Q261 / 270 · NDA 2023 (Phase 2) · Mathematics

If $r$ is the coefficient of correlation between x and y , then what is the correlation coefficient between $(3 x+4)$ and $(-3 y+3) ?$

  • Correct answer
    A
    $-r$
  • B
    $r$
  • C
    $\sqrt{3} r$
  • D
    $-\sqrt{3} r$

Solution

$\begin{aligned} & \because r=\frac{\operatorname{cor}(x, y)}{\sqrt{\operatorname{var}(x) \cdot \operatorname{var}(y)}} \\ & \operatorname{cor}(3 x \times 4,-3 y+3)=\sum\left(3 x-3 x^{\prime}\right)\left(-3 y+3 y^{\prime}\right) \\ & =-9 \operatorname{cor}(x, y) \\ & \therefore r(3 x+4,-3 y+3)=\frac{-9 \operatorname{cor}(x, y)}{\sqrt{9 \operatorname{cor}(x)-\sqrt{9 \operatorname{var}(y)}}} \\ & =\frac{-9 \operatorname{cor}(x, y)}{9 \sqrt{\operatorname{var}(x)} \cdot \sqrt{\operatorname{var}(y)}}=-r\end{aligned}$

Hence, the correct answer is option: 'A'.

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