Correlation and RegressionPYQ Jun 23Question 3719 of 188
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The regression lines will be perpendicular to each other when the value of r\displaystyle r is

Options

A1
B-1
C1/2
D0
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Correct Answer

✅ Option d — 0

All Options:

  • A1
  • B-1
  • C1/2
  • D0

Detailed Solution & Explanation

The angle θ\displaystyle \theta between the two regression lines of y\displaystyle y on x\displaystyle x and x\displaystyle x on y\displaystyle y is given by: tan⁡θ=1−r2∣r∣(σxσyσx2+σy2)\tan \theta = \frac{1 - r^2}{|r|} \left(\frac{\sigma_x \sigma_y}{\sigma_x^2 + \sigma_y^2}\right) For the regression lines to be perpendicular to each other, the angle θ\displaystyle \theta must be 90∘\displaystyle 90^\circ. Since tan⁡90∘=∞\displaystyle \tan 90^\circ = \infty, the denominator of the term 1−r2∣r∣\displaystyle \frac{1 - r^2}{|r|} must be zero (assuming σx≠0\displaystyle \sigma_x \ne 0 and σy≠0\displaystyle \sigma_y \ne 0). Therefore: ∣r∣=0  ⟹  r=0|r| = 0 \implies r = 0 When r=0\displaystyle r = 0, the two variables are uncorrelated (independent), and the regression lines are perpendicular to each other (parallel to the coordinate axes, intersecting at the means). Hence, **Option D** 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

Regression

A statistical method used for estimating the relationships between a dependent variable and one or more independent variables.

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