Question

When fishing off the shores of Florida, a spotted seatrout must be between 24 and 30 inches long before it can be kept; otherwise, it must be returned to the waters. In a region of the Gulf of Mexico, the lengths of the spotted seatrout that are caught, are normally distributed with a mean of 22 inches, and a standard deviation of 4 inches. What is the probability that a fisherman catches a spotted seatrout within the legal limits?

Answer 1

Answer: 0.2858

Explanation:

Given : When fishing off the shores of Florida, a spotted seatrout must be between 24 and 30 inches long before it can be kept.

In a region of the Gulf of Mexico, the lengths of the spotted seatrout that are caught, are normally distributed with
`\mu=22\text{ inches}` and
`\sigma=4\text{ inches}`.

Let x be the lengths of the spotted seatrout that are caught .

Using formula
`z=(x-\mu)/(\sigma)`

For x= 24

`z=(24-22)/(4)=0.5`

For x= 30

`z=(30-22)/(4)=2`

Then, by using the z-value table , the probability that a fisherman catches a spotted seatrout within the legal limits will be :-

`P(24<x<30)=P(0.5<z<2)\\\\=P(z<2)-P(z<0.5)\\\\=0.9772498-0.6914625\\\\=0.2857873\approx0.2858`

Hence, the probability that a fisherman catches a spotted seatrout within the legal limits = 0.2858