The first of the two samples in a group has 100
Question:

The first of the two samples in a group has 100 items with mean 15 and standard deviation 3 . If the whole group has 250 items with mean $15.6$ and standard deviation $\sqrt{13.44}$, then the standard deviation of the second sample is :

1. 8

2. 6

3. 4

4. 5

Correct Option: , 3

Solution:

$\sigma^{2}=\frac{\mathrm{n}_{1} \sigma_{1}^{2}+\mathrm{n}_{2} \sigma_{2}^{2}}{\mathrm{n}_{1}+\mathrm{n}_{2}}+\frac{\mathrm{n}_{1} \mathrm{n}_{2}}{\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)^{2}}\left(\overline{\mathrm{x}}_{1}-\overline{\mathrm{x}}_{2}\right)^{2}$

$\mathrm{n}_{2}=150, \quad \overline{\mathrm{x}}_{2}=16, \mathrm{~V}_{2}(\mathrm{x})=\sigma_{2}$

$13.44=\frac{100 \times 9+150 \times \sigma_{2}^{2}}{250}+\frac{100 \times 150}{(250)^{2}} \times 1$

$\Rightarrow \sigma_{2}^{2}=16 \Rightarrow \sigma_{2}=4$