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STAT 506 - Sampling Theory and Methods
•
ANGEL
•
Department of Statistics
•
Eberly College of Science
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Lesson 3: Unequal Probability Sampling
3.2 The Hansen-Hurwitz estimator
Submitted by gfj100 on Tue, 12/01/2009 - 09:37
Unit Summary
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Hansen-Hurwitz estimators (sampling with replacement)
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How to random sample with unequal probability (sampling with replacement)
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Compute the Hansen-Hurwitz estimator
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When and how to use p.p.s.
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Palm tree total example
Sampling is with replacement.
Think About It!
Come up with an answer to this question by yourself and then click on the icon to the left to
reveal the solution.
Why do we use or talk about sampling with replacement?
When sampling with replacement, the variances tend to be larger. However, formula for replacement are
simpler and easier to derive. When the sample size is small compared to
N
, with and without replacement
are not too different. We often use the easier formula derived for the with replacement to approximate
that for the without replacement.
Let
p
i
,
i
= 1, .
.. ,
N
denote the probability that a given population unit will be selected.
The Hansen-Hurwitz estimator is:

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** preview**
has intentionally

Since,
where
= the population total
thus,
and
is an unbiased estimator for τ.
Since
,
An unbiased estimator for
is:
and an approximate (1 - α) 100% confidence interval for τ is:
For population mean μ = τ/
N
one uses:
How do we perform unequal probability sampling according to given
p
i
?
Example 1
Estimate the total number of computer help requests for last year in a large firm.
The director of computer support department plans to sample 3 divisions of a large firm that has 10
divisions, with varying numbers of employees per division. Since number of computer support requests

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3.2 The Hansen-Hurwitz estimator STAT 506 - Sampling Theory and Methods

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