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  4. The mean and variance of 5 observations are 5 and 8 respectively. If 3 observations are 1,3,5, then the sum of cubes of the remaining two observations is

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Q. The mean and variance of $5$ observations are $5$ and $8$ respectively. If $3$ observations are $1,3,5$, then the sum of cubes of the remaining two observations is

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Solution:

$ \frac{1+3+5+a+b}{5}=5$
$ a+b=16 \ldots \ldots(1) $
$ \sigma^2=\frac{\sum x_1^2}{5}-\left(\frac{\sum x}{5}\right)^2 $
$8=\frac{1^2+3^2+5^2+a^2+b^2}{5}-25 $
$ a^2+b^2=130 \ldots \ldots(2) $
$ \text { by }(1),(2) $
$ a=7, b=9 $
$ \text { or } a=9, b=7$