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Chapter 7
Sampling and Sampling Distributions
Simple Random Sampling
Point Estimation
Introduction to Sampling Distributions
Sampling Distribution of x
n = 100
p
Sampling Distribution of
Sampling Methods
n = 30
1
Statistical Inference
The purpose of statistical inference is to obtain
information about a population from information
contained in a sample.
A population is the set of all the elements of interest.
A ________ is a subset of the population.
The sample results provide only ________ of the
values of the population characteristics.
A parameter is a numerical characteristic of a
population.
With proper sampling methods, the sample results
will provide “good” estimates of the population
characteristics.
2
Simple Random Sampling
Finite Population
– A simple random sample from a finite population
of size N is a sample selected such that each
possible sample of size n has the same probability
of being selected.
– Replacing each sampled element before selecting
subsequent elements is called sampling with
___________.
3
Simple Random Sampling
Finite Population
– Sampling without ___________ is the procedure
used most often.
– In large sampling projects, computer-generated
random numbers are often used to automate the
sample selection process.
4
Simple Random Sampling
Infinite Population
– A simple random sample from an infinite
population is a sample selected such that the
following conditions are satisfied.
• Each element selected comes from the same
population.
• Each element is selected independently.
5
Simple Random Sampling
Infinite Population
– The population is usually considered infinite if it
involves an ongoing process that makes listing or
counting every element impossible.
– The random number selection procedure cannot
be used for infinite populations.
6
Point Estimation
In point estimation we use the data from the sample
to compute a value of a sample statistic that serves as
an estimate of a population parameter.
We refer to x as the _____________ of the population
mean .
s is the point estimator of the population standard
deviation .
p is the point estimator of the population proportion
p.
7
Point Estimation
When the expected value of a point estimator is equal
to the population parameter, the point estimator is
said to be unbiased.
8
Sampling Error
The absolute difference between an unbiased point
estimate and the corresponding population
parameter is called the sampling error.
Sampling error is the result of using a subset of the
population (the sample), and not the entire
population to develop estimates.
The sampling errors are:
|x   | for sample mean
|s   | for sample standard deviation
| p  p| for sample proportion
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Example: St. Andrew’s
St. Andrew’s College receives 900 applications
annually from prospective students. The application
forms contain a variety of information including the
individual’s scholastic aptitude test (SAT) score and
whether or not the individual desires on-campus
housing.
10
Example: St. Andrew’s
The director of admissions would like to know the
following information:
– the average SAT score for the applicants, and
– the proportion of applicants that want to live on
campus.
11
Example: St. Andrew’s
We will now look at three alternatives for obtaining
the desired information.
– Conducting a census of the entire 900 applicants
– Selecting a sample of 30 applicants, using a random
number table
– Selecting a sample of 30 applicants, using
computer-generated random numbers
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Example: St. Andrew’s
Taking a Census of the 900 Applicants
– SAT Scores
• Population Mean
x


