Normal Distribution Third Step

Question

I am getting stuck at the 3rd stage in a process, of calculating based on Normal Distribution. The following equations are used:

1. $N(μ,σ)$
2. $f(x) = P(x \leq 3)$
3. $z = (x - μ) /σ$
4. $N(8, 3)$

So, I have calculated as follows:

$\mu= 8, σ = 3, z = (3 - 8 / 3)\simeq -1.67$

But, then I get completely stuck. As next answer, must be: $0.0475$ and $1 - z$ does not give this value. How, can I arrive at that value?

EDIT: $z-1.67 = 0.0475$, BUT how??

EDIT 2:

Where does '0.9525'? I do find the '0.0475' in the Normal Probability Table, looking up z-score, "Areas Under Normal Curve"

Answer 1

This notation means that $\sigma^2 = 3$, thus $\sigma = \sqrt{3}$.

$$P(X \leq 3)= P\left(\frac{X-8}{\sqrt{3}}\leq \frac{-5}{\sqrt{3}}\right)= P(Z \leq \frac{-5}{\sqrt{3}})= P(Z \leq -2.88675...)$$

where $Z \sim N(0,1)$.