• # question_answer The first of the two samples has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 156 and standard deviation $\sqrt{13.44},$ then the standard deviation of second group is A)  2                             B)  4       C)  3                             D)  5

Here we are given ${{n}_{1}}=100,{{\overline{x}}_{1}}=15$and${{\sigma }_{1}}=3$ ${{n}_{1}}+{{n}_{2}}=250,\overline{x}=15.6$and$\sigma =\sqrt{13.44}$ we want ${{\sigma }_{2}}$obviously, ${{n}_{2}}=150,$ Now$\overline{x}=\frac{{{n}_{1}}{{\overline{x}}_{1}}+{{n}_{2}}{{\overline{x}}_{2}}}{{{n}_{1}}+{{n}_{2}}}$ $15.6\times 250=100\times 15+150\times {{\overline{x}}_{2}}$ $\Rightarrow$${{\overline{x}}_{2}}=16$ Hence${{d}_{1}}-{{\overline{x}}_{1}}-\overline{x}=-0.6,{{d}_{2}}-{{\overline{x}}_{2}}-\overline{x}=0.4$ The variance of the combined group ${{\sigma }^{2}}$ is given by the formula $({{n}_{1}}+{{n}_{2}}){{\sigma }^{2}}={{n}_{1}}(\sigma _{1}^{2}+d_{1}^{2})+{{n}_{2}}(\sigma _{2}^{2}+d_{2}^{2})$ $250\times 13.44=100(9+0.36)+150(\sigma _{2}^{2}+0.16)$ $\Rightarrow$${{\sigma }_{2}}=4$