Answer to Question #161686 in Statistics and Probability for rock 206

Question

Biologists gather data on a sample of fishes in a large lake. They capture, measure the length of, and release 1,000 fishes. They find that the standard deviation is 5 centimeters, and the mean is 25 centimeters. the distribution( according to histogram) is normal. approximately what percantage of fish in the lake is likely to have an length within three standard deviations of the mean?

Answer 1

The length of the fish is normally distributed with mean $25$ cm and standard deviation $5$ cm.

Let $X$ be the random variable which denote the length of the fish.

Then $\mu =25$ and $\sigma =5$

Let $Z= \frac {X-\mu}{\sigma}$ .Then $Z= \frac {X-25}{5}$

Now we find the probability of the length of fish within three standard deviation. i.e $P(X<15)$.

$\therefore P(X<15)=P(Z<\frac{15-25}{5})$

$=P(Z<-2)$

$=0.5- P(0<Z<2)$

$=0.5-0.4772$ [from normal distribution table ]

$=0.023$ (approximately)

Therefore approximately $0.23$ percent of fish in the lake have length with three standard deviation of the mean.