Question

Scores on a test can be approximated with a normal distribution. If the average score was an 30 and the standard deviation was 4, find the probability that a randomly chosen test has a score of a) Below 40          b) between 30 and 35

Answer 1

We have a normal distribution, $\mu=30, \sigma=4.$

Let's convert it to the standard normal distribution, $z=\cfrac{x-\mu}{\sigma}.$

a) $z=\cfrac{40-30}{4} =2.5;$

$P(X<40)=P(Z<2.5)=$

$=0.9938$ ​ (from z-table).

b)

$z_1=\cfrac{30-30}{4}=0, \\ z_2=\cfrac{35-30}{4}=1.25,$

$P(30<X<35)=\\P(0<Z<1.25)=\\P(Z<1.25)-P(Z<0)=$

$=0.8944-0.5=0.3944$ (from z-table).