i
900
 990
• Population Standard Deviation

(x
i
  )2
900
 80
13
Example: St. Andrew’s
Taking a Census of the 900 Applicants
– Applicants Wanting On-Campus Housing
• Population Proportion
648
p
 .72
900
14
Example: St. Andrew’s
Take a Sample of 30 Applicants
Using a Random Number Table
Since the finite population has ____ elements, we
will need 3-digit random numbers to randomly select
applicants numbered from 1 to 900.
We will use the last three digits of the 5-digit
random numbers in the third column of the
textbook’s random number table (________________).
15
Example: St. Andrew’s
Take a Sample of 30 Applicants
Using a Random Number Table
The numbers we draw will be the numbers of the
applicants we will sample unless
• the random number is greater than 900 or
• the random number has already been used.
We will continue to draw random numbers until we
have selected _______ applicants for our sample.
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Example: St. Andrew’s
Use of Random Numbers for Sampling
3-Digit
Applicant
Random Number Included in Sample
744
No. 744
436
No. 436
865
No. 865
790
No. 790
835
No. 835
902
Number exceeds 900
190
No. 190
436
Number already used
etc.
etc.
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Example: St. Andrew’s
Sample Data
Random
No. Number
1
744
2
436
3
865
4
790
5
835
.
.
30
685
Applicant
SAT Score
Connie Reyman
1025
William Fox
950
Fabian Avante
1090
Eric Paxton
1120
Winona Wheeler
1015
.
.
Kevin Cossack
965
On-Campus
Yes
Yes
No
Yes
No
.
No
18
Example: St. Andrew’s
Take a Sample of 30 Applicants
Using Computer-Generated Random Numbers
– Excel provides a function for generating random
numbers in its worksheet.
– 900 random numbers are generated, one for each
applicant in the population.
– Then we choose the ____ applicants corresponding
to the 30 smallest random numbers as our sample.
– Each of the 900 applicants have the same
probability of being included.
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Using Excel to Select
a Simple Random Sample
Formula Worksheet
1
2
3
4
5
6
7
8
9
A
B
C
Applicant
Number
1
2
3
4
5
6
7
8
SAT
Score
1008
1025
952
1090
1127
1015
965
1161
On-Campus
Housing
Yes
No
Yes
Yes
Yes
No
Yes
No
D
Random
Number
=RAND()
=RAND()
=RAND()
=RAND()
=RAND()
=RAND()
=RAND()
=RAND()
Note: Rows 10-901 are not shown.
20
Using Excel to Select
a Simple Random Sample
n
Value Worksheet
1
2
3
4
5
6
7
8
9
A
B
C
D
Applicant
Number
1
2
3
4
5
6
7
8
SAT
Score
1008
1025
952
1090
1127
1015
965
1161
On-Campus
Housing
Yes
No
Yes
Yes
Yes
No
Yes
No
Random
Number
0.41327
0.79514
0.66237
0.00234
0.71205
0.18037
0.71607
0.90512
Note: Rows 10-901 are not shown.
21
Using Excel to Select
a Simple Random Sample
Value Worksheet (Sorted)
1
2
3
4
5
6
7
8
9
A
B
C
D
Applicant
Number
12
773
408
58
116
185
510
394
SAT
Score
1107
1043
991
1008
1127
982
1163
1008
On-Campus
Housing
No
Yes
Yes
No
Yes
Yes
Yes
No
Random
Number
0.00027
0.00192
0.00303
0.00481
0.00538
0.00583
0.00649
0.00667
Note: Rows 10-901 are not shown.
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Example: St. Andrew’s
Point Estimates
– x as Point Estimator of 
x

x
29,910

 997
30
30
i
– s as Point Estimator of 
s
2
(
x

x
)
 i
29

163,996
 75.2
29
– p as Point Estimator of p
p  20 30  .68
23
Example: St. Andrew’s
Point Estimates
Note: Different random numbers would have
identified a different sample which would have
resulted in different point estimates.
24
Sampling Distribution of
x
Process of Statistical Inference
Population
with mean
=?
The value of x is used to
make inferences about
the value of .
A simple random sample
of n elements is selected
from the population.
The sample data
provide a value for
the sample mean x.
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Sampling Distribution of
x
The sampling distribution of x is the probability
distribution of all possible values of the sample
mean x.
Expected Value of
x
E( x) = 
where:
 = the population mean
What does this imply?
– The sample mean (from a simple random sample)
is an unbiased estimator of population mean.
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Sampling Distribution of

 x : Standard Deviation of
•
x
x
referred to as the standard error of the mean.
Finite Population

N n
x  ( )
n N 1
Infinite Population
x 

n
• A finite population is treated as being
•
infinite if n/N < .05.
( N  n) / ( N  1) is the finite correction factor.
27
Sampling Distribution of
x
If we use a large (n > 30) simple random sample, the
central limit theorem enables us to conclude that the
sampling distribution of x can be approximated by a
normal probability distribution.
When the simple random sample is small (n < 30), the
sampling distribution of x can be considered normal
only if we assume the population has a normal
probability distribution.
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Example: St. Andrew’s
Sampling Distribution of
x for the SAT Scores

