$X \sim N(5, \sigma^2$). If $P(X < -1) = 0.1587$, what is the standard deviation $\sigma$ of $X$?

Question

$X \sim N(5, \sigma^2$). If $P(X < -1) = 0.1587$, what is the standard deviation $\sigma$ of $X$?

Standardizing,

$P(\frac{X - 5}{\sigma} < \frac{-1 - 5}{\sigma}) = 0.1587$

$P(Z < \frac{-6}{\sigma}) = 0.1587$

Now, $P(Z < -1) = 0.1587$,

Hence $\frac{-6}{\sigma} = -1$

I.e. $\sigma = 6$

Is that correct?

Answer 1

It's correct. Note that you are using that the CDF of $Z$, $F_Z(z)=P(Z\leq z)$, is injective to conclude that $$ F_Z(-1)=F_Z(-6/\sigma) \;\;\Longrightarrow \;\; -1=-6/\sigma. $$