tests.

The physical

impossibility of

checking all items in

the population.

The cost of studying

all the items in a

population.

The adequacy of sample results

in most cases.

The time-consuming aspect of contacting the whole population.

Probability Sampling/Methods

Systematic Random Sampling

The items or individuals of the

population are arranged in some

order A random starting point

is selected and then every kth

member of the population is

selected for the sample.

Simple Random Sample

A sample formulated so

that each item or person

in the population has

the same chance of

being included.

A probability sample is a sample selected such

that each item or person

in the population being studied has a known likelihood of being included in the sample.

Methods of Probability Sampling

Stratified Random

Sampling : A

population is first

divided into

subgroups, called

strata, and a sample

is selected from each

stratum

Cluster Sampling

Cluster Sampling: A population is first divided

into primary units then samples are selected from the primary units

Methods of Probability Sampling

The sampling error is the difference between

a sample statistic and its corresponding

population parameter.

In nonprobability

sample inclusion in

the sample is based

on the judgment of

the person selecting

the sample

The sampling distribution of the sample mean is

a probability distribution consisting of all

possible sample means of a given sample size

selected from a population

Example 1

Partner Hours Dunn 22 Hardy 26 Kiers 30 Malory 26 Tillman 22

The law firm of

Hoya and

Associates has five

partners At their

weekly partners

meeting each

reported the

number of hours

they billed clients

for their services

last week

If two partners are selected randomly, how many different samples are possible?

Example 1

10 )!

2 5

(

! 2

!

5

2





C

5 objects

taken 2 at

a time

A total of 10 different samples

Partners Total Mean

Example 1 continued

Frequency probability

As a sampling distribution

Example 1 continued

2

25 10

) 2 ( 28 )

3 ( 26 )

2 ( 24 )

1 (

22











X



Compute the mean of the sample means

Compare it with the population mean

The mean of the sample means

The population mean

2

25 5

22 26

30 26

22















Notice that the mean of the sample means is exactly equal to the population mean

Central Limit Theorem

 x = 

n

For a population with a mean  and a variance 2 the sampling distribution of the means of all possible

samples of size n generated from the population will be

approximately normally distributed

This approximation improves with larger samples

The mean of the sampling distribution equal to m and the variance equal to 2/n.

The standard error of the

mean is the standard

deviation of the standard

deviation of the sample

means given as:

Central Limit Theorem

Sample Means

the sample size is large enough even when the underlying population may be nonnormal

Sample means

follow the normal

probability

distribution under

two conditions:

the underlying population

follows the normal

distribution

OR

n s

X



standard deviation is known.

Sample Means

To determine the

probability that a sample

mean falls within a

particular region, use

Example 2

Suppose the mean selling

price of a gallon of gasoline

in the United States is $1.30

Further, assume the

distribution is positively

skewed, with a standard

deviation of $0.28 What is

the probability of selecting a

sample of 35 gasoline

stations and finding the

sample mean within $.08?

Example 2 continued

69

1 35

28 0

$

30 1

$ 38

1

$











n s

X

69

1 35

28

0

$

30

1

$ 22

1

$













n s

X

Step One : Find the z-values corresponding to

$1.24 and $1.36 These are the two points within

$0.08 of the population mean.

Example 2 continued

9090

) 4545 (.

2 )

69 1 69

1

P

Step Two: determine the probability of a z-value

between -1.69 and 1.69

We would expect about 91 percent of the sample means to be within $0.08 of

the population mean.