Question

The first of the two samples in a group has $100$ items with mean $15$ and standard deviation $3$ . If the whole group has $250$ items with mean $15.6$ and standard deviation $\sqrt{13.44}$, then the standard deviation of the second sample is:

• A. 8
• B. 6
• C. 4
• D. 5

Answer 1

Answer: (C)

$n_1=100m=250$
${\bar{X}}_1=15\bar{X}=15.6$
$V_1(x)=9\text{Var}(x)=13.44$
${\sigma }^2=\frac{n_1{\sigma }_1^2+n_2{\sigma }_2^2}{n_1+n_2}+\frac{n_1{n}_2}{{\left(n_1+n_2\right)}^2}{\left({\bar{x}}_1-{\bar{x}}_2\right)}^2$
$n_2=150,{\bar{x}}_2=16,V_2(x)={\sigma }_2$
$13.44=\frac{100\times 9+150\times {\sigma }_2^2}{250}+\frac{100\times 150}{(250{)}^2}\times 1$
$\Rightarrow {\sigma }_2^2=16$
$\Rightarrow {\sigma }_2=4$