Question

In an all boys school, the heights of the student body are normally distributed with a mean of 71 inches and a standard deviation of 3.5 inches. Out of the 1707 boys who go to that school, how many would be expected to be taller than 75 inches tall, to the nearest whole number?

Answer 1

The formula for the z score of a number is given by:

`z=\frac{x-\overline{x}}{\sigma}`

Where:

`\begin{gathered} x=\text{ the observed value} \\ \overline{x}=\text{ the mean} \\ \sigma=\text{ the standard deviation} \end{gathered}`

In this case,

`\begin{gathered} x=75 \\ \overline{x}=71 \\ \sigma=\text{ 3.5} \end{gathered}`

Therefore, the z score of x=75 is given by:

`z=(75-71)/(3.5)=(4)/(3.5)\approx1.143`

Therefore, the probability that a boy is taller than 75 inches is given by the area under the normal probability distribution curve between z=1.143 and z=∞, P(z > 1.143):

The area is approximately 0.1265.

Therefore, the required probability is 0.1265.

Convert the probability to percent by multiplying with 100:

`0.1265*100=12.65`

Hence, about 12.65 % of all the boys are taller than 75 inches.

Therefore, the total number of boys that are taller than 75 inches is given by:

`(12.65)/(100)*1707\approx216`

Therefore, the number of boys expected to be taller than 75 inches is approximately:

216