Question

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

• A. $0.2$
• B. $0.5$
• C. $0.66$
• D. $0.33$

Answer 1

The correct option is C

$0.66$

Explanation for correct option:

Given,

$\begin{array}{rcl}\sum {(x-\overline{x})}^2& =& 36,\\ \sum {(y-\overline{y})}^2& =& 25,\\ \sum (x-\overline{x})\sum (y-\overline{y})& =& 20\end{array}$

Karl Pearson relation.

$$\begin{gathered} {\mathit{r}}_{\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{y}\mathbf{)}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{}\frac{\mathbf{\sum }\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{}\overline{\mathbf{x}}\mathbf{}\mathbf{)}\mathbf{\sum }\mathbf{(}\mathbf{y}\mathbf{}\mathbf{-}\overline{\mathbf{y}}\mathbf{}\mathbf{)}\mathbf{}}{\sqrt{\mathbf{\sum }{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{}\overline{\mathbf{x}}\mathbf{}\mathbf{)}}^{\mathbf{2}}}\mathbf{\times }\sqrt{\mathbf{\sum }{\mathbf{(}\mathbf{y}\mathbf{}\mathbf{-}\overline{\mathbf{y}}\mathbf{}\mathbf{)}}^{\mathbf{2}}}} \\ Putallthevalue \\ {\mathit{r}}_{\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{y}\mathbf{)}}=\frac{20}{\sqrt{36}\times \sqrt{25}} \\ {\mathit{r}}_{\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{y}\mathbf{)}}\mathbf{}=\frac{20}{5\times 6} \\ {r}_{(\mathrm{x},\mathrm{y})}=\frac{2}{3} \\ {r}_{(\mathrm{x},\mathrm{y})}=0.66 \end{gathered}$$

Hence, correct option is $\left(C\right)$.