$P(E(X))$ for multiple IID Gamma Distributions

Question

Losses relating to a type of insurance policy follow a gamma distribution with mean $30,000$ and $\alpha = 2$. For a sample of $100$ policy claims, calculate an approximate probability that the mean loss size is less than $27,000$. Answer $= 0.0787$ but I don't know why or how.

Answer 1

Outline: 1) Let $X$ be the sample mean. Look up the variance $\sigma^2$ of the gamma. 2) Then $X$ has mean $30000$, variance $\frac{\sigma^2}{100}$. 3) Use the normal approximation to $X$ to approximate the probability that $X$ is less than $27000$.