2023-II-44

Let $\mu$ and $\sigma$ be the mean and the standard deviation of the distribution.

$\{\begin{array}{ll}{\displaystyle \frac{46-\mu }{\sigma }}=-3& \dots \text{①}\\ {\displaystyle \frac{x-\mu }{\sigma }}=1& \dots \text{②}\\ {\displaystyle \frac{86-\mu }{\sigma }}=2& \dots \text{③}\end{array}$

$\text{③}÷\text{①}$, we have

$\begin{array}{rcl}{\displaystyle \frac{86-\mu }{46-\mu }}& =& {\displaystyle \frac{2}{-3}}\\ -3(86-\mu )& =& 2(46-\mu )\\ -258+3\mu & =& 92-2\mu \\ 5\mu & =& 350\\ \mu & =& 70\end{array}$

Sub. $\mu =70$ into $\text{③}$, we have

$\begin{array}{rcl}{\displaystyle \frac{86-70}{\sigma }}& =& 2\\ \sigma & =& 8\end{array}$

Sub. $\mu =70$ and $\sigma =8$ into $\text{②}$, we have

$\begin{array}{rcl}{\displaystyle \frac{x-70}{8}}& =& 1\\ x-70& =& 8\\ x& =& 78\end{array}$