Two sets,each of $20$ observations,have the same standard deviation $5$. The first set has a mean $17$ and the second a mean $22$. Determine the standard deviation of the set obtained by combining the given two sets.

(D) Given,$n_{1}=20, \sigma_{1}=5, \bar{x}_{1}=17$ and $n_{2}=20, \sigma_{2}=5, \bar{x}_{2}=22$.
We know that the combined standard deviation $\sigma$ is given by:
$\sigma=\sqrt{\frac{n_{1} \sigma_{1}^{2}+n_{2} \sigma_{2}^{2}}{n_{1}+n_{2}}+\frac{n_{1} n_{2}(\bar{x}_{1}-\bar{x}_{2})^{2}}{(n_{1}+n_{2})^{2}}}$
Substituting the values:
$\sigma=\sqrt{\frac{20 \times(5)^{2}+20 \times(5)^{2}}{20+20}+\frac{20 \times 20(17-22)^{2}}{(20+20)^{2}}}$
$\sigma=\sqrt{\frac{500+500}{40}+\frac{400 \times (-5)^{2}}{40^{2}}}$
$\sigma=\sqrt{\frac{1000}{40}+\frac{400 \times 25}{1600}}$
$\sigma=\sqrt{25+\frac{10000}{1600}}$
$\sigma=\sqrt{25+6.25} = \sqrt{31.25} \approx 5.59$