Question

The first of two samples has $100$ items with mean $15$ and S.D. $3$. If the whole group has $250$ items with mean $15.6$ and $s.d=\sqrt{13.44}$, find the standard deviation of the second group.

• A. $5$
• B. $4$
• C. $6$
• D. $3.52$

Answer 1

The correct option is B $4$

Use ${\sigma }^2=\frac{n_1\times ({\sigma }_1^2+d_1^2)+n_2\times ({\sigma }_2^2+d_2^2)}{n_1+n_2}$

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 $15.6=\frac{100\times 15+150\times m_2}{250}$

$\Rightarrow$ $m_2=\frac{250\times 15.6-100\times 15}{150}$

$\Rightarrow$ $m_2=\frac{2400}{15}$

$m_2=16$

$\Rightarrow$ $13.44=\frac{100\times 9+150\times {\sigma }^2+100\times {\left(0.6\right)}^2+150\times {\left(0.4\right)}^2}{250}$

$\Rightarrow$ $\sigma =4.$ ans