Solution — 2017 Question 6 Normal Distribution

$$\begin{array}{cc}\mathrm{P}(X<1.6)& =0.2\\ \mathrm{P}(Z<\frac{1.6-\mu }{\sigma })& =0.2\\ \frac{1.6-\mu }{\sigma }& =-0.84162\end{array}$$

$$\begin{array}{cc}\mathrm{P}(X>1.75)& =0.3\\ \mathrm{P}(Z>\frac{1.75-\mu }{\sigma })& =0.3\\ \frac{1.75-\mu }{\sigma }& =0.52440\end{array}$$

$$\begin{array}{cccc}& \mu -0.8416\sigma & =1.6& \text{(1)}\end{array}$$

$$\begin{array}{cccc}& \mu +0.5244\sigma & =1.75& \text{(2)}\end{array}$$

$$\begin{array}{cc}\text{Mean}& =1.6924\\ & =1.69∎\text{ (to 3 s.f.)}\end{array}$$

$$\begin{array}{cc}\text{Variance}& =0.109{81}^2\\ & =0.0121∎\text{ (to 3 s.f.)}\end{array}$$