**Unformatted text preview: **Sampling and Sampling Distributions
AS&W – Chapter 7 p
Sampling Distribution of
Properties of Point Estimators Other Sampling Methods 1 Sampling Distribution ofp Making Inferences about a Population Proportion Population
with proportion
p=? The value of p is used
to make inferences
about the value of p. A simple random sample
of n elements is selected
from the population. The sample data
provide a value for
p
the
sample
proportion .
2 Sampling Distribution ofp
p
The sampling distribution of
is the probability
distribution of all possible values of the sample
proportion p .
Expected Value ofp E( p) p
where:
p = the population proportion 3 Sampling Distribution ofp
p
Standard Deviation of
Finite Population
p(1 p) N n
p n
N 1 Infinite Population
p(1 p)
p n p is referred to as the standard error of the
proportion.
• A finite population is treated as being
infinite if n/N < .05. 4 Form of the Sampling Distribution ofp p
The sampling distribution of
can be approximated
by a normal 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 5 Form of the Sampling Distribution ofp
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. 6 Sampling Distribution ofp Example: Robert’s College
Recall that 72% of the
prospective students applying
to Robert’s College desire
on-campus housing.
What is the probability that a simple random sample of 30 applicants will provide an estimate of the population proportion of applicant
desiring on-campus housing that is within plus or
minus .05 of the actual population proportion?
7 Sampling Distribution ofp
For our example, with n = 30 and p = .72,
the normal distribution is an acceptable
approximation because:
np = 30(.72) = 21.6 > 5
and
n(1 - p) = 30(.28) = 8.4 > 5
• The sample of 30 is treated as being infinite
since n/N = 30/900 < .05. 8 Sampling Distribution ofp Sampling
Distribution
of p E(p).72 .72(1 .72)
p .082
30 p 9 Sampling Distribution ofp Step 1: Calculate the z-value at the upper endpoint
the interval.
z = (.77 .72)/.082 = .61
Step 2: Find the area under the curve to the left of th
upper endpoint.
P(z < .61) = .7291 10 Sampling Distribution ofp
Cumulative Probabilities for
the Standard Normal Distribution
z
. .00
. .01
. .02
. .5 .6915 .6950 .6985
.6 .7257 .7291 .7324
.7 .7580 .7611 .7642
.8 .7881 .7910 .7939
.9 .8159 .8186 .8212
. . . . .03
. .04
. .05
. .06
. .7019 .7054 .7088 .7123
.7357 .7389 .7422 .7454
.7673 .7704 .7734 .7764
.7967 .7995 .8023 .8051 .07
. .08
. .09
. .7157 .7190 .7224
.7486 .7517 .7549
.7794 .7823 .7852
.8078 .8106 .8133 .8238 .8264 .8289 .8315 .8340 .8365 .8389
.
.
.
.
.
.
. 11 Sampling Distribution ofp Sampling
Distribution
of p p .082 Area = .7291 p
.72 .77 12 Sampling Distribution ofp Step 3: Calculate the z-value at the lower endpoint o
the interval.
z = (.67 .72)/.082 = - .61
Step 4: Find the area under the curve to the left of th
lower endpoint.
P(z < -.61) = P(z > .61)
= 1 P(z < .61)
= 1 . 7291
= .2709 13 Sampling Distribution ofp Sampling
Distribution
of p p .082 Area = .2709 p
.67 .72 14 Sampling Distribution ofp 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 applican
applica
wanting on-campus housing will be within +/-.05 of th
actual population proportion :
P(.67 < p< .77) = .4582 15 Sampling Distribution ofp Sampling
Distribution
of p p .082 Area = .4582 p
.67 .72 .77 16 Properties of Point Estimators Before using a sample statistic as a point estimator,
statisticians check to see whether the sample statistic has the
following properties associated with good point estimators.
Let θ be the population parameter of interest
Then
is the sample statistic or the point estimator of θ ˆ
Unbiased
Efficiency
Consistency
17 Properties of Point Estimators
Unbiased
If the expected value of the sample statistic is
equal to the population parameter being
estimated, the sample statistic is said to be an
unbiased estimator of the population parameter. ˆ
Sampling Distribution of ˆ Unbiased Estimator E (ˆ) ˆ Biased Estimator 18 Properties of Point Estimators
Efficiency
Given the choice of two unbiased estimators of
the same population parameter, we would prefer
to use the point estimator with the smaller
standard deviation, since it tends to provide
estimates closer to the population parameter.
The point estimator with the smaller standard
deviation is said to have greater relative
efficiency than the other. 19 Properties of Point Estimators
Efficiency Sampling Distribution of ˆ2 Sampling Distribution of ˆ1 ˆ First point estimator is relatively more efficient
20 Properties of Point Estimators
Consistency
A point estimator is consistent if the values
of the point estimator tend to become closer to
the population parameter as the sample size
becomes larger.
Recall the standard deviation (error)x of x n 21 Other Sampling Methods Stratified Random Sampling Cluster Sampling Systematic Sampling Convenience Sampling Judgment Sampling 22 Stratified Random Sampling
The
The population
population is
is first
first divided
divided into
into groups
groups of
of
elements
elements called
called strata
strata..
Each
Each element
element in
in the
the population
population belongs
belongs to
to one
one and
and
only
only one
one stratum.
stratum.
