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2.5. Consider a population of size N = 3, U = {1, 2, 3). Let s; = {1, 2}, S2 =* {1,3}, $3 = {2,3}, $4 = {1, 2, 3), p(1)
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2.5. Consider a population of size N = 3, U = {1, 2, 3). Let s; = {1, 2}, S2 =* {1,3}, $3 = {2,3}, $4 = {1, 2, 3), p(1) 0.4, p(32) * 0.3, p(53) = 0.2, and p(sa) = 0.1. (a) Calculate all the r, and all the Tiki- (b) Find the value of E(ns) in two ways: (i) by a direct calculation, using the definition and (ii) by use of the formula that expresses E(n.) as a function of the fixe 2.6. Consider the population and the design in Exercise 2.5 above. Let the values of the study variable y be y; = 16, y2 = 21, and yz = 18. Then we have t = 55. (a) Compute the expected value and the variance of the n estimator from the defini- tions in Section 2.7. (b) Compute the variance of the n estimator using Result 2.8.1. (c) Compute the coefficient of variation of the a estimator. (d) Compute a variance estimate () using the variance estimator (2.8.6) for each of the four possible samples. Determine the expected value of the variance estimator in the present situation using the definition of expected value in Section 2.7.