3. Suppose that m the stratification variance 2 i=1 i /ni is to be minimized subject to...
Question:
3. Suppose that m the stratification variance σ2 i=1 i /ni is to be minimized subject to the m constraint i=1 ni = n.
(a) Let the minimization be taken over all real positive numbers ni, not just the integers.
the m Show that optimal ni’s satisfy ni = nσi/
i=1 σi , i = 1, . . . , m. [Hint: Use a Lagrange multiplier, see section 7.6.1.] m
(b) Show that the resulting minimum value of the variance is σ2 i=1 i /n.
(c) Let σ2 = 4, σ2 = 9, σ2 1 2 3 = 25, and n = 50. Find the optimal sample sizes n1, n2 and n3. What is the resulting variance?
(d) Continuing (c), suppose that by mistake a person used σ2 = 4, σ2 2 1 2 = 16 and σ3 = 16 instead. How would such a person allocate the sample of size 50? If those allocations were used instead of the optimal ones calculated in
c, how much higher would the resulting variance be?
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