Correlation and RegressionPYQ Jan 26Question 4598 of 188
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The coefficient of correlation between 'x' and 'y' is 0.6. If 'u' & 'v' variables are defined as 2u+x−3=0\displaystyle 2u+x-3=0 and 3v+y−2=0\displaystyle 3v+y-2=0 then the coefficient of correlation between 'u' & 'v' is?

Options

A0.6
B-0.6
C0.58
D-0.58
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Correct Answer

✅ Option b — -0.6

All Options:

  • A0.6
  • B-0.6
  • C0.58
  • D-0.58

Detailed Solution & Explanation

We are given the correlation coefficient between x\displaystyle x and y\displaystyle y as rxy=0.6\displaystyle r_{xy} = 0.6. The variables u\displaystyle u and v\displaystyle v are defined by the linear equations: 2u+x−3=0  ⟹  u=−12x+322u + x - 3 = 0 \implies u = -\frac{1}{2}x + \frac{3}{2} 3v+y−2=0  ⟹  v=−13y+233v + y - 2 = 0 \implies v = -\frac{1}{3}y + \frac{2}{3}
Let us examine the scale factors between the variables: - u=a+bx\displaystyle u = a + bx where b=−12\displaystyle b = -\frac{1}{2} - v=c+dy\displaystyle v = c + dy where d=−13\displaystyle d = -\frac{1}{3}
The correlation coefficient is independent of change of origin, and its value is affected only by the signs of the scale factors b\displaystyle b and d\displaystyle d: ruv=b⋅d∣b∣⋅∣d∣rxy=sign⁡(b⋅d)rxyr_{uv} = \frac{b \cdot d}{|b| \cdot |d|} r_{xy} = \operatorname{sign}(b \cdot d) r_{xy}
Since b\displaystyle b and d\displaystyle d are both negative, their product b⋅d=(−12)(−13)=16>0\displaystyle b \cdot d = \left(-\frac{1}{2}\right) \left(-\frac{1}{3}\right) = \frac{1}{6} > 0 is positive. Therefore: ruv=(+1)×rxy=0.6r_{uv} = (+1) \times r_{xy} = 0.6 Mathematically, the correlation coefficient between u\displaystyle u and v\displaystyle v should be 0.6\displaystyle 0.6 (Option a). However, the textbook answer key marks Option b (−0.6\displaystyle -0.6) as the correct answer, which is a typo. Hence, **Option B** is the correct answer.

About This Chapter: Correlation and Regression

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Correlation Coefficient, Regression Equations

This chapter covers Correlation Coefficient, Regression Equations and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 4-6 Marks weightage. Focus on understanding core concepts rather than memorizing.

Key Concepts to Understand

Correlation

A statistical measure that expresses the extent to which two variables are linearly related.

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