Question

A group has $250$ items with mean and standard deviation equal to $15.6$ and $\sqrt{13.44}$ respectively. If the second group of two samples has $100$ items with mean equal to $15$ and standard deviation equal to $3$, then the standard deviation of first group is

Answer 1

From the given data,
$n=250,\overline{x}=15.6,\sigma =\sqrt{13.44}{n}_2=100,{\overline{x}}_2=15,{\sigma }_2=3\Rightarrow n_1=150$
We know that,
$\overline{x}=\frac{n_1\overline{x_1}+n_2\overline{x_2}}{n_1+n_2}\Rightarrow 15.6=\frac{150\left(\overline{x_1}\right)+100\left(15\right)}{250}\Rightarrow \overline{x_1}=16$

Using Combined Variance formula, we get
${\sigma }^2=\frac{1}{n_1+n_2}\left[n_1\right({\sigma }_1^2+d_1^2)+n_2({\sigma }_2^2+d_2^2\left)\right]$
Where
$d_1=\overline{x}-\overline{x_1}=15.6-16=-0.4d_2=\overline{x}-\overline{x_2}=15.6-15=0.6$

$\therefore 13.44=\frac{1}{250}\left[150\right({\sigma }_1^2+0.16)+100(9+0.36\left)\right]\Rightarrow {\sigma }_1^2=16\therefore {\sigma }_1=4$