The mean and variance of eight observations a | vedclass

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(A) Let the remaining two observations be $x$ and $y$.The sum of the eight observations is $8 	imes 9 = 72$.The sum of the six given observations is

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$4, 8$

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(A) Let the remaining two observations be $x$ and $y$.The sum of the eight observations is $8 	imes 9 = 72$.The sum of the six given observations is $6 + 7 + 10 + 12 + 12 + 13 = 60$.Thus,$x + y = 72 - 60 = 12$ ........... $(1)$The variance is given by $\sigma^2 = \frac{1}{n} \sum x_i^2 - (\bar{x})^2$.$9.25 = \frac{1}{8} (6^2 + 7^2 + 10^2 + 12^2 + 12^2 + 13^2 + x^2 + y^2) - 9^2$.$9.25 + 81 = \frac{1}{8} (36 + 49 + 100 + 144 + 144 + 169 + x^2 + y^2)$.$90.25 	imes 8 = 642 + x^2 + y^2$.$722 = 642 + x^2 + y^2 \Rightarrow x^2 + y^2 = 80$ ........... $(2)$From $(1)$,$(x + y)^2 = 144 \Rightarrow x^2 + y^2 + 2xy = 144$.$80 + 2xy = 144$ $\Rightarrow 2xy = 64$ $\Rightarrow xy = 32$.Since $x + y = 12$ and $xy = 32$,the quadratic equation $t^2 - 12t + 32 = 0$ gives the values.$(t - 8)(t - 4) = 0$.Thus,the remaining observations are $4$ and $8$.

The mean and variance of eight observations are $9$ and $9.25,$ respectively. If six of the observations are $6, 7, 10, 12, 12,$ and $13,$ find the remaining two observations.

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