Correlation and RegressionPYQ Jan 26Question 4542 of 188
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For the variables x and y, the regression equations are given as x+2y−5=0\displaystyle x + 2y - 5 = 0 and 2x+3y−8=0\displaystyle 2x + 3y - 8 = 0 and the arithmetic means of x and y are 1 and 2 respectively. Compute the correlation coefficient between x and y.

Options

A-0.87
B0
C0.87
D1
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Correct Answer

✅ Option a — -0.87

All Options:

  • A-0.87
  • B0
  • C0.87
  • D1

Detailed Solution & Explanation

Given the regression equations:
1) x+2y−5=0\displaystyle x + 2y - 5 = 0
2) 2x+3y−8=0\displaystyle 2x + 3y - 8 = 0

We need to identify which is the regression line of y\displaystyle y on x\displaystyle x and which is of x\displaystyle x on y\displaystyle y. The product of their slopes (byx×bxy=r2\displaystyle b_{yx} \times b_{xy} = r^2) must be in the range [0,1]\displaystyle [0, 1].

**Assumption 1:** Let Equation 1 be the regression line of x\displaystyle x on y\displaystyle y and Equation 2 be the regression line of y\displaystyle y on x\displaystyle x.
- From Equation 1: x=−2y+5  ⟹  bxy=−2\displaystyle x = -2y + 5 \implies b_{xy} = -2
- From Equation 2: 3y=−2x+8  ⟹  y=−23x+83  ⟹  byx=−23\displaystyle 3y = -2x + 8 \implies y = -\frac{2}{3}x + \frac{8}{3} \implies b_{yx} = -\frac{2}{3}
Checking r2\displaystyle r^2:
r2=bxy×byx=(−2)×(−23)=43≈1.33>1r^2 = b_{xy} \times b_{yx} = (-2) \times \left(-\frac{2}{3}\right) = \frac{4}{3} \approx 1.33 > 1
Since r2\displaystyle r^2 cannot exceed 1\displaystyle 1, this assumption is incorrect.

**Assumption 2:** Let Equation 1 be the regression line of y\displaystyle y on x\displaystyle x and Equation 2 be the regression line of x\displaystyle x on y\displaystyle y.
- From Equation 1: 2y=−x+5  ⟹  y=−12x+52  ⟹  byx=−0.5\displaystyle 2y = -x + 5 \implies y = -\frac{1}{2}x + \frac{5}{2} \implies b_{yx} = -0.5
- From Equation 2: 2x=−3y+8  ⟹  x=−32y+4  ⟹  bxy=−1.5\displaystyle 2x = -3y + 8 \implies x = -\frac{3}{2}y + 4 \implies b_{xy} = -1.5
Checking r2\displaystyle r^2:
r2=byx×bxy=(−0.5)×(−1.5)=0.75r^2 = b_{yx} \times b_{xy} = (-0.5) \times (-1.5) = 0.75
Since 0≤r2≤1\displaystyle 0 \le r^2 \le 1, this assumption is correct.

The correlation coefficient (r\displaystyle r) is the square root of r2\displaystyle r^2:
r=±0.75≈±0.866r = \pm \sqrt{0.75} \approx \pm 0.866
Since both regression coefficients byx\displaystyle b_{yx} and bxy\displaystyle b_{xy} are negative, the correlation coefficient r\displaystyle r must also be negative:
r=−0.866≈−0.87r = -0.866 \approx -0.87

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.

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