Let X1, X2, X3, and X4 be a random sample of observations from a population with...

Let X1, X2, X3, and X4 be a random sample of observations from a population with mean ? and variance ?2. Consider the following two point estimators of ?:

b1= 0.30 X1 + 0.30 X2 + 0.30 X3 + 0.30 X4 and

b2= 0.20 X1 + 0.40 X2 + 0.40 X3 + 0.20 X4 .

Which of the following constraints is true?

A. Var(b1)/Var(b2)=0.76

B. Var(b1)Var(b2)

C. Var(b1)=Var(b2)

D. Var(b1)>Var(b2)

Solutions

Expert Solution

We have, Var(b1) = Var(X1) + Var(X2)

+ Var(X3) + ? Var(X4) = 4 * = 0.36 .

[Since, X1, X2 , X3 and X4 are random samples, the covariance

terms will be 0].

Var(b2) = Var(X1) + Var(X2) + Var(X3) +

Var(X4) = 0.4 .

Hence, Var(b1) < Var(b2).

Hence, Option (B) is the correct choice. (Ans).

orchestra answered 1 year ago

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