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    Population Analysis

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    27. A random sample of n observations is taken from a population... SEE ATTACHMENT

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    a) since u^ = C1 X1 + C2 X2 + ... + Cn Xn
    then the expected value of u^ is:
    E(u^)= E( C1 X1 + C2 X2 + ... + Cn Xn )
    = E(C1 X1) + E(C2 X2) + ... + E(Cn Xn )
    = C1 E(X1) + C2 E(X2) + ... + Cn E(Xn )

    also, because E(Xi) = u for all i = 1, 2, ... n
    Then E(u^)= C1* u + C2*u + ... + Cn*u
    = (C1 + C2 + ... + Cn) u
    so for u^ to be unbiased, we need E(u^) = u, or
    (C1 + C2 + ... + Cn) u = u
    therefore, (C1 + C2 + ... + Cn) = 1 is the necessary and sufficient condition for u^ to be ...

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