Solution — 2021 Question 6 Normal Distribution
Let be the height of a randomly chosen plant, in metres.
$
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 > 1.72\right) &= 0.3 \\ \mathrm{P}\left(Z > \frac{1.72 - \mu}{\sigma}\right) &= 0.3 \\ \frac{1.72 - \mu}{\sigma} &= 0.524\,40 \end{align*}$$ ParseError: {align*} can be used only in display mode.
[apply standardization and inverse norm to find mu/sigma]
\begin{align*}
\mathrm{P}\left(X < 1.52\right) &= 0.1
\\ \mathrm{P}\left(Z < \frac{1.52 - \mu}{\sigma}\right) &= 0.1
\\ \frac{1.52 - \mu}{\sigma} &= -1.2816
\end{align*}$$
[apply standardization and inverse norm to find mu/sigma]
Cross-multiplying and rearranging,
\begin{align*}
\mu + 0.5244\sigma &= 1.72 \tag{1}
\end{align*}$$
\begin{align*}
\mu - 1.2816\sigma &= 1.52 \tag{2}
\end{align*}$$
Solving (1) and (2) with a GC,