Question

Means of two samles of sizes $50,100$ respectivly are $54.1,50.3$ and SD are $8$ and $7$. The combined SD of two samples is

• A. $7.56$
• B. $7.00$
• C. $7.28$
• D. $8$

Answer 1

The correct option is A $7.56$

$n_1=50$ & $n_2=100.$

${\overline{x}}_1=54.1$ & ${\overline{x}}_2=50.3$.

${\sigma }_1=8$ & ${\sigma }_2=7.$

$\overline{x}=\frac{n_1{\overline{x}}_1+n_2{\overline{x}}_2}{n_1+n_2}=\frac{50\times 54.1+100\times 50.3}{150}=51.567$.

${\sigma }_1={\overline{x}}_1-\overline{x}=54.1-51.56=2.53$.

${\sigma }_2={\overline{x}}_2-\overline{x}=50.3-51.56=-1.26$.

${\sigma }^2(n_1+n_2)=n_1({\sigma }_1^2+d_1^2)+n_2({\sigma }_2^2+d_2^2)=50(60+6.4)+100(49+1.58)$

${\sigma }^2\left(150\right)=3520+5058$

${\sigma }^2=\frac{8578}{150}=57.18$

$\sigma =7.56$