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

$\begin {array} {1 1} (A)\;5 & \quad (B)\;4 \\ (C)\;6 & \quad (D)\;3.52 \end {array}$

Toolbox:
• $\sigma^2= \large\frac{n_1( \sigma_1^2+d_1^2)+n_2(\sigma_2^2+d_2^2)}{n_1+n_2}$
Use $\sigma^2= \large\frac{n_1( \sigma_1^2+d_1^2)+n_2(\sigma_2^2+d_2^2)}{n_1+n_2}$
where $d_1=m_1-a$
$d_2=m_2-a$
$a$ = mean of the whole group
Let $m_2$ = mean of the second group
$\therefore 15.6 = \large\frac{100 \times 15+150 \times m_2}{250}$
$\Rightarrow\: 250 \times 15.6 = 100 \times 15 + 150 \times m_2$
$\Rightarrow\: \large\frac{250 \times15.6 - 100 \times 15}{150} $$= m_2 \Rightarrow\: \large\frac{3900-1500}{150}$$=m_2$
$\Rightarrow\: \large\frac{2400}{150}$$=m_2$
$\Rightarrow\:m_2=16$
$13.44=\large\frac{(100 \times 9 + 150 \times \sigma^2)+100 \times (0.6)^2+150 \times (0.4)^2}{250}$
$\Rightarrow \sigma = 4$
Hence Ans : (B)
edited Mar 26, 2014