Transcript Slides for Chapter 8

8- 1
Chapter
Eight
McGraw-Hill/Irwin
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.
8- 2
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
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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
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Why sample?
The physical
impossibility of
checking all items in
the population.
The cost of studying
all the items in a
population.
The destructive
nature of
certain tests.
The time-consuming
aspect of contacting
the whole population.
The adequacy of
sample results
in most cases.
Why Sample the Population?
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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.
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
Probability Sampling/Methods
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.
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Stratified Random
Sampling: A
population is first
divided into
subgroups, called
strata, and a sample
is selected from each
stratum.
Methods of Probability Sampling
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Cluster Sampling: A population is first divided
into primary units then samples are selected from
the primary units.
Cluster Sampling
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In nonprobability
sample inclusion in
the sample is based
on the judgment of
the person selecting
the sample.
The sampling error is
the difference between
a sample statistic and
its corresponding
population parameter.
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.
Methods of Probability Sampling
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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.
Partner
Hours
Dunn
22
Hardy
26
Kiers
30
Malory
26
Tillman
22
If two partners are
selected randomly, how
many different samples
are possible?
Example 1
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5 objects
taken 2 at
a time.
5!
 10
5 C2 
2! (5  2)!
Partners
1,2
1,3
1,4
1,5
2,3
2,4
2,5
3,4
3,5
4,5
Total
48
52
48
44
56
52
48
56
52
48
A total of 10
different
samples
Mean
24
26
24
22
28
26
24
28
26
24
Example 1
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As a sampling distribution
Sample Mean
Frequency
Relative
Frequency
probability
22
1
1/10
24
4
4/10
26
3
3/10
28
2
2/10
Example 1 continued
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Compute the mean of the sample means.
Compare it with the population mean.
The mean of the sample means
X
22 (1)  24 (2)  26 (3)  28 (2)

 25 .2
10
The population mean
22  26  30  26  22

 25 .2
5
Notice that the
mean of the
sample means is
exactly equal to
the population
mean.
Example 1 continued
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Central Limit Theorem
For a population with a mean  and a variance s2
the sampling distribution of the means of all possible
samples of size n generated from the population will be
approximately normally distributed.
The standard error of the
mean is the standard
deviation of the standard
deviation of the sample
means given as:
sx
=
s
n
This approximation improves
with larger samples.
The mean of the sampling
distribution equal to m and
the variance equal to s2/n.
Central Limit Theorem
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Sample means
follow the normal
probability
distribution under
two conditions:
the underlying population
follows the normal
distribution
OR
the sample size is large
enough even when the
underlying population
may be nonnormal
Sample Means
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To determine the probability
that a sample mean falls
within a particular region,
use
z
X 
s
n
Use s in place of s if the population
standard deviation is known.
Sample Means
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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
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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.
z
X 
s
z
n
X 
s

n
$1.38  $1.30
$0.28

35
$1.22  $1.30
$0.28
 1.69
 1.69
35
Example 2 continued
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Step Two: determine the probability of a z-value
between -1.69 and 1.69.
P(1.69  z  1.69)  2(.4545 )  .9090
We would expect about 91
percent of the sample
means to be within $0.08 of
the population mean.
Example 2 continued