Answer to Question #332444 in Statistics and Probability for Sushi

Question

The heights of grade 11 male students are normally distributed with a mean of 65 inches and a standard deviation 2.25 inches. (a) Find the values of 45.75 inches and 72.5 inches. (b) What is the probability that a randomly chosen member of the group has height x berween 60 inches and 70 inches?

Answer 1

We have a normal distribution, $\mu=65, \sigma=2.25.$

Let's convert it to the standard normal distribution,

$z=\cfrac{x-\mu}{\sigma}.$

$\text{(a) }z_1=\cfrac{45.75-65}{2.25}=-8.56;\\ z_2=\cfrac{72.5-65}{2.25}=3.33.\\ \text{(b) }z_3=\cfrac{60-65}{2.25}=-2.22;\\ z_4=\cfrac{70-65}{2.25}=2.22.\\ P(60<X<70)=\\ =P(-2.22<Z<2.22)=\\ =P(Z<2.22)-P(Z<-2.22)=\\ =0.9868-0.0132=0.9736 \text{ (from z-table).}$