Exponential distribution question for n items

Question

The length of a piece of string extruded from a machine without breakage is an exponential random variable with mean 30 metres. What is the (approximate) probability that the average length of 60 pieces will be between 28 and 34 metres?

I know how to do it if it was for a string. I do not know how to go about it with respect to the 60 strings.

For one string, between 28 and 34 I can simply integrate the pdf to get the probability.

Answer 1

Hint/outline:

1. Consider the $i$th string. Its length $X_i$ follows a _____ distribution?
2. Do you have just one string? No, you have 60. Assume they are all independent.
3. Are you interested in their sum? No, you want the average, call it $\bar X$.
4. Thus you want $P(20<\bar X< 38)$ Since you have 60 strings, then $\bar X$ is approximately normal by the CLT. So you need the mean of $\bar X$ and variance of $\bar X$. What are they?
5. Calculate like you would a regular normal distribution.