I'm trying to solve the following questions
Suppose that the lifetimes of light bulbs are independent, exponentially distributed random variables with a mean of $2000$ hours each.
1) Calculate the probability that a randomly chosen light bulb survives for more than $2500$ hours.
For this I got $e^{-2500/2000}$ (I don't know if this is needed for the other parts but thought I'd mention it).
I'm stuck on the other two parts
Suppose that the bulbs come in boxes of $100$.
2) Approximately calculate the probability that the average lifetime of all the bulbs in a particular box exceeds $2500$ hours.
3) Approximately calculate the probability that the sum of the lifetimes of all the bulbs in a particular box exceeds $220,000$ hours.
I know I need to use the central limit theorem somehow for both so I need to calculate the mean and variance for both parts. For number 2, I have no clue what to do. For number 3, I calculated:
$\text{mean} = 2000\times100=200000$
$\text{variance} = 2000\times100^2=20000000$
Then I did
$$P\left(z>\frac{220000-\text{mean}}{\text{sq. root of variance}}\right) = p(z>4.47)$$
but since you can't find probabilities for values greater than $3$ in the $z$ table, I got stuck and assumed I did it wrong.
So I really don't know how to solve either. Any help would be great! Thanks.