Question

	Brand 1	Brand 2
No. of bulbs	$50$	$100$
mean	$800$	$770$
variance	$100$	$81$

Find combined mean and SD.

• A. $756,90$
• B. $780,90.5$
• C. $789,15$
• D. $780,16.95$

Answer 1

The correct option is D $780,16.95$

$n_1=50,{\overline{x}}_1=800,{\sigma }_1=10,{\sigma }_1^2=100$ $n_2=100,{\overline{x}}_2=770,{\sigma }_2=9,{\sigma }_1^2=81$

$\overline{x}=\frac{n_1{\overline{x}}_1+n_2{\overline{x}}_2}{n_1+n_2}=\frac{50\times 800+100\times 770}{150}=780$

$d_1={\overline{x}}_1-\overline{x}=800-780=20$

$d_2={\overline{x}}_2-\overline{x}=770-780=-10$

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

${\sigma }^2\left(150\right)=50\times (100+400)+100\times (81+100)$

${\sigma }^2=287.33$

$\sigma =16.95$