# The lengths of a certain species of fish were found to be normally distributed. The mean length is 93 cm with a standard deviation of 12 cm. In a school of 360 of these fish, about how many would be longer than 81 cm?

Apr 21, 2017

Number of fish having a length greater than 81 cm out of 360 $= 304$

#### Explanation:

Given -
Mean length of fish $= \overline{x} = 93$ cm
SD $\sigma = 12 c m$

Number of fish having a length greater than 81 cm out of 360.

Find the value of $z$ at $x = 81$

$z = \frac{x - \overline{x}}{\sigma} = \frac{81 - 93}{12} = \frac{- 12}{12} = - 1$

Find the Area between $z = - 1$ and $z = \infty$

Area between $z = - 1$ and $z = \infty$ = Area between $z = - 1$ and $z = 0 +$ Area between $z = 0$ and $z = \infty$
Area between[ $z = - 1$ and z=oo]= 0.3413 + 0.5 = 0.8413

Number of fish having a length greater than 81 cm out of 360 $= 0.8413 \times 360 = 303.868 = 304$