Starting with a population that is normally distributed with a mean of 100 and a standard deviation of 12, what is the probability of drawing a score greater than 104?
I need to know how to work on problems similar to this, my professor has not given us any examples, and I feel stuck.
Best Answer
Let random variable $X$ have normal distribution with mean $\mu$ and standard deviation $\sigma$. Then $$\Pr(X\gt a)=\Pr(X-\mu\gt a-\mu)=\Pr\left(\frac{X-\mu}{\sigma}\gt \frac{a-\mu}{\sigma}\right)=\Pr\left(Z\gt \frac{a-\mu}{\sigma}\right),$$ where $Z$ is standard normal.
In our case, we have $\mu=100$, and $\sigma=12$, and $a=104$. So we want $$\Pr\left(Z\gt \frac{4}{12}\right).\tag{1}$$
Remark: To evaluate (1) numerically, one can use software, or, if one is old-fashioned, one uses tables of the standard normal distribution. In fact we do not really need to work with the standard normal $Z$, for there are programs, and online normal distribution calculators, that will evaluate $\Pr(X\gt a)$ directly if you input $\mu$, $\sigma$, and $a$.