Question

the lengths of a certain species of fish was found to be normally distributed. the mean length is 78 cm with a standard deviation of 14 cm. in a school of 370 of these fish, how many would be longer than than 92 cm? 50 361 59 126

Explanation:

Step1: Calculate the z - score

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x = 92$ cm (the value we are interested in), $\mu=78$ cm (the mean), and $\sigma = 14$ cm (the standard deviation).
$z=\frac{92 - 78}{14}=\frac{14}{14}=1$

Step2: Find the proportion of fish longer than the z - score

Using the standard normal distribution table, the proportion of data to the left of $z = 1$ is $0.8413$. So the proportion of data to the right of $z = 1$ (fish longer than 92 cm) is $1-0.8413 = 0.1587$.

Step3: Calculate the number of fish

Multiply the proportion by the total number of fish. Let $n$ be the number of fish longer than 92 cm. $n=0.1587\times370\approx59$.

Answer:

59