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 (subpopulations).
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: Y_{11 }= 10, Y_{12} = 13, Y_{13} = 12, Y_{14} = 17, Y_{15 }= 08
Stratum 2: Y_{21 }= 15, Y_{22} = 11, Y_{23} = 16, Y_{24} = 12, Y_{25 }= 10, Y_{26 }= 14
Stratum 3: Y_{31 }= 14, Y_{32} = 15, Y_{33} = 15, Y_{34} = 11, Y_{35 }= 13, Y_{36 }= 16, Y_{37 }= 14

Compute the population total, mean, and variance based on these hypothetical data,

Compute the three strata totals, means, and variances,

Exhibit the symbolic relationship betweens the population total, mean, and variance with those of the strata,

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 Y_{11 }= 10, and Y_{14} = 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 Y_{22} = 11, Y_{25 }= 10, and Y_{26 }= 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 Y_{32} = 15, Y_{33} = 15, Y_{35 }= 13, and Y_{37 }= 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:
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.
