This question is part of my exam study guide (and not for marks). I am given a solution. However, I am having trouble following it.
The solution given is:
$V[\bar{X}]=E[\bar{X}^2]-E[\bar{X}]^2$
$\Longrightarrow \frac{\sigma^{2}}{n}=E[\bar{X}^2]-\mu^2$
$ \Longrightarrow E[\bar{X}^2]= \frac{\sigma^{2}}{n} + \mu^2 $
We know $\sigma^2=1$ $ \Longrightarrow E[\bar{X}^2]= \frac{1}{n} + \mu^2$
Let $S(X)=\bar{X}^{2}-\frac{1}{n}$. It is unbiased, and a function of $\bar{X}$, the complete sufficient statistic. Therefore, it is UMVUE.
Done.
I am having trouble understanding the solution. Particularly how S(X) was chosen, and how it followed that is is UMVUE. I understand that Lehman-Scheffe was used, however I would like an explanation to help me understand exactly what went into this solution.
Thanks in advance.