Download Sampling Distribution of

Slides Prepared by
JOHN S. LOUCKS
St. Edward’s University
© 2004 Thomson/South-Western
Slide 1
Chapter 7
Sampling and Sampling Distributions
Simple Random Sampling
Point Estimation
Introduction to Sampling Distributions
Sampling Distribution of x
Sampling Distribution of p
Sampling Methods
© 2004 Thomson/South-Western
Slide 2
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 sample is a subset of the population.
© 2004 Thomson/South-Western
Slide 3
Statistical Inference
The sample results provide only estimates of the
values of the population characteristics.
With proper sampling methods, the sample results
can provide “good” estimates of the population
characteristics.
A parameter is a numerical characteristic of a
population.
© 2004 Thomson/South-Western
Slide 4
Simple Random Sampling:
Finite Population

Finite populations are often defined by lists such as:
• Organization membership roster
• Credit card account numbers
• Inventory product numbers

A simple random sample of size n 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.
© 2004 Thomson/South-Western
Slide 5
Simple Random Sampling:
Finite Population
 Replacing each sampled element before selecting
subsequent elements is called sampling with
replacement.
 Sampling without replacement is the procedure
used most often.
 In large sampling projects, computer-generated
random numbers are often used to automate the
sample selection process.
© 2004 Thomson/South-Western
Slide 6
Simple Random Sampling:
Infinite Population


Infinite populations are often defined by an ongoing
process whereby the elements of the population
consist of items generated as though the process would
operate indefinitely.
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.
© 2004 Thomson/South-Western
Slide 7
Simple Random Sampling:
Infinite Population
 In the case of infinite populations, it is impossible to
obtain a list of all elements in the population.
 The random number selection procedure cannot be
used for infinite populations.
© 2004 Thomson/South-Western
Slide 8
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
mean .
x as the point estimator of the population
s is the point estimator of the population standard
deviation .
p is the point estimator of the population proportion p.
© 2004 Thomson/South-Western
Slide 9
Sampling Error
 When the expected value of a point estimator is equal
to the population parameter, the point estimator is said
to be unbiased.
 The absolute value of the 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.
 Statistical methods can be used to make probability
statements about the size of the sampling error.
© 2004 Thomson/South-Western
Slide 10
Sampling Error

The sampling errors are:
|x   | for sample mean
|s   | for sample standard deviation
| p  p| for sample proportion
© 2004 Thomson/South-Western
Slide 11
Example: St. Andrew’s
St. Andrew’s College receives
900 applications annually from
prospective students. The
application form contains
a variety of information
including the individual’s
scholastic aptitude test (SAT) score and whether or not
the individual desires on-campus housing.
© 2004 Thomson/South-Western
Slide 12
Example: St. Andrew’s
The director of admissions
would like to know the
following information:
• the average SAT score for
the 900 applicants, and
• the proportion of
applicants that want to live on campus.
© 2004 Thomson/South-Western
Slide 13
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 Excel
© 2004 Thomson/South-Western
Slide 14
Conducting a Census

If the relevant information for the entire 900 applicants
was in the college’s database, the population
parameters of interest could be calculated using the
formulas presented in Chapter 3.

We will assume for the moment that conducting a
census is practical in this example.
© 2004 Thomson/South-Western
Slide 15
Conducting a Census


Population Mean SAT Score
xi


 990
900
Population Standard Deviation for SAT Score


2
(
x


)
 i
 80
900
Population Proportion Wanting On-Campus Housing
648
p
 .72
900
© 2004 Thomson/South-Western
Slide 16
Simple Random Sampling
 Now suppose that the necessary information on
the current year’s applicants was not yet entered
in the college’s database.
 Furthermore, the Director of Admissions must obtain
estimates of the population parameters of interest for
a meeting taking place in a few hours.
 She decides a sample of 30 applicants will be used.
 The applicants were numbered, from 1 to 900, as
their applications arrived.
© 2004 Thomson/South-Western
Slide 17
Simple Random Sampling:
Using a Random Number Table

Taking a Sample of 30 Applicants
• Since the finite population has 900 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, and continue
into the fourth column as needed.
© 2004 Thomson/South-Western
Slide 18
Simple Random Sampling:
Using a Random Number Table

Taking a Sample of 30 Applicants
•
•
•
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 30 applicants for our sample.
(We will go through all of column 3 and part of
column 4 of the random number table, encountering
in the process five numbers greater than 900 and
one duplicate, 835.)
© 2004 Thomson/South-Western
Slide 19
Simple Random Sampling:
Using a Random Number Table

