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.

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    $\begingroup$ Apply normalization and lookup in normal probability distribution table. $\endgroup$ – Pieter21 Oct 6 '14 at 18:19

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 $a$.


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