Correlation and RegressionMCQPYQ Sep 24Question 3725 of 188
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If the regression lines are 3x−4y+8=0\displaystyle 3x - 4y + 8 = 0 and 4x−3y=1\displaystyle 4x - 3y = 1, then the correlation coefficient between x\displaystyle x and y\displaystyle y is __________.

Options

A3/4\displaystyle 3/4
B3/8\displaystyle 3/8
C4/8\displaystyle 4/8
D1/4\displaystyle 1/4
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Correct Answer

✅ Option a — 3/4\displaystyle 3/4

All Options:

  • A3/4\displaystyle 3/4
  • B3/8\displaystyle 3/8
  • C4/8\displaystyle 4/8
  • D1/4\displaystyle 1/4

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Detailed Solution & Explanation

To find the correlation coefficient r\displaystyle r, we identify the regression lines of y\displaystyle y on x\displaystyle x and x\displaystyle x on y\displaystyle y from the given equations:\n\n1. Assume 3x−4y+8=0\displaystyle 3x - 4y + 8 = 0 is the regression line of y\displaystyle y on x\displaystyle x:\n4y=3x+8impliesy=frac34x+24y = 3x + 8 \\implies y = \\frac{3}{4}x + 2\nThis gives the regression coefficient of y\displaystyle y on x\displaystyle x as:\nbyx=frac34b_{yx} = \\frac{3}{4}\n\n2. Assume 4x−3y=1\displaystyle 4x - 3y = 1 is the regression line of x\displaystyle x on y\displaystyle y:\n4x=3y+1impliesx=frac34y+frac144x = 3y + 1 \\implies x = \\frac{3}{4}y + \\frac{1}{4}\nThis gives the regression coefficient of x\displaystyle x on y\displaystyle y as:\nbxy=frac34b_{xy} = \\frac{3}{4}\n\n3. We verify the assumption using the relation r2=byxtimesbxy\displaystyle r^2 = b_{yx} \\times b_{xy}:\nr2=frac34timesfrac34=frac916le1r^2 = \\frac{3}{4} \\times \\frac{3}{4} = \\frac{9}{16} \\le 1\nSince r2\displaystyle r^2 is less than or equal to 1\displaystyle 1, our assumption is correct. Since both regression coefficients are positive, the correlation coefficient r\displaystyle r must also be positive:\nr=sqrtfrac916=frac34r = \\sqrt{\\frac{9}{16}} = \\frac{3}{4}\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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