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
836
No. 836
. . . and so on
© 2004 Thomson/South-Western
Slide 20
Simple Random Sampling:
Using a Random Number Table

Sample Data
Random
No. Number
1
744
2
436
3
865
4
790
5
835
.
.
.
.
30
498
Applicant
Conrad Harris
Enrique Romero
Fabian Avante
Lucila Cruz
Chan Chiang
.
.
SAT
Score
1025
950
1090
1120
930
.
.
Live OnCampus
Yes
Yes
No
Yes
No
.
.
Emily Morse
1010
No
© 2004 Thomson/South-Western
Slide 21
Using Excel to Select
a Simple Random Sample

Taking a Sample of 30 Applicants
• 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 30 applicants corresponding
to the 30 smallest random numbers as our sample.
• Each of the 900 applicants has the same probability
of being included.
© 2004 Thomson/South-Western
Slide 22
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.
© 2004 Thomson/South-Western
Slide 23
Using Excel to Select
a Simple Random Sample

Value Worksheet
A
1
2
3
4
5
6
7
8
9
B
C
Applicant
SAT
On-Campus
Number
Score
Housing
1
1008
Yes
2
1025
No
3
952
Yes
4
1090
Yes
5
1127
Yes
6
1015
No
7
965
Yes
8
1161
No
Note: Rows 10-901 are not shown.
© 2004 Thomson/South-Western
D
Random
Number
0.84891
0.30181
0.35709
0.99433
0.33072
0.54548
0.46758
0.33167
Slide 24
Using Excel to Select
a Simple Random Sample

Put Random Numbers in Ascending Order
Step 1 Select cells A2:A901
Step 2 Select the Data menu
Step 3 Choose the Sort option
Step 4 When the Sort dialog box appears:
Choose Random Numbers in the
Sort by text box
Choose Ascending
Click OK
© 2004 Thomson/South-Western
Slide 25
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.
© 2004 Thomson/South-Western
Slide 26
Point Estimates

x as Point Estimator of 
x

x
i
30

29, 910
 997
30
s as Point Estimator of 
s


 (x
i
 x )2
29

163, 996
 75.2
29
p as Point Estimator of p
p  20 30  .68
Note: Different random numbers would have
identified a different sample which would have
resulted in different point estimates.
© 2004 Thomson/South-Western
Slide 27
Summary of Point Estimates
Obtained from a Simple Random Sample
Population
Parameter
Parameter
Value
 = Population mean
990
x = Sample mean
997
 = Population std.
80
s = Sample std.
deviation for
SAT score
75.2
p = Population proportion wanting
campus housing
.72
p = Sample pro-
.68
SAT score
deviation for
SAT score
© 2004 Thomson/South-Western
Point
Estimator
Point
Estimate
SAT score
portion wanting
campus housing
Slide 28
Sampling Distribution of x

Process of Statistical Inference
Population
with mean
=?
The value of x is used to
make inferences about
the value of .
© 2004 Thomson/South-Western
A simple random sample
of n elements is selected
from the population.
The sample data
provide a value for
the sample mean x .
Slide 29
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
© 2004 Thomson/South-Western
Slide 30
Sampling Distribution of x
Standard Deviation of x
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.
•  x is referred to as the standard error of the
mean.
© 2004 Thomson/South-Western
Slide 31
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.
© 2004 Thomson/South-Western
Slide 32
Sampling Distribution of x for SAT Scores
Sampling
Distribution
of x
E( x )  990
© 2004 Thomson/South-Western
x 

