Correlation and RegressionPYQ Sept 25Question 4493 of 188
All Questions

If the regression coefficient byx\displaystyle b_{yx} is greater than one, then the regression coefficient bxy\displaystyle b_{xy}

Options

Acannot be less than one
Bcannot be greater than one
Ccan be equal to one
Dcan be equal to zero
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Correct Answer

✅ Option b — cannot be greater than one

All Options:

  • Acannot be less than one
  • Bcannot be greater than one
  • Ccan be equal to one
  • Dcan be equal to zero

Detailed Solution & Explanation

Let us analyze the mathematical relationship between the two regression coefficients from first principles:
1. **Relationship with Correlation Coefficient**:
The product of the regression coefficient of y\displaystyle y on x\displaystyle x (byx\displaystyle b_{yx}) and the regression coefficient of x\displaystyle x on y\displaystyle y (bxy\displaystyle b_{xy}) is equal to the square of the correlation coefficient (r2\displaystyle r^2):
byx×bxy=r2b_{yx} \times b_{xy} = r^2
2. **Properties of Correlation Coefficient**:
The correlation coefficient r\displaystyle r always lies between −1\displaystyle -1 and 1\displaystyle 1, inclusive (i.e., −1≤r≤1\displaystyle -1 \le r \le 1). Therefore, its square must satisfy:
0≤r2≤10 \le r^2 \le 1
3. **Application of the Inequality**:
Using the relation byx×bxy=r2\displaystyle b_{yx} \times b_{xy} = r^2, we obtain the inequality:
byx×bxy≤1b_{yx} \times b_{xy} \le 1
If one of the regression coefficients is greater than one, say byx>1\displaystyle b_{yx} > 1:
bxy≤1byx<1b_{xy} \le \frac{1}{b_{yx}} < 1
This mathematically proves that if one regression coefficient is greater than 1, the other must be less than 1. Therefore, bxy\displaystyle b_{xy} cannot be greater than one.
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

Regression

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

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