Find $x$ such thtat $P(|X-3|<x)=0.99$

Question

Given:$ \mu =3$ and $\sigma =2$

With my calculation I reached, $P(Z<\frac{x}{2})=0.995$

The problem is that in the table there are two close values to 0.995, which are 0.9951 and 0.9949

Which one should I choose ?

Answer 1

Your equation is $\frac{x}2=\Phi^{-1}(0.995)$, right? Then you are right that the most tables show

$\Phi(2.57)=0.9949 $ and $\Phi(2.58)=0.9951$

Here you can apply the method of linear interpolation:

$\frac{x}{2}\approx 2.57+\frac{2.58-2.57}{0.9951-0.9949}\cdot (0.995-0.9949)=2.575$

Here it is the same as calculating the mean of the two z-values.