JEE MAIN - Mathematics (2023 - 29th January Evening Shift - No. 17)
Let $$X=\{11,12,13,....,40,41\}$$ and $$Y=\{61,62,63,....,90,91\}$$ be the two sets of observations. If $$\overline x $$ and $$\overline y $$ are their respective means and $$\sigma^2$$ is the variance of all the observations in $$\mathrm{X\cup Y}$$, then $$\left| {\overline x + \overline y - {\sigma ^2}} \right|$$ is equal to ____________.
Answer
603
Explanation
$$x = \{ 11,12,13\,....,40,41\} $$
$$y = \{ 61,62,63\,....,90,91\} $$
$$\overline x = {{{{31} \over 2}(11 + 41)} \over {31}} = {1 \over 2} \times 52 = 26$$
$$\overline y = {{{{31} \over 2}(61 + 91)} \over {31}} = {1 \over 2} \times 152 = 76$$
$${\sigma ^2} = {{\sum {x_i^2 + \sum {y_i^2} } } \over {62}} - {\left( {{{\sum {x + \sum y } } \over {62}}} \right)^2}$$
$$ = 705$$
Now,
$$\left| {\overline x + \overline y - {\sigma ^2}} \right|$$
$$ = |26 + 76 - 705| = 603$$
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