Question

Assume the random variable x is normally distributed with mean μ=50 and standard deviation σ equals=7. find the 20h percentile.

Answer 1

The 20th percentile is the cutoff value $k$ such that $\mathbb P(X\le k)=0.20$.

Because $X\sim\mathcal N(50,7)$, you can write

$\mathbb P(X\le k)=\mathbb P\left(\frac{X-50}7\le\frac{k-50}7\right)=\mathbb P\left(Z\le\frac{k-50}7\right)=0.20$

Consult a table of $z$ scores to find the corresponding critical value and you'll find this value to be about $-0.8416$. So, you have

$\frac{k-50}7=-0.8416\implies k\approx44.11$