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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.

$\begin{array}{1 1}(A)\;0.99\\(B)\;5.59\\(C)\;8\\(D)\;1.25\end{array} $

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  • Formula used to solve this problem are :
  • Combined SD $=\sqrt {\large\frac{n_1\sigma_1 ^2 + n_2 \sigma_2 ^2}{n_1+n_2} + \frac{n_1n_2 (\bar{x_1} - \bar {x_2})^2}{(n_1+n_2)}}$
  • Given $n_1=20 \qquad n_2=20$
  • $\sigma_1=5 \qquad \sigma_2=20$
  • $\bar {x_1}=17 \qquad \bar{x_2} =22$
Substituting the values in the formula
Combined SD $= \sqrt { \large\frac{20 \times 5^2+ 20 \times 5^2}{20+20}+\frac{20 \times 20 (17-22)^2}{(20+20)^2}}$
$\qquad= \sqrt {\large\frac{1000}{40}+\frac{400 \times 5^2}{40^2}}$
$\qquad= \sqrt {\large\frac{1000}{40} +\frac{400 \times 5^2}{40^2}}$
$\qquad= \sqrt {\large\frac{1000}{40} +\large\frac{10000}{40 \times 40}}$
$\qquad= \large\frac{1}{40} $$ \sqrt { 40000+10000}$
$\qquad= \large\frac{1}{40}$$ \sqrt {50000}$
$\qquad= \large\frac{223.606}{40}$
$\qquad= 5.59$
$\therefore $ The combined standard deviation of the set obtained are $5.59$
Hence B is the correct answer.
answered Jul 3, 2014 by meena.p

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