Delta Method for the Log of a Gamma Sample Mean

Let $X_1, \ldots, X_n$ be i.i.d.\ $\mathrm{Gamma}(2, 1)$ (shape $2$, rate $1$), so $E[X_i] = 2$ and $\mathrm{Var}(X_i) = 2$.

Define $W_n = \ln(\bar{X}_n)$.

(a) Using the delta method, determine the asymptotic distribution of $\sqrt{n}(W_n - \ln 2)$.

(b) For $n = 200$, approximate $P(W_n < 0.6)$.

You may use $\ln 2 \approx 0.6931$ and $\Phi(-1.86) \approx 0.0314$.