The mean weight of 150 students is 60 Kgs. The mean weight of boys is 70 and that of girls is 55 with standard deviations 10 and 15 respectively. The combined variance is

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$\frac{700}{3}$

$\frac{300}{7}$

$\frac{500}{3}$

$\frac{400}{7}$

answer is A.

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Detailed Solution

$n = n_{1} + n_{2} = 150 (n_{1}$ is number of boys $n_{2}$ is number
of girls)

$\begin{array}{l} 60 = \frac{n_{1} {\bar{x}}_{1} + n_{2} {\bar{x}}_{2}}{n_{1} + n_{2}} \Rightarrow 70 n_{1} + 55 n_{2} = 150 \times 60 = 9000 \\ \Rightarrow 14 n_{1} + 11 n_{2} = 1800 \\ \Rightarrow 14 n_{1} + 11 (150 - n_{1}) = 1800 \\ \Rightarrow n_{1} = 50 so n_{2} = 100 \\ d_{1} = {\bar{x}}_{1} - x = 70 - 60 = 10 \\ d_{2} = {\bar{x}}_{2} - x = 55 - 60 = - 5 \\ σ^{2} = \frac{1}{150} (n_{1} (σ_{1}^{2} + d_{1}^{2}) + n_{2} (σ_{2}^{2} + d_{2}^{2})) \\ = \frac{1}{150} (50 (100 + 100) + 100 (225 + 25)) \\ = \frac{700}{3} . \end{array}$