Correlation and RegressionPYQ Sept 25Question 4492 of 188
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If m+3x=10\displaystyle m + 3x=10 and 2y+5n=25\displaystyle 2y + 5n=25 and regression coefficient of y\displaystyle y on x\displaystyle x is 0.80, what is the regression coefficient of n\displaystyle n on m\displaystyle m?

Options

A-0.106
B9.375
C0.106
D0.0106
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Correct Answer

✅ Option c — 0.106

All Options:

  • A-0.106
  • B9.375
  • C0.106
  • D0.0106

Detailed Solution & Explanation

Let us analyze the regression coefficient under a change of origin and scale from first principles:
1. **Given Equations**:
m+3x=10  ⟹  m=10−3xm + 3x = 10 \implies m = 10 - 3x
2y+5n=25  ⟹  5n=25−2y  ⟹  n=5−0.4y2y + 5n = 25 \implies 5n = 25 - 2y \implies n = 5 - 0.4y
Let u=m\displaystyle u = m and v=n\displaystyle v = n represent the new variables.
2. **Determine the change of scale**:
We can express the relation between the variables as:
u=−3x+10  ⟹  x=−13u+103u = -3x + 10 \implies x = -\frac{1}{3}u + \frac{10}{3}
v=−0.4y+5  ⟹  y=−2.5v+12.5v = -0.4y + 5 \implies y = -2.5v + 12.5
Standard deviation changes with scale only. Taking the standard deviation on both sides of the equations:
σu=∣−3∣σx=3σx  ⟹  σm=3σx\sigma_u = |-3|\sigma_x = 3\sigma_x \implies \sigma_m = 3\sigma_x
σv=∣−0.4∣σy=0.4σy  ⟹  σn=0.4σy\sigma_v = |-0.4|\sigma_y = 0.4\sigma_y \implies \sigma_n = 0.4\sigma_y
3. **Determine correlation coefficient (r\displaystyle r)**:
Let rxy\displaystyle r_{xy} be the correlation coefficient between x\displaystyle x and y\displaystyle y, and rmn\displaystyle r_{mn} be the correlation coefficient between m\displaystyle m and n\displaystyle n.
Since m=−3x+10\displaystyle m = -3x + 10 has a negative coefficient for x\displaystyle x and n=−0.4y+5\displaystyle n = -0.4y + 5 has a negative coefficient for y\displaystyle y, the product of their scale changes is (−3)×(−0.4)=1.2>0\displaystyle (-3) \times (-0.4) = 1.2 > 0.
Since the product is positive, the sign of the correlation coefficient does not change: rmn=rxy\displaystyle r_{mn} = r_{xy}.
4. **Calculate regression coefficient of n\displaystyle n on m\displaystyle m (bnm\displaystyle b_{nm})**:
The regression coefficient of y\displaystyle y on x\displaystyle x is given by:
byx=rxyσyσx=0.80b_{yx} = r_{xy} \frac{\sigma_y}{\sigma_x} = 0.80
The regression coefficient of n\displaystyle n on m\displaystyle m is given by:
bnm=rmnσnσmb_{nm} = r_{mn} \frac{\sigma_n}{\sigma_m}
Substitute the relations σn=0.4σy\displaystyle \sigma_n = 0.4\sigma_y and σm=3σx\displaystyle \sigma_m = 3\sigma_x:
bnm=rxy0.4σy3σx=0.43(rxyσyσx)b_{nm} = r_{xy} \frac{0.4\sigma_y}{3\sigma_x} = \frac{0.4}{3} \left(r_{xy} \frac{\sigma_y}{\sigma_x}\right)
bnm=0.43byx=0.43×0.80=0.323≈0.1067b_{nm} = \frac{0.4}{3} b_{yx} = \frac{0.4}{3} \times 0.80 = \frac{0.32}{3} \approx 0.1067
Rounding to three decimal places yields 0.106\displaystyle 0.106.
Hence, **Option C** 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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