We would like to select n₁ and n₂ to minimize f(n₁, n2) subject to the constraint g(n1, n2) = B2 4' where B is a desired error bound. (a) Define (4) Νισι h(n1, n2) = (√n₁₁ - N101) 2 + (√22 - N202) 2 N√√√n2 (5) Here is an expression not depending on n₁ and n2 and which will be specified later. Show that h(n1, n2) = f(n1, n2) - AA + X² (B² + 10 + 2012). N2 (6) for some expression A not depending on n₁ and (b) Show that f(nu,2) > A4 – A( + Nơi Noi), - N101 + N202), (7) and that equality is achieved in (7) when n1 = N√√√c₁ AN₁₁ and n₂ = AN202 N√√√C2 We have shown in (8) that n; is proportional to Noi √ci To find out what you should choose, substitute (8) into (2) and (4). 3. Recall the formula in page 8 of Lecture 6. If we would like to minimize V(st) for a fixed cost, then C). ni = n Niσi/√√ci Σκαι Νεσκ/VC k=1 (1) Here n is the sample size of stratum i, N; is the population size, σ; is the standard deviation and c; is the cost of one sample. n = n₁ + ··· + nL is the total sample size. What (1) is saying is that we should select ni proportional to Niσi/√√ci. This exercise is to prove the above statement. Consider for concreteness L = 2. The estimator is Yst = 1 + 2. №1 N² Assuming that N₁ - 1≈ Ni as in the lecture notes, g(n1, n2) = Var(yst) = Total cost f(n1, n2) = n1c1 +n2c2. N2n1 No 1 (1 - 1) + №2012 (1),(2) (3)

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