# The score of adults on an IQ test are approximately Normal with mean 100 and standart deviation 15. Alysha scores 135 on such a test. She scores highter than what percent of all adults? a. About 5% b. About 95% c. About 99%

Probability
The score of adults on an IQ test are approximately Normal with mean 100 and standart deviation 15. Alysha scores 135 on such a test. She scores highter than what percent of all adults?

Let X be the random variable of IQ test. Then $$\displaystyle{X}\sim{N}{\left({100},{15}^{{2}}\right)}$$
which means that $$\displaystyle\frac{{{X}-{100}}}{{15}}\sim{N}{\left({0},{1}\right)}$$
We need to find $$\displaystyle{P}{\left({X}\le{135}\right)}$$
We write this is a way to get the standard normal variable: $$\displaystyle{P}{\left({\left(\frac{{{X}-{100}}}{{15}}\right)}\le\frac{{{135}-{100}}}{{15}}\right)}={P}{\left({\left(\frac{{{X}-{100}}}{{15}}\right)}\le\frac{{7}}{{3}}\right)}$$
Let Ф denote the standard normal distribution, so $$\displaystyle{P}{\left({\left(\frac{{{X}-{100}}}{{15}}\right)}\le\frac{{7}}{{3}}\right)}=Ф{\left(\frac{{7}}{{3}}\right)}\simФ{\left({2.33}\right)}\sim{0.9901}$$