Sampling Methods and Estimators_9703

8/3/2019 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

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Sampling Methods and Estimators_9703