Question

Consider the statistics of two sets of observations as follows :
$\begin{array}{cccc}& \text{Size}& \text{Mean}& \text{Variance}\\ \text{Observation I}& 10& 2& 2\\ \text{Observation II}& n& 3& 1\end{array}$

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

Answer 1

For $\text{Observation I}:$
$\frac{\sum x_i}{10}=2\Rightarrow \sum x_i=20$
$\frac{\sum x_i^2}{10}-(2{)}^2=2\Rightarrow \sum x_i^2=60$

For $\text{Observation II}:$
$\frac{\sum y_i^2}{n}-3^2=1\Rightarrow \sum y_i^2=10n$

Now, combined variance
${\sigma }^2=\frac{\sum (x_i^2+y_i^2)}{10+n}-{\left(\frac{\sum (x_i+y_i)}{10+n}\right)}^2$
$\Rightarrow \frac{17}{9}=\frac{60+10n}{10+n}-\frac{(20+3n{)}^2}{(10+n{)}^2}$
$\Rightarrow 17(n^2+20n+100)=9(n^2+40n+200)$
$\Rightarrow 8n^2-20n-100=0$
$\Rightarrow 2n^2-5n-25=0$
$\Rightarrow n=5$