Correlation and RegressionMCQPYQ June 19Question 3702 of 188
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A.M. of regression coefficient is

Options

AEqual to r\displaystyle r
BGreater than or equal to r\displaystyle r
CHalf of r\displaystyle r
DNone of these
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Correct Answer

✅ Option b — Greater than or equal to r\displaystyle r

All Options:

  • AEqual to r\displaystyle r
  • BGreater than or equal to r\displaystyle r
  • CHalf of r\displaystyle r
  • DNone of these

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Detailed Solution & Explanation

Let the two regression coefficients be byx\displaystyle b_{yx} and bxy\displaystyle b_{xy}. Their Arithmetic Mean (AM) is: AM=byx+bxy2\text{AM} = \frac{b_{yx} + b_{xy}}{2} Their Geometric Mean (GM) is related to the correlation coefficient r\displaystyle r by: GM=byx×bxy=∣r∣\text{GM} = \sqrt{b_{yx} \times b_{xy}} = |r| By the mathematical inequality of AM and GM (which states that AM≥GM\displaystyle \text{AM} \ge \text{GM} for any non-negative numbers): ∣byx∣+∣bxy∣2≥∣byx∣×∣bxy∣\frac{|b_{yx}| + |b_{xy}|}{2} \ge \sqrt{|b_{yx}| \times |b_{xy}|} ∣byx∣+∣bxy∣2≥∣r∣\frac{|b_{yx}| + |b_{xy}|}{2} \ge |r| Since regression coefficients byx\displaystyle b_{yx}, bxy\displaystyle b_{xy}, and r\displaystyle r must all share the same sign, we can generalize this to: byx+bxy2≥r(for r>0)\frac{b_{yx} + b_{xy}}{2} \ge r \quad (\text{for } r > 0) In all valid scenarios, the arithmetic mean of the two regression coefficients is always greater than or equal to the correlation coefficient (r\displaystyle r). *Note: The textbook/source file incorrectly lists Option A (Equal to r\displaystyle r) as the correct answer. By standard mathematical principles, the AM is greater than or equal to the GM, which means AM≥r\displaystyle \text{AM} \ge r (Option B).* 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.

Related Comparison Tables

Correlation vs Regression: Statistics for CA Foundation

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