Ans: (a) $a=4$, $b=21$, $c=3$ (b) Mean $=3.276\text{ kg}$, standard deviation $=0.299\text{ kg}$
- Note that the frequency of the class $2.6\text{ kg}-2.8\text{ kg}$ is equal to the cumulative frequency of weight less than $2.85\text{ kg}$. Hence, we have $a=4$.
Consider the cumulative frequency of weight less than $3.45\text{ kg}$.
$\begin{array}{rcl}
a + 12 + b & = & 37 \\
4 + 12 + b & = & 37 \\
b & = & 21
\end{array}$Since the total frequency is $50$, then we have
$\begin{array}{rcl}
a + 12 + b + 10 + c & = & 50 \\
4 + 12 + 21 + 10 + c & = & 50 \\
c & = & 3
\end{array}$ - The estimate of the mean
$\begin{array}{cl}
= & \dfrac{4 \times 2.7 + 12 \times 3.0 + \cdots + 3 \times 3.9}{50} \\
= & 3.276 \text{ kg}
\end{array}$The estimate of the standard deviation
$\begin{array}{cl}
= & \sqrt{ \dfrac{4(2.7-3.276)^2 +\cdots + 3 (3.9-3.276)^2}{50}} \\
= & 0.299~038~459 \\
\approx & 0.299 \text{ kg}
\end{array}$
