Transcript Chap008 - jimakers.com

8- 1
Chapter
Eight
McGraw-Hill/Irwin
© 2006 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
8- 3
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
8- 4
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?
8- 5
Simple Random Sample A sample selected so that
each item or person in the population has the same
chance of being included.
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.
(i.e. 1 in every 10 items)
Probability Sampling/Methods
8- 6
Stratified Random
Sampling: A
population is first
divided into
subgroups, called
strata, and a sample
is selected from each
stratum.
Methods of Probability Sampling
8- 7
Cluster Sampling: A population is first divided
into primary units then samples are selected from
the primary units.
Cluster Sampling
8- 8
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
8- 9
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
8- 11
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
8- 12
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
8- 13
Central Limit Theorem
If all samples of a particular size are selected from any
population, the sampling distribution of the sample
mean is approximately a normal distribution.
The standard error of the
mean is the standard
deviation of the
population means
divided by the square
root of n 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
8- 14
Central Limit Theorem

States that any distribution of sample means x from a large
population approaches the normal distribution as n increases to
infinity


The mean of the population of means is always equal to the mean of the
parent population.
The standard deviation of the population of means is always equal to the
standard deviation of the parent population divided by the square root of the
sample size (N).
If you chart the x values, the values will have less variation than the
individual measurements
 This is true if the sample size is sufficiently large.


What does this mean?
Central Limit Theorem
http://www.chem.uoa.gr/applets/appletcentrallimit/appl_centrallimit2.html
VII-5
8- 15
Central Limit Theorem
For almost all populations, the
sampling distribution of the mean
can be closely approximated by a
normal distribution, provided the
sample is sufficiently large.
Collect many x children,
(assumption is infinite number of
samples), create histograms.
Central Limit Theorem
VII-6
8- 16
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
8- 18
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
8- 19
Step One : Find the z-values corresponding to
$1.22 and $1.38. These are the two points within
$0.08 of the population mean.
z
X 
s
z
n
X 
s

n
$1.38  $1.30
 1.69
$0.28 35

$1.22  $1.30
$0.28
 1.69
35
Example 2 continued
8- 20
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