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 (integer)
603
Solution
<p>$x = \{ 11,12,13\,....,40,41\}$</p>
<p>$y = \{ 61,62,63\,....,90,91\}$</p>
<p>$$\overline x = {{{{31} \over 2}(11 + 41)} \over {31}} = {1 \over 2} \times 52 = 26$$</p>
<p>$$\overline y = {{{{31} \over 2}(61 + 91)} \over {31}} = {1 \over 2} \times 152 = 76$$</p>
<p>$${\sigma ^2} = {{\sum {x_i^2 + \sum {y_i^2} } } \over {62}} - {\left( {{{\sum {x + \sum y } } \over {62}}} \right)^2}$$</p>
<p>$= 705$</p>
<p>Now,</p>
<p>$\left| {\overline x + \overline y - {\sigma ^2}} \right|$</p>
<p>$= |26 + 76 - 705| = 603$</p>
About this question
Subject: Mathematics · Chapter: Statistics · Topic: Measures of Central Tendency
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