80

 14.6
n
30
x
Slide 33
Sampling Distribution of x for 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 +/10 of
the actual population mean  ?
In other words, what is the probability that x will be
between 980 and 1000?
© 2004 Thomson/South-Western
Slide 34
Sampling Distribution of x for SAT Scores
Step 1: Calculate the z-value at the upper endpoint of
the interval.
z = (1000 - 990)/14.6= .68
Step 2: Find the area under the curve to the left of the
upper endpoint.
P(z < .68) = .7517
© 2004 Thomson/South-Western
Slide 35
Sampling Distribution of x for SAT Scores
Cumulative Probabilities for
the Standard Normal Distribution
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
.
.
.
.
.
.
.
.
.
.
.
.5
.6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224
.6
.7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549
.7
.7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852
.8
.7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133
.9
.8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389
.
.
.
.
.
© 2004 Thomson/South-Western
.
.
.
.
.
.
Slide 36
Sampling Distribution of x for SAT Scores
Sampling
Distribution
of x
 x  14.6
Area = .7517
x
990 1000
© 2004 Thomson/South-Western
Slide 37
Sampling Distribution of x for SAT Scores
Step 3: Calculate the z-value at the lower endpoint of
the interval.
z = (980 - 990)/14.6= - .68
Step 4: Find the area under the curve to the left of the
lower endpoint.
P(z < -.68) = P(z > .68)
= 1 - P(z < .68)
= 1 - . 7517
= .2483
© 2004 Thomson/South-Western
Slide 38
Sampling Distribution of x for SAT Scores
Sampling
Distribution
of x
 x  14.6
Area = .2483
x
980 990
© 2004 Thomson/South-Western
Slide 39
Sampling Distribution of x for SAT Scores
Step 5: Calculate the area under the curve between
the lower and upper endpoints of the interval.
P(-.68 < z < .68) = P(z < .68) - P(z < -.68)
= .7517 - .2483
= .5034
The probability that the sample mean SAT score will
be between 980 and 1000 is:
P(980 <
x < 1000) = .5034
© 2004 Thomson/South-Western
Slide 40
Sampling Distribution of x for SAT Scores
Sampling
Distribution
of x
 x  14.6
Area = .5034
980 990 1000
© 2004 Thomson/South-Western
x
Slide 41
Relationship Between the Sample Size
and the Sampling Distribution of x
 Suppose we select a simple random sample of 100
applicants instead of the 30 originally considered.
 E(x ) =  regardless of the sample size. In our
example, E( x) remains at 990.
 Whenever the sample size is increased, the standard
error of the mean  x is decreased. With the increase
in the sample size to n = 100, the standard error of the
mean is decreased to:

80
x 

 8.0
n
© 2004 Thomson/South-Western
100
Slide 42
Relationship Between the Sample Size
and the Sampling Distribution of x
With n = 100,
x  8
With n = 30,
 x  14.6
E( x )  990
© 2004 Thomson/South-Western
x
Slide 43
Relationship Between the Sample Size
and the Sampling Distribution of x
 Recall that when n = 30, P(980 <
x < 1000) = .5034.
 We follow the same steps to solve for P(980 < x < 1000)
when n = 100 as we showed earlier when n = 30.
 Now, with n = 100, P(980 <
x < 1000) = .7888.
 Because the sampling distribution with n = 100 has a
smaller standard error, the values of x have less
variability and tend to be closer to the population
mean than the values of x with n = 30.
© 2004 Thomson/South-Western
Slide 44
Relationship Between the Sample Size
and the Sampling Distribution of x
Sampling
Distribution
of x
x  8
Area = .7888
x
980 990 1000
© 2004 Thomson/South-Western
Slide 45
Sampling Distribution of p

