Solution — 2021 Question 6 Normal Distribution

Let $X$ be the height of a randomly chosen plant, in metres.

$X \sim N (μ, σ^{2})$

\begin{matrix} P (X > 1.72) & = 0.3 \\ P (Z > \frac{1.72 - μ}{σ}) & = 0.3 \\ \frac{1.72 - μ}{σ} & = 0.524 40 \end{matrix}

apply standardization and inverse norm to find mu/sigma

\begin{matrix} P (X < 1.52) & = 0.1 \\ P (Z < \frac{1.52 - μ}{σ}) & = 0.1 \\ \frac{1.52 - μ}{σ} & = - 1.2816 \end{matrix}

apply standardization and inverse norm to find mu/sigma

Cross-multiplying and rearranging,

\begin{matrix} μ + 0.5244 σ & = 1.72 & (1) \end{matrix}

\begin{matrix} μ - 1.2816 σ & = 1.52 & (2) \end{matrix}

Solving (1) and (2) with a GC, $μ = 1.66 ∎, σ = 0.111 ∎$