Question

Suppose that the fish in my pond have mean length 20 inches with a standard deviation of 4 inches. What is the probability that in an independent random sample, the total length of 5 fish caught from this pond will be between 95 and 110 inches in length?

Assume we know nothing else about the distribution of lengths for this population of fish.

Answer 1

`~~.532`

Explanation:

Via the Law of Eexpectations we know the following

Let `S_n = X_1+X_2+...+X_n` be an independent trail process, then

`E(S_n) = nmu = 100`
`sigma(S_n) = sigmasqrt(n) = 8.9442719099992`

we know that 1 standard deviation from the mean is about `68%` so `[91,109]` is about that much so the results should be fairly close.

looking at the chart we see that from the range of `[-.5,1]` equivalent to about `[95.5,109]` is a pretty good approximation of `[95,110]`. The resulting amount is `53.2%`