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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Statistics

NDA 2023 · Mathematics · Statistics

If r is the coefficient of correlation between x and y , then what...

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

Options

  1. A−r-r
  2. Brr
  3. C3r\sqrt{3} r
  4. D−3r-\sqrt{3} r

Correct answer

A. −r-r

Step-by-step solution

∵r=cor⁡(x,y)var⁡(x)⋅var⁡(y)cor⁡(3x×4,−3y+3)=∑(3x−3x′)(−3y+3y′)=−9cor⁡(x,y)∴r(3x+4,−3y+3)=−9cor⁡(x,y)9cor⁡(x)−9var⁡(y)=−9cor⁡(x,y)9var⁡(x)⋅var⁡(y)=−r\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}
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