Approximating distribution and delta method

Question

Suppose i.i.d r.v $X$ has density of $\frac{3x+1}{8}$ on interval $(0,2)$, define $Y = \pi X^2$. Questions are:

1. If $Z = \pi \bar{X}^2$, where $\bar{X}$ is arithmetic mean of all $X_s$

2. If $\bar{Y}$ defined to be arithmetic mean of all $Y_s$

For question 1, we could find the approximate distribution of $Z$ using CLT and Delta method. For question 2, we could also find the approximate distribution of $\bar{Y}$.

Will these two be the same in the end? Why or why not?

Thanks!