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)

$n_{1}=100\,\, m=250$
$\bar{X}_{1}=15\,\, \bar{X}=15.6$
$V_{1}(x)=9\,\, \text{Var}(x)=13.44$
$\sigma^{2}=\frac{n_{1} \sigma_{1}^{2}+n_{2} \sigma_{2}^{2}}{n_{1}+n_{2}}+\frac{n_{1} n_{2}}{\left(n_{1}+n_{2}\right)^{2}}\left(\bar{x}_{1}-\bar{x}_{2}\right)^{2}$
$n_{2}=150, \bar{x}_{2}=16, V_{2}(x)=\sigma_{2}$
$13.44=\frac{100 \times 9+150 \times \sigma_{2}^{2}}{250}+\frac{100 \times 150}{(250)^{2}} \times 1$
$\Rightarrow \sigma_{2}^{2}=16$
$\Rightarrow \sigma_{2}=4$