80
x 

 14.6
n
30
E ( x )    990
x
29
Example: St. Andrew’s
Sampling Distribution of x for the SAT Scores
What is the probability that a simple random
sample of 30 applicants will provide an estimate of
the population mean SAT score that is within plus or
minus 10 of the actual population mean  ?
In other words, what is the probability that x
will be between _____ and ______?
30
Example: St. Andrew’s
Sampling Distribution of
Area = ?
x for the SAT Scores
Sampling
distribution
of x
x
980 990 1000
31
Example: St. Andrew’s
Sampling Distribution of
x for the SAT Scores
Using the standard normal probability table with
z = 10/14.6= .68, we have area = (.2517)(2) =
The probability is _______ that the sample mean will
be within +/-10 of the actual population mean.
32
Example: St. Andrew’s
Sampling Distribution of
Area = 2(.2517) = .5034
x for the SAT Scores
Sampling
distribution
of x
x
980 990 1000
33
Sampling Distribution of
p
The sampling distribution of p is the probability
distribution of all possible values of the sample
proportion p.
Expected Value of
p
E ( p)  p
where:
p = the population proportion
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Sampling Distribution of
Standard Deviation of
p
Finite Population
p 
p
p(1  p) N  n
n
N 1
Infinite Population
p(1  p)
p 
n
–  p is referred to as the standard error of the
proportion.
35
Sampling Distribution of
p
The sampling distribution of pcan be approximated
by a normal probability distribution whenever the
sample size is large.
The sample size is considered large whenever these
conditions are satisfied:
np > 5
and
n(1 – p) > 5
36
Sampling Distribution of
n
n
p
For values of p near .50, sample sizes as small as 10
permit a normal approximation.
With very small (approaching 0) or large
(approaching 1) values of p, much larger samples are
needed.
37
Example: St. Andrew’s
Sampling Distribution of
On-campus Housing
p for Applicants Desiring
The normal probability distribution is an
acceptable approximation because:
np = 30(.72) = 21.6 > 5
and
n(1 - p) = 30(.28) = 8.4 > 5.
38
Example: St. Andrew’s
Sampling Distribution of
On-campus Housing
p for Applicants Desiring
.72(1  .72)
p 
 .082
30
E( p )  .72
39
Example: St. Andrew’s
Sampling Distribution of p for Applicants Desiring
On-campus Housing
What is the probability that a simple random
sample of 30 applicants will provide an estimate of the
population proportion of applicants desiring oncampus housing that is within plus or minus .05 of the
actual population proportion?
In other words, what is the probability that p
will be between ______ and ______?
40
Example: St. Andrew’s
Sampling Distribution of
On-campus Housing
p for Applicants Desiring
Sampling
distribution
of p
Area = ?
0.67 0.72 0.77
p
41
Example: St. Andrew’s
Sampling Distribution of
On-campus Housing
p for Applicants Desiring
For z = .05/.082 = .61, the area = (.2291)(2) = ______.
The probability is _______ that the sample proportion
will be within +/-.05 of the actual population
proportion.
42
Example: St. Andrew’s
Sampling Distribution of
Area = 2(.2291) = .4582
p for In-State Residents
Sampling
distribution
of p
0.67 0.72 0.77
p
43
Sampling Methods
Stratified Random Sampling
Cluster Sampling
Systematic Sampling
Convenience Sampling
Judgment Sampling
44
Stratified Random Sampling
The population is first divided into groups of
elements called strata.
Each element in the population belongs to one and
only one stratum.
Best results are obtained when the elements within
each stratum are as much alike as possible (i.e.
homogeneous group).
A simple random sample is taken from each stratum.
Example: The basis for forming the strata might be
department, location, age, industry type, etc.
45
Stratified Random Sampling
Advantage: If strata are homogeneous, this method is
as “precise” as simple random sampling but with a
smaller total sample size.
Formulas are available for combining the stratum
sample results into one population parameter
estimate.
46
Cluster Sampling
The population is first divided into separate groups of
elements called clusters.
Ideally, each cluster is a representative small-scale
version of the population (i.e. heterogeneous group).
A simple random sample of the clusters is then taken.
All elements within each sampled (chosen) cluster
form the sample.
Example: A primary application is area sampling,
where clusters are city blocks or other well-defined
areas.
… continued
47
Cluster Sampling
Advantage: The close proximity of elements can be
cost effective (I.e. many sample observations can be
obtained in a short time).
Disadvantage: This method generally requires a larger
total sample size than simple or stratified random
sampling.
48
Systematic Sampling
If a sample size of n is desired from a population
containing N elements, we might sample one element
for every N/n elements in the population.
We randomly select one of the first N/n elements
from the population list.
We then select every N/nth element that follows in
the population list.
This method has the properties of a simple random
sample, especially if the list of the population
elements is a random ordering.
… continued
49
Systematic Sampling
Advantage: The sample usually will be easier to
identify than it would be if simple random sampling
were used.
Example: Selecting every 100th listing in a telephone
book after the first randomly selected listing.
50
Convenience Sampling
It is a nonprobability sampling technique. Items are
included in the sample without known probabilities
of being selected.
The sample is identified primarily by convenience.
Advantage: Sample selection and data collection are
relatively easy.
Disadvantage: It is impossible to determine how
representative of the population the sample is.
Example: A professor conducting research might use
student volunteers to constitute a sample.
51
Judgment Sampling
The person most knowledgeable on the subject of the
study selects elements of the population that he or
she feels are most representative of the population.
It is a nonprobability sampling technique.
Advantage: It is a relatively easy way of selecting a
sample.
Disadvantage: The quality of the sample results
depends on the judgment of the person selecting the
sample.
Example: A reporter might sample three or four
senators, judging them as reflecting the general
opinion of the senate.
52
End of Chapter 7
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