Question

In a Math test, the mean score is 45 and the standard deviation is 4.

Assuming normality, what is the probability that a score picked at random will lie

a. above score 50?

b. below score 38?

c. between 35 and 53?

Answer 1

$\mu=45\\\sigma=4$

Let the random variable $X$ represent a Math test score.

a)

We determine

$p(x\gt 50)=p({x-\mu\over \sigma}\gt{50-\mu\over\sigma})=p(Z\gt {50-45\over4})=p(Z\gt 1.25)=1-p(Z\lt 1.25)=1-0.8944=0.1056$

b) We find

$p(x\lt38)=p(Z\lt{38-45\over4})=p(Z\lt -1.75)=\phi(-1.75)=0.0401$  

c)

$p(35\lt x\lt 53)=p({35-45\over4}\lt Z\lt{53-45\over 4})=p(-2.5\lt Z\lt 2)=\phi(2)-\phi(-2.5)=0.9772-0.0062=0.971$