Solution — 2017 Question 6 Normal Distribution
Let be the height, in m, of a randomly chosen adult male.
$
ParseError: Can't use function '$' in math mode at position 1:
$̲ X \sim \mathrm{N} \left( \mu, \sigma^2 \right)
\begin{align*} \mathrm{P}\left(X < 1.6\right) &= 0.2 \\ \mathrm{P}\left(Z < \frac{1.6 - \mu}{\sigma}\right) &= 0.2 \\ \frac{1.6 - \mu}{\sigma} &= -0.841\,62 \end{align*}$$ ParseError: {align*} can be used only in display mode.
[apply standardization and inverse norm to find mu/sigma]
\begin{align*}
\mathrm{P}\left(X > 1.75\right) &= 0.3
\\ \mathrm{P}\left(Z > \frac{1.75 - \mu}{\sigma}\right) &= 0.3
\\ \frac{1.75 - \mu}{\sigma} &= 0.524\,40
\end{align*}$$
[apply standardization and inverse norm to find mu/sigma]
Cross-multiplying and rearranging,
\begin{align*}
\mu - 0.8416\sigma &= 1.6 \tag{1}
\end{align*}$$
\begin{align*}
\mu + 0.5244\sigma &= 1.75 \tag{2}
\end{align*}$$
Solving (1) and (2) with a GC,
\begin{align*}
\text{Mean} &= 1.6924
\\ &= 1.69\; \QED \text{ (to 3 s.f.)}
\end{align*}$$
\begin{align*}
\text{Variance} &= 0.109\,81^2
\\ &= 0.0121\; \QED \text{ (to 3 s.f.)}
\end{align*}$$