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 $\frac{17}{9}$ , then the value of n is equal to_______

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answer is 5.

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Detailed Solution

$\begin{array}{l} σ^{2} = \frac{\sum (x_{i}^{2} + z_{i}^{2})}{10 + n} - {(\frac{\sum (x_{i} + z_{i})}{10 + n})}^{2} \\ \frac{17}{9} = \frac{60 + 10 n}{10 + n} - \frac{{(20 + 3 n)}^{2}}{{(10 + n)}^{2}} \\ \frac{17}{9} = \frac{(60 + 10 n) (10 + n)}{{(10 + n)}^{2}} - \frac{{(20 + 3 n)}^{2}}{{(10 + n)}^{2}} \\ 17 (n^{2} + 20 n + 100) = 9 (n^{2} + 40 n + 200) \\ 8 n^{2} - 20 n - 100 = 0 \\ 2 n^{2} - 5 n - 25 = 0 \\ (2 n - 5) (n - 5) = 0 \\ n = 5 \end{array}$