Statistics 431
Home Work # 2
(Professor S. Hedayat)
Consider a finite population U consisting of 6 distinct units labeled by 1, 2,3,4,5, and 6.
A survey statistician recommends the following sampling plan d (design) for the study of
the population mean.
Sampling plan d:
Support:
Probability distribution
Over the support
s1
s2
s3
s4
{1,4,5}
{2,3,6}
{1,2,6}
{3,4,5}
1/6
1/6
1/6
3/6
1- What will be the H-T estimator of the population mean? Remember that you should
exhibit H-T estimator for each of the 4 samples.
2-Design an algorithm for implementing this sampling plan based on rolling a fair 6sided die. And roll your die and report what sample was selected. Report H-T estimator
for your selected (probability) sample. By the way, we call the selected sample a
probability sample since we used a prob. distribution over the support to select the
sample. Remark: later we shall learn that certain probability samples are called random
samples.
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3- Keep the support as it is. But change, if possible, the probability distribution over the
support so that H-T estimator of the population mean becomes sample mean for each of
the 4 samples (it might be appealing to estimate the population mean by the sample
mean).
Support:
Probability distribution
Over the support
s1
s2
s3
s4
{1,4,5}
{2,3,6}
{1,2,6}
{3,4,5}
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4- By doing problem 3 you should be able to reach a conclusion that if a sampling plan
has certain features then H-T estimator of the population mean is the sample mean for
each sample. Describe your discovery.
5- In class we learned that for any sampling plan we can construct an infinite number of
unbiased estimators of the population total (as well for the population mean). H-T
estimator is an example in this class. Construct an unbiased estimator of the population
mean based on the original sampling plan above so that it differs from H-T estimator.
Exhibit your estimator. Again recall that you should say what we should do for each of
the sample in case selected.
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