Best
Best results
results are
are obtained
obtained when
when the
the elements
elements within
within
each
each stratum
stratum are
are as
as much
much alike
alike as
as possible
possible
(i.e.
(i.e. aa homogeneous
homogeneous group
group).
). 23 Stratified Random Sampling
A
A simple
simple random
random sample
sample is
is taken
taken from
from each
each stratum.
stratum.
Formulas
Formulas are
are available
available for
for combining
combining the
the stratum
stratum
sample
sample results
results into
into one
one population
population parameter
parameter
estimate.
estimate.
Advantage
Advantage:: IfIf strata
strata are
are homogeneous,
homogeneous, this
this method
method
is
is as
as “precise”
“precise” as
as simple
simple random
random sampling
sampling but
but with
with
aa smaller
smaller total
total sample
sample size.
size.
Example
Example:: The
The basis
basis for
for forming
forming the
the strata
strata might
might be
be
department,
department, location,
location, age,
age, industry
industry type,
type, and
and so
so on.
on.
24 Cluster Sampling
The
The population
population is
is first
first divided
divided into
into separate
separate groups
groups
of
of elements
elements called
called clusters
clusters..
Ideally,
Ideally, each
each cluster
cluster is
is aa representative
representative small-scale
small-scale
version
version of
of the
the population
population (i.e.
(i.e. heterogeneous
heterogeneous group).
group). A
A simple
simple random
random sample
sample of
of the
the clusters
clusters is
is then
then taken.
taken
All
All elements
elements within
within each
each sampled
sampled (chosen)
(chosen) cluster
cluster
form
form the
the sample.
sample. 25 Cluster Sampling
Example
Example:: A
A primary
primary application
application is
is area
area sampling,
sampling,
where
where clusters
clusters are
are city
city blocks
blocks or
or other
other well-defined
well-defined
areas.
areas.
Advantage
Advantage:: The
The close
close proximity
proximity of
of elements
elements can
can be
be
cost
cost effective
effective (i.e.
(i.e. many
many sample
sample observations
observations can
can be
be
obtained
obtained in
in aa short
short time).
time).
Disadvantage
Disadvantage:: This
This method
method generally
generally requires
requires aa
larger
larger total
total sample
sample size
size than
than simple
simple or
or stratified
stratified
random
random sampling.
sampling. 26 Systematic Sampling
IfIf aa sample
sample size
size of
of nn is
is desired
desired from
from aa population
population
containing
containing N
N elements,
elements, we
we might
might sample
sample one
one
element
element for
for every
every N
N//nn elements
elements in
in the
the population.
population.
We
We randomly
randomly select
select one
one of
of the
the first
first N
N//nn elements
elements
from
from the
the population
population list.
list.
We
We then
then select
select every
every N
N//nn th
th element
element that
that follows
follows in
in
the
the population
population list.
list. 27 Systematic Sampling
This
This method
method has
has the
the properties
properties of
of aa simple
simple random
random
sample,
sample, especially
especially ifif the
the list
list of
of the
the population
population
elements
elements is
is aa random
random ordering.
ordering.
Advantage
Advantage:: The
The sample
sample usually
usually will
will be
be easier
easier to
to
identify
identify than
than it
it would
would be
be ifif simple
simple random
random sampling
sampling
were
were used.
used. Example
Example:: Selecting
Selecting every
every 100
100thth listing
listing in
in aa telephone
telephone
book
book after
after the
the first
first randomly
randomly selected
selected listing
listing 28 Convenience Sampling
It
It is
is aa nonprobability
nonprobability sampling
sampling technique
technique.. Items
Items are
are
included
included in
in the
the sample
sample without
without known
known probabilities
probabilities
of
of being
being selected.
selected.
The
The sample
sample is
is identified
identified primarily
primarily by
by convenience
convenience.. Example
Example:: A
A professor
professor conducting
conducting research
research might
might use
use
student
student volunteers
volunteers to
to constitute
constitute aa sample.
sample. 29 Convenience Sampling Advantage
Advantage:: Sample
Sample selection
selection and
and data
data collection
collection are
are
relatively
relatively easy.
easy.
Disadvantage
Disadvantage:: It
It is
is impossible
impossible to
to determine
determine how
how
representative
representative of
of the
the population
population the
the sample
sample is.
is. 30 Judgment Sampling The
The person
person most
most knowledgeable
knowledgeable on
on the
the subject
subject of
of the
the
study
study selects
selects elements
elements of
of the
the population
population that
that he
he or
or
she
she feels
feels are
are most
most representative
representative of
of the
the population.
population.
It
It is
is aa nonprobability
nonprobability sampling
sampling technique
technique..
Example
Example:: A
A reporter
reporter might
might sample
sample three
three or
or four
four
senators,
senators, judging
judging them
them as
as reflecting
reflecting the
the general
general
opinion
opinion of
of the
the senate.
senate. 31 Judgment Sampling
Advantage
Advantage:: It
It is
is aa relatively
relatively easy
easy way
way of
of selecting
selecting aa
sample.
sample. Disadvantage
Disadvantage:: The
The quality
quality of
of the
the sample
sample results
results
depends
depends on
on the
the judgment
judgment of
of the
the person
person selecting
selecting the
the
sample.
sample. 32 ...

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