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Question

# For the given data, the calculation corresponding to all value of pairs $\left(x,y\right)$ is following $\sum {\left(x-\overline{x}\right)}^{2}=36,\sum {\left(y-\overline{y}\right)}^{2}=25,\sum \left(x-\overline{x}\right)\sum \left(y-\overline{y}\right)=20$. Then the Karl Pearson’s correlation coefficient is

A

$0.2$

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B

$0.5$

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C

$0.66$

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D

$0.33$

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Solution

## The correct option is C $0.66$Explanation for correct option:Given,$\begin{array}{rcl}\sum {\left(x-\overline{x}\right)}^{2}& =& 36,\\ \sum {\left(y-\overline{y}\right)}^{2}& =& 25,\\ \sum \left(x-\overline{x}\right)\sum \left(y-\overline{y}\right)& =& 20\end{array}$Karl Pearson relation.${\mathbit{r}}_{\mathbf{\left(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{y}\mathbf{\right)}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{}\frac{\mathbf{\sum }\mathbf{\left(}\mathbf{x}\mathbf{-}\mathbf{}\overline{\mathbf{x}}\mathbf{}\mathbf{\right)}\mathbf{\sum }\mathbf{\left(}\mathbf{y}\mathbf{}\mathbf{-}\overline{\mathbf{y}}\mathbf{}\mathbf{\right)}\mathbf{}}{\sqrt{\mathbf{\sum }{\mathbf{\left(}\mathbf{x}\mathbf{-}\mathbf{}\overline{\mathbf{x}}\mathbf{}\mathbf{\right)}}^{\mathbf{2}}}\mathbf{×}\sqrt{\mathbf{\sum }{\mathbf{\left(}\mathbf{y}\mathbf{}\mathbf{-}\overline{\mathbf{y}}\mathbf{}\mathbf{\right)}}^{\mathbf{2}}}}\phantom{\rule{0ex}{0ex}}Putallthevalue\phantom{\rule{0ex}{0ex}}{\mathbit{r}}_{\mathbf{\left(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{y}\mathbf{\right)}}=\frac{20}{\sqrt{36}×\sqrt{25}}\phantom{\rule{0ex}{0ex}}{\mathbit{r}}_{\mathbf{\left(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{y}\mathbf{\right)}}\mathbf{}=\frac{20}{5×6}\phantom{\rule{0ex}{0ex}}{r}_{\left(\mathrm{x},\mathrm{y}\right)}=\frac{2}{3}\phantom{\rule{0ex}{0ex}}{r}_{\left(\mathrm{x},\mathrm{y}\right)}=0.66\phantom{\rule{0ex}{0ex}}$Hence, correct option is $\left(C\right)$.

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