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7-1
Chapter 7
Created by Bethany Stubbe and Stephan Kogitz
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
Chapter Seven
7-2
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 often the only feasible way to learn
something 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.
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
Chapter Seven
7-3
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.
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-4
Why Sample the Population?
To contact the whole population would
often be time consuming.
The cost of studying all the items in a
population is often prohibitive.
The adequacy of sample results.
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-5
Why Sample the Population
The destructive nature of certain tests.
The physical impossibility of checking all
items in the population.
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-6
Probability Sampling
A probability sample is a sample
selected such that each member of
the population being studied has a
known likelihood of being included in
the sample.
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-7
Methods of Probability Sampling
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: A random starting
point is selected and then every kth member of
the population is selected.
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-8
Methods of Probability Sampling
Stratified Random Sampling: A population is
divided into subgroups, called strata, and a
sample is randomly selected from each stratum.
Cluster Sampling: A population is divided into
clusters using naturally occurring geographic or
other boundaries. Then, clusters are randomly
selected and a sample is collected by randomly
selecting from each cluster.
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-9
Methods of Probability Sampling
In nonprobability sample inclusion in
the sample is based on the judgment
of the person selecting the sample.
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-10
The Sampling Distribution of the
Sample Mean
The sampling error is the difference between
a sample statistic and its corresponding
population parameter.
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-11
Sampling Distribution of the Sample
Mean
The sampling distribution of the sample
mean is a probability distribution of all
possible sample means of a given
sample size.
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-12
EXAMPLE 1
The law firm of Tybo 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
1.
Dunn
22
2.
Hardy
26
3.
Kiers
30
4.
Malinowski
26
5.
Tillman
22
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-13
Example 1
If two partners are selected randomly, how many different
samples are possible?
There are 10 different samples. This is the
combination of 5 objects taken 2 at a time.
5!
 10
5 C2 
2! (5  2)!
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-14
Example 1 continued
Partners
Total
Mean
1,2
48
24
1,3
52
26
1,4
48
24
1,5
44
22
2,3
56
28
2,4
52
26
2,5
48
24
3,4
56
28
3,5
52
26
4,5
48
24
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-15
EXAMPLE 1
continued
Organize the sample means into a sampling
distribution.
S a m p le
M ean
F re q u e n c y
22
1
R e la tiv e
F re q u e n c y
p ro b a b ility
1 /1 0
24
4
4 /1 0
26
3
3 /1 0
28
2
2 /1 0
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-16
EXAMPLE 1
continued
Compute the mean of the sample means.
Compare it with the population mean.
The mean of the sample means is 25.2
hours.
X
22 (1)  24 (2)  26 (3)  28 (2)

 25 .2
10
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-17
Example 1 continued
The population mean is also 25.2 hours.
22  26  30  26  22

 25 .2
5
Notice that the mean of the sample means is
exactly equal to the population mean.
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-18
Central Limit Theorem
If all samples of a specified size are selected from
any population, the sampling distribution of the
sample mean is approximately a normal
distribution. This approximation improves with
larger samples.
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-19
Mean of the Sample Means
The mean of the distribution of the sample mean
will be exactly equal to the population mean if we
are able to select all possible samples of a
particular size from the a given population.

McGraw-Hill-Ryerson
X
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-20
Standard Error of the Mean
If the standard deviation of the population is σ,
the standard deviation of the distribution of the
sample mean is
  / n
x
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-21
Sampling from a Normal Population
If a population follows the normal
distribution, the sampling distribution of
the sample mean will also follow the
normal distribution.
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-22
Finding the z Value of X When σ is
known
If the population is normally distributed
Assume the standard deviation, σ, is
known.
 To determine the probability a sample
mean falls within a particular region, use:

z
McGraw-Hill-Ryerson
X 

n
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-23
Finding the z Value of X When σ is
Unknown
If the population is not normally distributed,
but the sample is at least 30 observations,
the sampling distribution of the sample
mean is approximately normal.
Assume the population standard deviation is
not known, use the sample standard
deviation.
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-24
Finding the z Value of X When σ is
Unknown continued
If the population standard deviation, σ, is
unknown:
To determine the probability a sample mean
falls within a particular region, use:
z
X 
s
McGraw-Hill-Ryerson
n
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-25
Example 2
The mean selling price of 100 ml. Tube of
toothpaste is $1.30. The distribution is
positively skewed, with a standard
deviation of $0.28. What is the probability
of selecting a sample of 35 stores and
finding the sample mean within $.08?
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-26
Example 2 continued
The first step is to 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
McGraw-Hill-Ryerson
n

$1.38  $1.30
$0.28
 1.69
35
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-27
Example 2
z
X 
s n
McGraw-Hill-Ryerson

continued
$1.22  $1.30
 1.69
$0.28 35
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.
7-28
Example 2
continued
Next we determine the probability of a zvalue between -1.69 and 1.69. It is:
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.
McGraw-Hill-Ryerson
© The McGraw-Hill Companies, Inc., 2004 All Rights Reserved.