Question

The first of two groups has $100$ items with mean $15$ and combined group has $250$ items with mean $15.6$ and SD $\sqrt{13.34}$, find the SD of second group if SD of first is $3$.

• A. $3.5$
• B. $4$
• C. $5$
• D. $6$

Answer 1

Answer: (B)

$4$

$n_1=100,n_1+n_2=250.$ $n_2=150$

${\overline{x}}_1=15,$ $\overline{x}=15.6$

$\sigma =3.65,$ ${\sigma }_1=3$

$\overline{x}=\frac{n_1{\overline{x}}_1+n_2{\overline{x}}_2}{n_1+n_2}$

$15.6=\frac{100\times 15+150\times {\overline{x}}_2}{250}$

$3900=1500+150{\overline{x}}_2$

${\overline{x}}_2=16$

$d_1={\overline{x}}_1-\overline{x}=15-15.6=-0.6$

$d_2={\overline{x}}_2-\overline{x}=16-15.6=0.4$

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

$13.34\times 250=100\times (9+0.36)+150({\sigma }_2^2+0.16)$

$3335=936+150({\sigma }_2^2+0.16)$

$\frac{2399}{150}={\sigma }_2^2+0.16$

${\sigma }_2^2=3.979\approx 4$