Making Inferences about a Population Proportion
Population
with proportion
p=?
The value of p is used
to make inferences
about the value of p.
© 2004 Thomson/South-Western
A simple random sample
of n elements is selected
from the population.
The sample data
provide a value for the
sample proportion p.
Slide 46
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
© 2004 Thomson/South-Western
Slide 47
Sampling Distribution of p
Standard Deviation of p
Finite Population
p 
p(1  p) N  n
n
N 1
Infinite Population
p 
p(1  p)
n
 p is referred to as the standard error of the
proportion.
© 2004 Thomson/South-Western
Slide 48
Sampling Distribution of p
The sampling distribution of p can 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
© 2004 Thomson/South-Western
and
n(1 – p) > 5
Slide 49
Sampling Distribution of p
For values of p near .50, sample sizes as small as 10
permit a normal approximation.
With very small (approaching 0) or very large
(approaching 1) values of p, much larger samples
are needed.
© 2004 Thomson/South-Western
Slide 50
Sampling Distribution of p for the Proportion
of Applicants Wanting On-Campus Housing
For our example, with n = 30 and p = .72, the
normal probability distribution is an acceptable
approximation because:
np = 30(.72) = 21.6 > 5
and
n(1 - p) = 30(.28) = 8.4 > 5
© 2004 Thomson/South-Western
Slide 51
Sampling Distribution of p for the Proportion
of Applicants Wanting On-Campus Housing
Sampling
Distribution
of p
.72(1  .72)
p 
 .082
30
E( p )  .72
© 2004 Thomson/South-Western
p
Slide 52
Sampling Distribution of p for the Proportion
of Applicants Wanting 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 on-campus 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 .67 and .77?
© 2004 Thomson/South-Western
Slide 53
Sampling Distribution of p for the Proportion
of Applicants Wanting On-Campus Housing
Step 1: Calculate the z-value at the upper endpoint of
the interval.
z = (.77 - .72)/.082 = .61
Step 2: Find the area under the curve to the left of the
upper endpoint.
P(z < .61) = .7291
© 2004 Thomson/South-Western
Slide 54
Sampling Distribution of p for the Proportion
of Applicants Wanting On-Campus Housing
Cumulative Probabilities for
the Standard Normal Distribution
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
.
.
.
.
.
.
.
.
.
.
.
.5
.6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224
.6
.7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549
.7
.7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852
.8
.7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133
.9
.8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389
.
.
.
.
.
© 2004 Thomson/South-Western
.
.
.
.
.
.
Slide 55
Sampling Distribution of p for the Proportion
of Applicants Wanting On-Campus Housing
Sampling
Distribution
of p
 p  .082
Area = .7291
p
.72 .77
© 2004 Thomson/South-Western
Slide 56
Sampling Distribution of p for the Proportion
of Applicants Wanting On-Campus Housing
Step 3: Calculate the z-value at the lower endpoint of
the interval.
z = (.67 - .72)/.082 = - .61
Step 4: Find the area under the curve to the left of the
lower endpoint.
P(z < -.61) = P(z > .61)
= 1 - P(z < .61)
= 1 - . 7291
= .2709
© 2004 Thomson/South-Western
Slide 57
Sampling Distribution of p for the Proportion
of Applicants Wanting On-Campus Housing
Sampling
Distribution
of p
 p  .082
Area = .2709
p
.67 .72
© 2004 Thomson/South-Western
Slide 58
Sampling Distribution of p for the Proportion
of Applicants Wanting On-Campus Housing
Step 5: Calculate the area under the curve between
the lower and upper endpoints of the interval.
P(-.61 < z < .61) = P(z < .61) - P(z < -.61)
= .7291 - .2709
= .4582
The probability that the sample proportion of applicants
wanting on-campus housing will be within +/-.05 of the
actual population proportion :
P(.67 < p < .77) = .4582
© 2004 Thomson/South-Western
Slide 59
Sampling Distribution of p for the Proportion
of Applicants Wanting On-Campus Housing
Sampling
Distribution
of p
 p  .082
Area = .4582
p
.67
© 2004 Thomson/South-Western
.72
.77
Slide 60
Sampling Methods





Stratified Random Sampling
Cluster Sampling
Systematic Sampling
Convenience Sampling
Judgment Sampling
© 2004 Thomson/South-Western
Slide 61
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. a homogeneous group).
© 2004 Thomson/South-Western
Slide 62
Stratified Random Sampling
A simple random sample is taken from each stratum.
Formulas are available for combining the stratum
sample results into one population parameter
estimate.
Advantage: If strata are homogeneous, this method
is as “precise” as simple random sampling but with
a smaller total sample size.
Example: The basis for forming the strata might be
department, location, age, industry type, and so on.
© 2004 Thomson/South-Western
Slide 63
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.
© 2004 Thomson/South-Western
Slide 64
Cluster Sampling
Example: A primary application is area sampling,
where clusters are city blocks or other well-defined
areas.
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.
© 2004 Thomson/South-Western
Slide 65
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.
© 2004 Thomson/South-Western
Slide 66
Systematic Sampling
This method has the properties of a simple random
sample, especially if the list of the population
elements is a random ordering.
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
© 2004 Thomson/South-Western
Slide 67
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.
Example: A professor conducting research might use
student volunteers to constitute a sample.
© 2004 Thomson/South-Western
Slide 68
Convenience Sampling
Advantage: Sample selection and data collection are
relatively easy.
Disadvantage: It is impossible to determine how
representative of the population the sample is.
© 2004 Thomson/South-Western
Slide 69
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.
Example: A reporter might sample three or four
senators, judging them as reflecting the general
opinion of the senate.
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Judgment Sampling
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.
© 2004 Thomson/South-Western
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End of Chapter 7
© 2004 Thomson/South-Western
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