How to find $\Phi^{-1}(\beta)$

Question

I need to find $\Phi^{-1}(\beta)$ when $\beta=0.1$ (or any number but for example) but I'm not quite sure how to find it using the normal table inversely like this. I've tried googling and looking through notes and while it should be relatively simple I'm a bit lost. Can anyone provide assistance?

Answer 1

There are several versions of the standard normal table, and I do not know which one you are using.

The most common version gives $\Phi(z)$ for $z$ ranging from $z=0$ to about $z=3.5$. You can check whether you are using this sort of table by looking up the entry under $z=0$. It should be $0.5$.

If we look in the body of this table, we find that $\Phi(z)=0.9$ at $z\approx 1.28$. So the area in the right tail from $z=1.28$ on is about $0.1$.

By symmetry, the area in the left tail from $-\infty$ to $-1.28$ is about $0.1$. Thus $\Phi(0.1)\approx -1.28$.