Question

If the mean height of a group of students equals 170cm with a standard deviation of 10 cm, then the probability that a student is between 160cm and 180cm is .

Answer 1

We have, $\mu=170,\space \sigma=10$

We are required to find the probability

$p(160\lt x\lt 180)$

Since we have the values of $\mu$ and $\sigma$ and this probability can be standardized and the desired probability found using the standard normal tables as follows.

$p((160-\mu)/\sigma\lt (x-\mu)/\sigma\lt (180-\mu)/\sigma)=p((160-\mu)/\sigma\lt Z\lt (180-\mu)/\sigma)$ Substituting for $\mu$ and $\sigma$ gives,

$p((160-170)/10\lt Z\lt (180-170)/10)=p(-1\lt Z \lt 1)=\phi(1)-\phi(-1)$

From the standard normal tables, $\phi(1)=0.8413$ and $\phi(-1)=0.1587$ . Therefore, $\phi(1)-\phi(-1)=0.8413-0.1587=0.6826$

Therefore, the probability that the height of a student is between 160cm and 180cm is 0.6826.