Statistics 431
Home Work # 7
(Professor S. Hedayat)
In this home work we shall concentrate on survey sampling problems related to
populations whose units have been stratified and our interests are to estimate parameters
of the population itself as well as the parameters of the strata (sub-populations).
To do a good job in this assignment you should know the sampling materials in Chapters
3 and 4 in the text book along with the sampling materials on stratified sampling plans we
presented and discussed in the class. Please prepare your answer in an organized and
clean format.
The population has 18 survey sampling units which are stratified into three strata of sizes
5, 6 and 7.
Problem 1:
For the sake of learning let us pretend that we know the values of these 18 survey
variables as they are given below:
Stratum 1: Y11 = 10, Y12 = 13, Y13 = 12, Y14 = 17, Y15 = 08
Stratum 2: Y21 = 15, Y22 = 11, Y23 = 16, Y24 = 12, Y25 = 10, Y26 = 14
Stratum 3: Y31 = 14, Y32 = 15, Y33 = 15, Y34 = 11, Y35 = 13, Y36 = 16, Y37 = 14
1.1. Compute the population total, mean, and variance based on these hypothetical data,
1.2. Compute the three strata totals, means, and variances,
1.3. Exhibit the symbolic relationship betweens the population total, mean, and variance
with those of the strata,
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1.4. Evaluate your formulas in 1.3 using the computations you carried out in 1.1 and 1.2.
Problem 2:
Now assume that we do not know the values of the 18 survey variables and plan to study
the population and its strata via a stratified simple random sampling plan exhibited
below. However, upon sampling we shall know the corresponding values as listed above
in the selected probability samples.
For Stratum 1: We adopted the SRS (5, 2) plan and upon its implementation sample {11,
14} were selected. We carried the interviews and obtained Y11 = 10, and Y14 = 17.
For Stratum 2: We adopted the SRS (6, 3) plan and upon its implementation sample {22,
25, 26} were selected. We carried the interviews and obtained Y22 = 11, Y25 = 10, and Y26
= 14.
For Stratum 3: We adopted the SRS (7, 4) plan and upon its implementation sample {32,
33, 35, 37} were selected. We carried the interviews and obtained Y32 = 15, Y33 = 15, Y35
= 13, and Y37 = 14.
Use HT methodologies and carry the following estimations using the data obtained via
the survey:
2.1. Estimate the three strata totals, means, and variances,
2.2. Estimate the standard deviations of your estimates in 2.1,
2.3. Estimate the population total, mean, and variance,
2.4. Estimate the standard deviations of your estimates in 2.3.
Problem 3:
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Before the implementation of the stratified simple random sampling plan as in Problem 2
prove that your HT methodologies you will be using as you used in Problem 2 will
provide unbiased estimators of the parameters as specified in problem 2.
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