Correlation and RegressionMCQMTP June 24 Series IQuestion 3666 of 188
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If r is the Karl Pearson's coefficient of correlation in a bivariate distribution the two regression lines are at right angles when

Options

Ar=pm1\displaystyle r = \\pm 1
Br=0\displaystyle r = 0
Cr=pm0.5\displaystyle r = \\pm 0.5
DNone
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Correct Answer

✅ Option b — r=0\displaystyle r = 0

All Options:

  • Ar=pm1\displaystyle r = \\pm 1
  • Br=0\displaystyle r = 0
  • Cr=pm0.5\displaystyle r = \\pm 0.5
  • DNone

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Detailed Solution & Explanation

**Condition for regression lines to be perpendicular:** The two regression lines: - Regression of Y\displaystyle Y on X\displaystyle X: slope = byx=rcdotfracsigmaysigmax\displaystyle b_{yx} = r \\cdot \\frac{\\sigma_y}{\\sigma_x} - Regression of X\displaystyle X on Y\displaystyle Y: expressed as Y\displaystyle Y on X\displaystyle X, slope = frac1bxy=fracsigmayrcdotsigmax\displaystyle \\frac{1}{b_{xy}} = \\frac{\\sigma_y}{r \\cdot \\sigma_x} **For perpendicular lines**, product of slopes = −1\displaystyle -1: byxtimesfrac1bxy=−1b_{yx} \\times \\frac{1}{b_{xy}} = -1 rcdotfracsigmaysigmaxtimesfracsigmayrcdotsigmax=−1r \\cdot \\frac{\\sigma_y}{\\sigma_x} \\times \\frac{\\sigma_y}{r \\cdot \\sigma_x} = -1 fracsigmay2sigmax2=−1\\frac{\\sigma_y^2}{\\sigma_x^2} = -1 This is only possible when r=0\displaystyle r = 0 (no correlation), in which case the regression lines are: - Y\displaystyle Y on X\displaystyle X line: Y=barY\displaystyle Y = \\bar{Y} (horizontal line) - X\displaystyle X on Y\displaystyle Y line: X=barX\displaystyle X = \\bar{X} (vertical line) Horizontal and vertical lines are **perpendicular (at right angles)** to each other. 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.

Regression

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

Related Comparison Tables

Correlation vs Regression: Statistics for CA Foundation

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