U consisting of 5 distinct units labeled by 1, 2, 3, 4, and Consider the following sampling plan d (design) for the study of the population total, t = Y1 + Y2 + Y3 + Y4 + sampling plan d: Support: s1 s2 s3 s4 {1,5} {2,3,4} {1,2,3

(Professor S. Hedayat) In this home work we shall concentrate on (design) unbiased quadratic estimation which includes variance in the population as well as variances of the estimated parameters such as the total and the mean of the population.

Consider the following sampling plan d (design) for the study of the population total,

T = Y_{1 }+ Y_{2} + Y_{3} + Y_{4} + Y_{5}.

Sampling plan d:
Support: s_{1} s_{2} s_{3} s_{4} _{ } {1,5} {2,3,4} {1,2,3,5} {5}
Probability distribution

Over the support 1/6 1/6 1/6 3/6

1- What will be the variance of the H-T estimator of the population total? Remember that you should exhibit H-T estimator for each of the 4 samples and for each estimator you should indicate what will be the associated variance for each estimator.

2- Suppose upon the implementation of d the sample s_{2} is selected for the study. Upon the interview the data are Y_{2} = 35, Y_{3 }= 42, and Y_{4} = 23. What will be the H-T estimate of T and the related variance for these data?

3- Provide an unbiased estimate of the variance of H-T estimator of T based on the data obtained in problem 2- above.

4- Modify, if possible, the above survey design by adding another sample, s_{5}, of the smallest possible size so that under the uniform probability distribution over the new support you can unbiasedly estimate the variance of H-T estimator of T if sample s_{4} is selected for interview. Provide point estimates of H-T estimator of T and the related unbiased estimator of the variance if Y 5 = 40.

5- In problem 4 we had uniform probability over the support. Is this a necessary or just a sufficient condition to be able to carry our statistical tasks as we listed in problem 4? Justify your answer.