Correlation and RegressionMCQMTP Nov 18Question 3734 of 188
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In a bivariate population, the linear regression lines 2x+y=0\displaystyle 2x + y = 0 and y=x\displaystyle y = x then the coefficient of correlation is

Options

A0\displaystyle 0
B13\displaystyle \frac{1}{3}
C−13\displaystyle -\frac{1}{3}
D−13\displaystyle -\frac{1}{\sqrt{3}}
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Correct Answer

✅ Option a — 0\displaystyle 0

All Options:

  • A0\displaystyle 0
  • B13\displaystyle \frac{1}{3}
  • C−13\displaystyle -\frac{1}{3}
  • D−13\displaystyle -\frac{1}{\sqrt{3}}

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Detailed Solution & Explanation

We are given the regression lines 2x+y=0\displaystyle 2x + y = 0 and y=x\displaystyle y = x.\n\n1. Let us rewrite the lines to find the potential regression coefficients:\n - From 2x+y=0\displaystyle 2x + y = 0, we have y=−2x\displaystyle y = -2x, which suggests a negative slope of −2\displaystyle -2.\n - From y=x\displaystyle y = x, we have x=y\displaystyle x = y, which suggests a positive slope of 1\displaystyle 1.\n2. For any bivariate distribution, both regression coefficients (byx\displaystyle b_{yx} and bxy\displaystyle b_{xy}) must have the same sign (either both positive or both negative), as they both share the sign of the correlation coefficient r\displaystyle r:\n byx=rfracsigmaysigmaxquadtextandquadbxy=rfracsigmaxsigmayb_{yx} = r \\frac{\\sigma_y}{\\sigma_x} \\quad \\text{and} \\quad b_{xy} = r \\frac{\\sigma_x}{\\sigma_y}\n3. Because the slopes of the given lines have opposite signs (−2\displaystyle -2 and +1\displaystyle +1), it is statistically impossible for these to represent valid regression lines with non-zero correlation. \n4. The only way this configuration can be resolved is if there is no correlation between the variables, meaning r=0\displaystyle r = 0.\n\nThus, the coefficient of correlation is 0\displaystyle 0, which corresponds to Option A.\n\nHence, **Option A** 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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