Question

CLUSTER SAMPLING WITH ESTIMATION Suppose a population of size N is divided into K- N/M groups of size M. We select a sample of size n -km the following way: » First we select k groups out of K groups by simple random sampling . We then select m units in each group selected on the first step by simple random sampling . The estimate of the population mean is the average Y of the sample. Let μί be the population average in the i-th group for i 1.2, K, and let σ e the population variance in the i-th group for i-1, 2, , K a. Show that we can write the estimator as where 1 if the i-th group is selected 0 otherwise and Yi is the sample average in the i-th group for i = 1, 2, . . . . K. Argue that t is reasonable to assume that the random variables Y1,. ..,YK are independent and independent from ., IK. Show that Y is an unbiased estimator of the population mean μ and show that the variance of Y is M- m k(M-1)m K K-k1 var(Y) b. Suggest an estimate for the quantity Is your estimate unbiased? Can you modify it to be an unbiased esti- mate?

Answer 1

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