Stratified Simple Random Sampling
(Chapter 5, Textbook, Barnett, V., 1991)
Consider another sampling method:
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Definition: Stratified Random Sample
A stratified random sample is obtained by dividing the
population elements into non-overlapping groups, called
strata and then selecting a random sample directly and
independently from each stratum.
A stratified SRS is a special case of stratified sampling
that uses SRS for selecting units from each stratum.
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Examples of stratification
1.For some types of income and expenditure surveys on
households in urban areas, states, provinces, counties,
and districts may be considered as the strata.
2. For business surveys on production, and sales,
stratification is usually based on industrial classifications
like industry type and employment size.
.
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Reasons for using stratified sampling
•
•
•
•
Allow sub-estimates: they can then be combined to
give an overall estimate, e.g. we estimate the
income level at district level as well as the whole HK.
Administrative convenience.
Allow different sampling fractions and methods: they
may be implemented in different sub-population, e.g.
small/large business, private/government housing,
urban/rural households.
More efficient estimates: if a heterogeneous
population is divided into strata that are internally
homogeneous.
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Some Notations
To estimate the population mean of a finite population, we assume
that the population is stratified, that is to say it has been divided into
k non-overlapping groups, or strata, of sizes:
The stratum means and variances are denoted by
and
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Estimation of Population Characteristics
in Stratified Populations
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Taking a stratified random sample
Sample mean and variance for ith stratum are denoted by
In each stratum, we have a sampling fraction:
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Estimating
The stratified sample mean is defined as
Here we assume the weights Wi=Ni /N is given (known).
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The mean and variance of
Note that
Since
Because it is assumed that “sampling in different strata are independent”, that is
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An unbiased estimator of
where
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Some Special cases of
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Estimator of the “pooled variance”
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Example: Advertising firm
An advertising firm conduct a sample survey to estimate the average
number of hours each week that households watch TV. The county
contains 2 towns, A and B, and a rural area. Town A is built around a
factory and contain mostly factory workers and school-aged children.
Town B is an suburb of a city and contains older residents with few
children at home.
There are 155 households in town A, 62 in town B, and 93 in the
rural area. The advertising firm interview n = 40 households with
random samples of size n1 = 20 from town A, n2 = 8 from town B,
and n3 = 12 from the rural area with proportional allocation. The
measurements of TV-viewing time in hours per week, are shown
below:
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Example: Advertising firm
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Example: Advertising firm
(a) Estimate the average TV-viewing time, in hours per
week, for all households in the county.
(b) In the study, the families of town A tend to be younger
and have more children than those of town B. Estimate
the difference between the average TV-viewing time, in
hours per week, for families of these 2 towns.
In both cases, provide an estimate of standard error for
the estimation.
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Example: Advertising firm
Solution (a): The population of households falls into 3 groups, 2
towns and a rural area with
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Example: Advertising firm
Solution (b):
Since the SRSs chosen within each stratum are independent, the
variance of the difference between 2 independent random variables
is the sum of their respective variances. The estimate of the
difference is
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Example: Estimation of the population total
(c) Estimate the total number of hours each week that households
view TV. Provide an estimate of s.e. for the estimation.
Solution:
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Simple random sampling
Stratified sampling with proportional allocation
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(a) When stratum size is large enough:

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Ni
N
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(b) When stratum size is not large enough:
The stratified sample mean will be more efficient than the s.r. sample mean
If and only if variation between the stratum means is sufficiently large
compared with within-strata variation!
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A B C
D
E
I
II
*
*
*
*
*
*
*
*
*
*
III
IV
*
*
*
*
*
*
*
*
*
*
V
*
*
*
*
*
(15 males and 10 females)
Take a stratified random sample with size 5 in each case, that is:
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var(y )  12.87
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Optimum Choice of Sample Size


To achieve required precision of estimation
Some cost limitation
The simplest form assumes that there is some overhead cost, c0 of administering
The survey, and that individual observations from the ith stratum each cost an
Amount ci. Thus the total cost is:
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I. Minimum variance for fixed cost
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I. Minimum variance for fixed cost (Cont.)
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I. Minimum variance for fixed cost (Cont.)
Then
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II. Minimum cost for fixed variance
Consider to satisfy
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for the minimum possible total cost.
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II. Minimum cost for fixed variance (Cont.)
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Given wi , ni  nwi
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Comparison of proportional allocation and
optimum allocation
Thus the extent of the potential gain from optimum (Neyman) allocation
Compared with proportional allocation depends on the variability of the
stratum variances: the larger this is, the greater the relative advantage
Of optimum allocation.
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dT  420*10%  42
d   dT / 243  0.1728
V (
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)  (0.0882) 2
z
36
For optimum allocation:
ni
wi  
n
Wi Si / ci
 (W S
i
i
/ ci )
The sample weights are about (0.527, 0.348, 0.124).
The required total sample size is now 31,
consisting of 16, 11, and 4 for each stratum.
By using simple random sampling, we will need 62 samples!
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Double sampling for stratification





Some practical considerations:
Unknown of Ni and Si2
Double sampling is a two-phase sampling.
For example, we may call many voters to identify income level
(phase 1 sample), when only a few could be interviewed (phase 2
sample) for purposes of completing a detailed questionnaire.
Quota Sampling
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Double sampling for stratification
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Post-hoc stratification
Suppose plans have been drawn up to conduct a sample survey on a
stratified population, and that stratum sizes and stratum variances are
known. However, we may not be able to determine in which stratum an
observation belongs, until it has been drawn.
For example, where strata correspond to different personal details on
people-such as their religious beliefs, income levels, and so on.
For such factors, published national reports may provide a clear
indication of stratum weights (sizes) and variances, but it can be
most difficult to sample Individuals from specific strata.
Sometimes we may have to draw our sample and stratify it subsequently:
that is, carry out a post-hoc stratification.
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Post-hoc stratification
Another possible use of post-hoc stratification is to correct “obvious
lack of representativeness” in a s.r. sample.
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Post-hoc stratification
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Conclusions on Stratified Sampling:
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Conclusions on Stratified Sampling:
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