The mean and variance of a set of 15 numbers are 12 and 14 respectively. The mean and variance of another set of 15 numbers are 14 and σ2 respectively. If the variance of all the 30 numbers in the two sets is 13, then σ2 is equal to:
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Official solution
We know that if n1,n2 are the sizes, Xˉ1,Xˉ2 are the means and σ1,σ2 are the standard deviation of the series, then the combine variance of the series.
\[
\[\begin{gathered} \sigma^2=\frac{n_1 \sigma_1^2+n_2 \sigma_2^2}{n_1+n_2}+\frac{n_1 \cdot n_2}{\left(n_1+n_2\right)^2}\left(\bar{X}_1-\bar{X}_2\right)^2 \\ \Rightarrow 13=\frac{15 \times 14+15 \times \sigma^2}{15+15}+\frac{15 \times 15}{(15+15)^2}(12-14)^2 \\ \Rightarrow 13=\frac{14+\sigma^2}{2}+\frac{1}{4} \times 4 \\ \Rightarrow 14+\sigma^2=2 \times 12 \\ \Rightarrow \sigma^2=10 \end{gathered}\]
\]