Ans: (a) $6$ (b) $\dfrac{3}{10}$
- Since there are $20$ letters, the inter-quartile range of the distribution
$\begin{array}{cl}
= & \dfrac{\text{the 15th datum} + \text{the 16th datum}}{2} – \dfrac{\text{the 5th datum} + \text{the 6th datum}}{2} \\
= & \dfrac{38 + 38}{2} – \dfrac{23 + 23}{2} \\
= & 15 \text{ g}
\end{array}$The range of the distribution
$\begin{array}{cl}
= & \text{the largest datum} – \text{the smallest datum} \\
= & 50 + w – 11 \\
= & (39 + w) \text{ g}
\end{array}$Hence, we have
$\begin{array}{rcl}
39 + w & = & 3 \times 15 \\
39 + w & = & 45 \\
w & = & 6
\end{array}$ - Note that the mode of the distribution is $38\text{ g}$.
Then the required probability
$\begin{array}{cl}
= & \dfrac{6}{20} \\
= & \dfrac{3}{10}
\end{array}$
