Correlation and RegressionMCQPYQ Nov 18Question 3700 of 188
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If the two regression lines are 3X=Y\displaystyle 3X=Y and 8Y=6X\displaystyle 8Y=6X, then the value of correlation coefficient is

Options

A0.5\displaystyle 0.5
B−0.5\displaystyle -0.5
C0.75\displaystyle 0.75
D−0.80\displaystyle -0.80
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Correct Answer

✅ Option a — 0.5\displaystyle 0.5

All Options:

  • A0.5\displaystyle 0.5
  • B−0.5\displaystyle -0.5
  • C0.75\displaystyle 0.75
  • D−0.80\displaystyle -0.80

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Detailed Solution & Explanation

Given the two regression equations: 1. 3X=Y⇒X=13Y\displaystyle 3X = Y \Rightarrow X = \frac{1}{3}Y 2. 8Y=6X⇒Y=68X=34X\displaystyle 8Y = 6X \Rightarrow Y = \frac{6}{8}X = \frac{3}{4}X We choose the lines such that the product of their slopes is less than or equal to 1\displaystyle 1. - Let X=13Y\displaystyle X = \frac{1}{3}Y be the regression line of X\displaystyle X on Y\displaystyle Y. This gives the regression coefficient of X\displaystyle X on Y\displaystyle Y: bxy=13b_{xy} = \frac{1}{3} - Let Y=34X\displaystyle Y = \frac{3}{4}X be the regression line of Y\displaystyle Y on X\displaystyle X. This gives the regression coefficient of Y\displaystyle Y on X\displaystyle X: byx=34b_{yx} = \frac{3}{4} **Step 1: Calculate the coefficient of correlation (r\displaystyle r)** The coefficient of correlation r\displaystyle r is the geometric mean of the two regression coefficients: r=±byx×bxyr = \pm \sqrt{b_{yx} \times b_{xy}} Substituting the values: r2=byx×bxy=34×13=14=0.25r^2 = b_{yx} \times b_{xy} = \frac{3}{4} \times \frac{1}{3} = \frac{1}{4} = 0.25 r=±0.25=±0.5r = \pm \sqrt{0.25} = \pm 0.5 **Step 2: Determine the sign of r\displaystyle r** The sign of the correlation coefficient r\displaystyle r must match the signs of the regression coefficients byx\displaystyle b_{yx} and bxy\displaystyle b_{xy}. Since both byx=0.75\displaystyle b_{yx} = 0.75 and bxy=0.333\displaystyle b_{xy} = 0.333 are positive, r\displaystyle r must be positive: r=0.5r = 0.5 *Note: The textbook/source file incorrectly lists the correct option as Option B (−0.5\displaystyle -0.5). Since both regression slopes are positive, the correlation coefficient must be positive (0.5\displaystyle 0.5, Option A).* Hence, **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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