2 We want to learn the proportion of Pennsylvania voters that want to vote for Al Gore.
Pennsylvania has 8,448,674 registered voters. Let p denote the proportion of Pennsylvania voters that want to vote for Gore. p is the relevant parameter of interest here.
We randomly choose with replacement 3 voters from the population of registered voters, and we ask them if they plan to vote for Gore. Let Y1, Y2, and Y3 denote the answer of each of the three polled voters (1 is yes, 0 otherwise). For the benoulli distribution, we know E(Y1) = E(Y2) = E(Y3) = p and Var(Y₁) = Var(Y2) = Var(Y3) = p(1-p). Moreover Y1, Y2, and Y3 are independent.
a. Is 5Y1-4Y2 + Y3 a biased or unbiased estimator of p? Justify precisely each step in your derivation to answer that question.
b. Is Y₁ +0.25Y2 0.25Y3 a biased or unbiased estimator of p? Justify precisely each step in your derivation to answer that question.
c. Compute Var (Y1+0.25Y2-0.25Y3)
d. Show that for any real numbers a and b, aY1 + bY2+(1-a-b)Y3 is an unbiased estimator of p
e. Show that Var(aY1+bY2+(1-a-b)Y3) = p(1-p)(2a²+262 + 2ab - 2b+1)
Hint: you can use the following equality without proving it:
(1-ab)²= a² + b²+1+2ab - 2a - 2b
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