Find standard deviation in normal distribution

Question

X follows normal distribution X~(μ,σ) and P(x<20)=0.75 and P(x<10)=0.25. What is μ and σ? I know the mean is 15, however, I can't find the standard deviation. Any help will be greatly appreciated.

Answer 1

\begin{align} 0.25 = {} & \Pr(Z<-0.675) \qquad (\text{from the table}) \\[8pt] = {} & \Pr(X<10) = \Pr(X-\mu<-5) \\[8pt] = {} & \Pr\left( \frac{X-\mu} \sigma < \frac{-5}\sigma \right) \\[8pt] = {} & \Pr\left( Z < \frac{-5}\sigma \right) \\[8pt] \text{So } -0.675 = {} & \frac{-5}\sigma. \end{align}

Answer 2

For a standard normal $Z \sim N(0, 1)$, you can look up (from a table or a computer) what $z_{0.25}$ satisfies $P(Z < z_{0.25}) = 0.25$. Then just note that $\frac{X - 15}{\sigma}$ is standard normal, so you can equate this to $z_{0.25}$ and solve for $\sigma$.