Solution Found!
In Example D of Section 8.4, the method of moments
Chapter , Problem 13(choose chapter or problem)
In Example D of Section 8.4, the method of moments estimate was found to be \(\hat{\alpha}=3 \bar{X}\). In this problem, you will consider the sampling distribution of \(\hat{\alpha}\).
a. Show that \(E(\hat{\alpha})=\alpha\)- that is, that the estimate is unbiased.
b. Show that \(\operatorname{Var}(\hat{\alpha})=\left(3-\alpha^{2}\right) / n\). [Hint: What is \(\operatorname{Var}(\bar{X})\)?]
c. Use the central limit theorem to deduce a normal approximation to the sampling distribution of \(\hat{\alpha}\). According to this approximation, if n = 25 and \(\alpha=0\), what is the \(P(|\hat{\alpha}|>.5)\)?
Questions & Answers
QUESTION:
In Example D of Section 8.4, the method of moments estimate was found to be \(\hat{\alpha}=3 \bar{X}\). In this problem, you will consider the sampling distribution of \(\hat{\alpha}\).
a. Show that \(E(\hat{\alpha})=\alpha\)- that is, that the estimate is unbiased.
b. Show that \(\operatorname{Var}(\hat{\alpha})=\left(3-\alpha^{2}\right) / n\). [Hint: What is \(\operatorname{Var}(\bar{X})\)?]
c. Use the central limit theorem to deduce a normal approximation to the sampling distribution of \(\hat{\alpha}\). According to this approximation, if n = 25 and \(\alpha=0\), what is the \(P(|\hat{\alpha}|>.5)\)?
ANSWER:Step 1 of 4
Given:
An i.i.d. sample of random variables 
It is given that 