Consider the statistics of two sets of observations as follows :

	Size	Mean	Variance
Observation I	10	2	2
Observation II	n	3	1

If the variance of the combined set of these two observations is ${{17} \over 9}$, then the value of n is equal to ___________.

For group - 1 : ${{\sum {{x_i}} } \over {10}} = 2 \Rightarrow \sum {{x_i}} = 20$<br><br>${{\sum {{x_i^2}} } \over {10}} - {(2)^2} = 2 \Rightarrow \sum {x_i^2} = 60$<br><br>For group - 2 : ${{\sum {{y_i}} } \over n} = 3 \Rightarrow \sum {{y_i}} = 3n$<br><br>${{\sum {y_i^2} } \over n} - {3^2} = 1 \Rightarrow \sum {y_i^2} = 10n$<br><br>Now, combined variance<br><br>$${\sigma ^2} = {{\sum {\left( {x_i^2 + y_i^2} \right)} } \over {10 + n}} - {\left( {{{\sum {\left( {{x_i} + {y_i}} \right)} } \over {10 + n}}} \right)^2}$$<br><br>$$ \Rightarrow {{17} \over 9} = {{60 + 10n} \over {10 + n}} - {{{{(20 + 3n)}^2}} \over {{{(10 + n)}^2}}}$$<br><br>$\Rightarrow$ 17 (n² + 20n + 100) = 9(n² + 40n + 200)<br><br>$\Rightarrow$ 8n² $-$ 20n $-$ 100 = 0<br><br>$\Rightarrow$ 2n² $-$ 5n $-$ 25 = 0 $\Rightarrow$ n = 5