Correlation and RegressionMCQPYQ Jan. 21Question 3705 of 188
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The intersecting point of the two regression lines: y\displaystyle y on x\displaystyle x and x\displaystyle x on y\displaystyle y is

Options

A(0,0)\displaystyle (0,0)
B(x,y)\displaystyle (x,y)
C(xˉ,yˉ)\displaystyle (\bar{x}, \bar{y})
D(1,1)\displaystyle (1,1)
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Correct Answer

✅ Option c — (xˉ,yˉ)\displaystyle (\bar{x}, \bar{y})

All Options:

  • A(0,0)\displaystyle (0,0)
  • B(x,y)\displaystyle (x,y)
  • C(xˉ,yˉ)\displaystyle (\bar{x}, \bar{y})
  • D(1,1)\displaystyle (1,1)

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Detailed Solution & Explanation

The two lines of regression are: 1. Regression line of y\displaystyle y on x\displaystyle x: y−yˉ=byx(x−xˉ)y - \bar{y} = b_{yx}(x - \bar{x}) 2. Regression line of x\displaystyle x on y\displaystyle y: x−xˉ=bxy(y−yˉ)x - \bar{x} = b_{xy}(y - \bar{y}) Here, xˉ\displaystyle \bar{x} represents the mean of the variable x\displaystyle x and yˉ\displaystyle \bar{y} represents the mean of the variable y\displaystyle y. If we substitute the point of means (centroid) (xˉ,yˉ)\displaystyle (\bar{x}, \bar{y}) into both regression equations, we get: For y\displaystyle y on x\displaystyle x: yˉ−yˉ=byx(xˉ−xˉ)  ⟹  0=0\bar{y} - \bar{y} = b_{yx}(\bar{x} - \bar{x}) \implies 0 = 0 For x\displaystyle x on y\displaystyle y: xˉ−xˉ=bxy(yˉ−yˉ)  ⟹  0=0\bar{x} - \bar{x} = b_{xy}(\bar{y} - \bar{y}) \implies 0 = 0 Since the point of arithmetic means (xˉ,yˉ)\displaystyle (\bar{x}, \bar{y}) satisfies both equations, it must be their point of intersection. Therefore, the two regression lines always intersect at (xˉ,yˉ)\displaystyle (\bar{x}, \bar{y}). Hence, **Option C** 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.

Related Comparison Tables

Correlation vs Regression: Statistics for CA Foundation

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