I am 13 years old trying to teach myself about standard deviation and was wondering how this problem would look like. I know I am young to be learning this but I was reading about this and got interested. I understand it a little but I wanted to see more of a solution to this problem. I have an idea of what the deviation is like, but I would like to see the written out solution.
In a certain school, the heights of the population of girls are normally distributed, with a mean of $63$ inches and a standard deviation of 2 inches. If there are $450$ girls in the school, determine how many of the girls are shorter than 60 inches. Round the answer to the nearest integer.
You need to use a calculator, there's no way to do this by hand. If the standard deviation is $\sigma = 2$ inches, and you want to know how many girls are shorter than $60$ inches, then since the mean is $63$ inches, then since $63-60 = 3$, then you want to know how many girls are at most $-1.5$ standard deviations below the mean. So you need to use a calculator (or look it up in a table) that tells you what proportion corresponds to $<-1.5 \sigma$. For example, you may use this website.
If you use that website and put in $z = -1.5$ and click "calculate", it tells you that the "one-tailed" probability for $z$ is $0.0668$. This means that the probability that somebody is less than $-1.5$ times the standard deviation below the mean is $0.0668$. Then you multiply this probability by the total number of girls, so you get
$$ 450 \times 0.0668 = 30.06 \approx 30$$