# 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.

Jul 17, 2016

$\approx .532$

#### Explanation:

Via the Law of Eexpectations we know the following

Let ${S}_{n} = {X}_{1} + {X}_{2} + \ldots + {X}_{n}$ be an independent trail process, then

$E \left({S}_{n}\right) = n \mu = 100$
$\sigma \left({S}_{n}\right) = \sigma \sqrt{n} = 8.9442719099992$

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

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