Solution — 2024 Question 6 Normal Distribution

$$\begin{array}{cc}\mathrm{P}(\mu -1.5<X<\mu +1.5)& =0.7\\ \mathrm{P}(\frac{\mu -1.5-\mu }{\sigma }<Z<\frac{\mu +1.5-\mu }{\sigma })& =0.7\\ \mathrm{P}(-\frac{1.5}{\sigma }<Z<\frac{1.5}{\sigma })& =0.7\\ \frac{1.5}{\sigma }& =1.0364\\ 1.0364\sigma & =1.5\\ \sigma & =1.4473\\ & =1.45∎\text{ (to 3 s.f.)}\end{array}$$

$$\begin{array}{cc}\mathrm{P}(X\le 17.3)& =0.9\\ \mathrm{P}(Z\le \frac{17.3-\mu }{1.4473})& =0.9\\ \frac{17.3-\mu }{1.4473}& =1.2816\\ 17.3-\mu & =1.8548\\ \mu & =15.445\\ & =15.4∎\text{ (to 3 s.f.)}\end{array}$$

$$\begin{array}{cc}\mathrm{P}(a<X<b)& =0.98\end{array}$$