Download Document

Slides Prepared by
JOHN S. LOUCKS
St. Edward’s University
© 2003 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
n = 100
Sampling Methods
n = 30
© 2003 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.
The sample results provide only estimates 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.
© 2003 Thomson/South-Western
Slide 3
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
replacement.
© 2003 Thomson/South-Western
Slide 4
Simple Random Sampling

Finite Population
• 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.
© 2003 Thomson/South-Western
Slide 5
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.
© 2003 Thomson/South-Western
Slide 6
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.
© 2003 Thomson/South-Western
Slide 7
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 point estimator of the population
mean .
s is the point estimator of the population standard
deviation .
p is the point estimator of the population proportion
p.
© 2003 Thomson/South-Western
Slide 8
Point Estimation

When the expected value of a point estimator is equal
to the population parameter, the point estimator is
said to be unbiased.
© 2003 Thomson/South-Western
Slide 9
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
© 2003 Thomson/South-Western
Slide 10
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.
© 2003 Thomson/South-Western
Slide 11
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.
© 2003 Thomson/South-Western
Slide 12
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
© 2003 Thomson/South-Western
Slide 13
Example: St. Andrew’s

Taking a Census of the 900 Applicants
• SAT Scores
• Population Mean
x


i
900
 990
• Population Standard Deviation

(x
© 2003 Thomson/South-Western
i
  )2
900
 80
Slide 14
Example: St. Andrew’s

Taking a Census of the 900 Applicants
• Applicants Wanting On-Campus Housing
• Population Proportion
648
p
 .72
900
© 2003 Thomson/South-Western
Slide 15
Example: St. Andrew’s

Take a Sample of 30 Applicants
Using a Random Number Table
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.
© 2003 Thomson/South-Western
Slide 16
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 30 applicants for our sample.
© 2003 Thomson/South-Western
Slide 17
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.
© 2003 Thomson/South-Western
Slide 18
Example: St. Andrew’s

Sample Data
Random
No. Number
1
744
2
436
3
865
4
790
5
835
.
.
30
685
Applicant
SAT Score On-Campus
Connie Reyman
1025
Yes
William Fox
950
Yes
Fabian Avante
1090
No
Eric Paxton
1120
Yes
Winona Wheeler
1015
No
.
.
.
Kevin Cossack
965
No
© 2003 Thomson/South-Western
Slide 19
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 30 applicants corresponding
to the 30 smallest random numbers as our sample.
• Each of the 900 applicants have the same
probability of being included.
© 2003 Thomson/South-Western
Slide 20
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.
© 2003 Thomson/South-Western
Slide 21
Using Excel to Select
a Simple Random Sample

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.
© 2003 Thomson/South-Western
Slide 22
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.
© 2003 Thomson/South-Western
Slide 23
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
© 2003 Thomson/South-Western
Slide 24
Example: St. Andrew’s

Point Estimates
Note: Different random numbers would have
identified a different sample which would have
resulted in different point estimates.
© 2003 Thomson/South-Western
Slide 25
Sampling Distribution of x

Process of Statistical Inference
Population
with mean
=?
The value of x is used to
make inferences about
the value of .
© 2003 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 26
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
© 2003 Thomson/South-Western
Slide 27
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.
© 2003 Thomson/South-Western
Slide 28
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.
© 2003 Thomson/South-Western
Slide 29
Example: St. Andrew’s

Sampling Distribution of
x
for the SAT Scores

80
x 

 14.6
n
30
E ( x )    990
© 2003 Thomson/South-Western
x
Slide 30
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 980 and 1000?
© 2003 Thomson/South-Western
Slide 31
Example: St. Andrew’s

Sampling Distribution of
x
for the SAT Scores
Sampling
distribution
of x
Area = ?
x
980 990 1000
© 2003 Thomson/South-Western
Slide 32
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 = (.2518)(2) = .5036
The probability is .5036 that the sample mean will
be within +/-10 of the actual population mean.
© 2003 Thomson/South-Western
Slide 33
Example: St. Andrew’s

Sampling Distribution of
Area = 2(.2518) = .5036
x
for the SAT Scores
Sampling
distribution
of x
x
980 990 1000
© 2003 Thomson/South-Western
Slide 34
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
© 2003 Thomson/South-Western
Slide 35
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.
© 2003 Thomson/South-Western
Slide 36
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
and
n(1 – p) > 5
© 2003 Thomson/South-Western
Slide 37
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 large
(approaching 1) values of p, much larger samples are
needed.
© 2003 Thomson/South-Western
Slide 38
Example: St. Andrew’s

Sampling Distribution of p for In-State Residents
The normal probability distribution is an
acceptable approximation because:
np = 30(.72) = 21.6 > 5
and
n(1 - p) = 30(.28) = 8.4 > 5.
© 2003 Thomson/South-Western
Slide 39
Example: St. Andrew’s

Sampling Distribution of p for In-State Residents
.72(1  .72)
p 
 .082
30
E( p )  .72
© 2003 Thomson/South-Western
Slide 40
Example: St. Andrew’s

Sampling Distribution of p for In-State Residents
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 .67 and .77?
© 2003 Thomson/South-Western
Slide 41
Example: St. Andrew’s

Sampling Distribution of p for In-State Residents
Sampling
distribution
of p
Area = ?
0.67 0.72 0.77
© 2003 Thomson/South-Western
p
Slide 42
Example: St. Andrew’s

Sampling Distribution of p for In-State Residents
For z = .05/.082 = .61, the area = (.2291)(2) = .4582.
The probability is .4582 that the sample proportion
will be within +/-.05 of the actual population
proportion.
© 2003 Thomson/South-Western
Slide 43
Example: St. Andrew’s

Sampling Distribution of p for In-State Residents
Area = 2(.2291) = .4582
Sampling
distribution
of p
0.67 0.72 0.77
© 2003 Thomson/South-Western
p
Slide 44
Sampling Methods





Stratified Random Sampling
Cluster Sampling
Systematic Sampling
Convenience Sampling
Judgment Sampling
© 2003 Thomson/South-Western
Slide 45
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.
Formulas are available for combining the stratum
sample results into one population parameter
estimate.
© 2003 Thomson/South-Western
Slide 46
Stratified Random Sampling


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, etc.
© 2003 Thomson/South-Western
Slide 47
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.
… continued
© 2003 Thomson/South-Western
Slide 48
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.
Example: A primary application is area sampling,
where clusters are city blocks or other well-defined
areas.
© 2003 Thomson/South-Western
Slide 49
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
© 2003 Thomson/South-Western
Slide 50
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.
© 2003 Thomson/South-Western
Slide 51
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.
© 2003 Thomson/South-Western
Slide 52
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.
© 2003 Thomson/South-Western
Slide 53
End of Chapter 7
© 2003 Thomson/South-Western
Slide 54