2019-I-12

$\begin{array}{rcl}
\dfrac{72 + 72}{2} – \dfrac{60 + c + 60 + c}{2} & = & 8 \\
72 – 60 – c & = & 8 \\
c & = & 4
\end{array}$

1. Note that the mean of the distribution is $69\text{ s}$. Then we have

   $\begin{array}{rcl}
   \dfrac{50 + a + 60 + 60 + \cdots + 79 + 80 + b}{20} & = & 69 \\
   a + b + 1373 & = & 1380 \\
   a + b & = & 7 \\
   a & = & 7 – b \ \ldots \unicode{x2460}
   \end{array}$

   Note that the range of the distribution exceeds $34\text{ s}$. Then we have

   $\begin{array}{rcl}
   (80 + b) – (50 + a) & > & 34 \\
   b – a & > & 4 \ \ldots \unicode{x2461}
   \end{array}$

   Sub. $\unicode{x2460}$ into $\unicode{x2461}$, we have

   $\begin{array}{rcl}
   b – (7 – b) & > & 4 \\
   2b & > & 11 \\
   b & > & \dfrac{11}{2}
   \end{array}$

   Since $a$ and $b$ are non negative integers, then we have $\left\{ \begin{array}{l} a = 0 \\ b = 7 \end{array} \right.$ or $\left\{ \begin{array}{l} a = 1 \\ b = 6 \end{array} \right.$.

2. For the least possible standard deviation, we should take $a = 1$ and $b = 6$.

   Hence, the least possible standard deviation of the distribution is $7.34\text{ s}$.