Join Complete Sufficient Statistic

Let X1,...,Xn be from a $\displaystyle N(\mu,4 \sigma^2)$

A) Find the Joint complete sufficient statistics for $\displaystyle \mu$ and $\displaystyle 4 \sigma^2$

B)Find an MVUE for $\displaystyle 4 \sigma^2$

C)Find an MVUE for $\displaystyle \mu^2$

a)From a previous problem we did in class, I know $\displaystyle Y_1 = \Sigma X^2_i$ and $\displaystyle Y_2 = \Sigma X_i$ are Joint complete sufficient statistics for $\displaystyle \mu$ and $\displaystyle 4 \sigma^2$

b)Let $\displaystyle Z_2 = \frac{Y_1 - Y^2_2/n}{n-1} = \frac{\Sigma(X_i - \overline{X})^2}{n-1}$ which is the sample variance.

Would $\displaystyle E(Z_2) = 4 \sigma^2$ or would it be $\displaystyle E(Z_2) = \sigma^2$

c)Let $\displaystyle Z_1 = \frac{Y_2}{n} = \overline{X}$ and $\displaystyle E(Z_1)=\mu$

$\displaystyle E(Z^2_1) = E(Z_1)^2 + Var(Z_1) = \mu^2 +\frac{4 \sigma^2}{n}$

Therefore $\displaystyle Z^2_1 - \frac{Z_2}{n}$ is an MVUE for $\displaystyle \mu^2$ according to Lehmann and Scheffe Theroem.

Did I make a mistake somewhere, and would $\displaystyle E(Z_2) = 4 \sigma^2$ or would it be $\displaystyle E(Z_2) = \sigma^2$