MoM of Uniform Distribution
Solution 1:
Note that we do not have $\bar{x} = \frac{z}3$. From your working,
$$\hat{z}_{MOM}= \frac{3}n\sum_{i=1}^n X_i^2$$
\begin{align} E[\hat{z}_{MON}]=\frac3n \sum_{i=1}^n E[X_i^2]=\frac3n\cdot n\left( \frac{z}3\right)=z \end{align}