Question

A random variable ℎ has a normal distribution with mean 7 and standard deviation
2.Calculate the probability that :
a)ℎ > 9
b)ℎ < 6
c)5 < ℎ < 8

Answer 1

We are given,

$\mu=7$

$\sigma=2$

To find these probabilities, the values are standardized and the probabilities read from the standard normal tables as below.

a.

$p(h\gt9)=p(((h-\mu)/\sigma)\gt(9-\mu)/\sigma)=p(Z\gt(9-7)/2)=p(Z\gt1)$

This can also be written as,

$1-p(Z\lt1)=1-0.8413=0.1587$

Therefore $p(h\gt9)=0.1587$

b.

$p(h\lt6)=p(((h-\mu)/\sigma)\lt(6-\mu)/\sigma)=p(Z\lt(6-7)/2)=p(Z\lt-0.5)$

From the standard normal tables,

$p(Z\lt-0.5)=0.3085$

Hence, $p(h\lt6)=0.3085$

c.

$p(5\lt h\lt8)$. On standardizing this we have,

$p((5-7)/2\lt Z\lt(8-7)/2)=p(-1\lt Z\lt 0.5)$

This probability can be written as,

$p(Z\lt0.5)-p(Z\lt-1)=0.6915-0.1587=0.5328$

Therefore, $p(5\lt h\lt 8)=0.5328$ .