Sampling Methods and Estimators_9703
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Transcript of Sampling Methods and Estimators_9703
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Sampling Methods and Estimators
PS II
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Probability Sampling Methods Simple Random Sampling
Stratified Random Sampling
Cluster Sampling
Systematic Sampling
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Stratified Random Sampling Population is divided into Hstrata
A SRS of size nh
is drawn from each stratum h
n =nh is used to develop estimates of populationparameters
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Why Stratified Random Sampling? Leads to more precise estimates of mean when
the population within each strata is relatively homogenous butconsiderable variation exists across strata
Useful when estimates are re uired for sub o ulations
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Sample Mean for SSRS Point estimator of population mean
H
x W x=1h =
/ , population fraction in stratum
Sample mean in stratum
h h
h
W N N h
x h
=
=
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Sample Variance for SSRS
2 2 2
1
1
( ), (1 / )(1 )1st
Hh
st x h h hh h
n
Var x W n N =
=
2 2 2
1
(1 / )(1 )st
Hh
x h h h
h h
ns W n s
N==
s ma e ar ance
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Sample Population for SSRS Point estimator of population total
stX N x=
Estimated variance of the point estimator of population total
2 2 2
stx xs N s=
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Allocation Methods in SSRS If availability of resources dictate that only a sample of size n
could be drawn, then how to allocate n to each stratum?
Equal Allocation
Proportional Allocation
O timal Allocation
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Proportional Allocation The sample size in each stratum is proportional to the total
number of elements in the stratum.
hh h
Nn nW n= =
Sample size for stratum
Total sample
Population units in stratum
Total Population
h
h
n h
n
N h
N
=
=
=
=
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Allocation Methods in SSRS Optimal Allocation (Neyman Allocation)
Choose n1, n
2,,n
hto minimize Var ( ) subject to the
condition that:stx
H
The allocation that satisfies the condition above is:
1h
h
n n=
=
1
h hh H
h h
h
W
n nW
=
=
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Allocation Methods in SSRS Optimal Allocation
The sample size in each stratum be proportional to the product
of the number of population elements in stratum and the
standard deviation of the characteristic being measured in the
stratum. Typically, the population standard deviation for individual strata
is unknown so the practical utility of optimal allocation is
limited
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Allocation Methods in SSRS Stratified random sampling with proportional allocation gives
a smaller variance in the mean estimate than SRS. It can be
shown that:
21H
=
1
sp h h
hn =
Sample mean under SRS
Sample mean under stratified proportional allocation
/Population mean in stratum
Overall population mean
sp
h h
h
x
x
W N Nh
=
=
=
=
=
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Cluster Sampling Divide the population into clusters and sample all the units
from a set of randomly selected clusters
Works well when clusters are heterogeneous
Each cluster is a small-scale version of the entire population
Feasibility and Economy
Sampling in large populations
Single-stage and Multi-stage cluster sampling
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Systematic Sampling Conceptually similar to simple random sampling
Easier to take the sample than SRS Procedure
Randomly order the population units
a cu a e e samp e n erva n
Randomly start with a kth element (1k I) from the firstinterval and pick every Ith element that follows in the frame