The first of two samples has 100 items with mean 15 and SD = 3. If the whole group has 250 items with mean 15.6 and $\mathrm{SD}=\sqrt{13.44}$, the SD of the second group is

Solution:

We know ,${\mathrm{\sigma }}^2=\frac{{\mathrm{n}}_1\left({\mathrm{\sigma }}_1^2+{\mathrm{d}}_1^2\right)+{\mathrm{n}}_2\left({\mathrm{\sigma }}_2^2+{\mathrm{d}}_2^2\right)}{{\mathrm{n}}_1+{\mathrm{n}}_2}$

where ${\mathrm{d}}_1={\mathrm{m}}_1-\mathrm{a},{\mathrm{d}}_2={\mathrm{m}}_2-\mathrm{a},\mathrm{a}$  being the mean of the whole group.

$\begin{array}{l}\therefore 15.6=\frac{100\times 15+150\times {\mathrm{m}}_2}{250}\Rightarrow {\mathrm{m}}_2=16\\ \text{ Thus, }13.44=\frac{\left[\begin{array}{c}\left(100\times 9+150\times {\mathrm{\sigma }}^2\right)+\\ 100\times (0.6{)}^2+150\times (0.4{)}^2\end{array}\right]}{250}\Rightarrow \mathrm{\sigma }=4\end{array}$