Correlation and RegressionMCQPYQ Sep 24Question 3650 of 188
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The variance of two variables 'x\displaystyle x' and 'y\displaystyle y' are $16\displaystyle and $25 and covariance between 'x\displaystyle x' and 'y\displaystyle y' is $18.5$. Another two variables '$u$' and '$v$' are defined as $u = (x-3)/2and\displaystyle andv = (y-2)/3$, then coefficient of correlation between '$u$' and '$v$' is

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option d — 0.925

All Options:

  • A0.800
  • B0.85
  • C0.90
  • D0.925

Detailed Solution & Explanation

**Correlation is invariant under linear transformation:** Given: - sigmax2=16Rightarrowsigmax=4\displaystyle \\sigma_x^2 = 16 \\Rightarrow \\sigma_x = 4 - sigmay2=25Rightarrowsigmay=5\displaystyle \\sigma_y^2 = 25 \\Rightarrow \\sigma_y = 5 - textCov(x,y)=18.5\displaystyle \Cov\displaystyle \text{Cov}(x,y) = 18.5 **Key property:** Correlation coefficient is unchanged by change of origin and scale. For u=fracx−32\displaystyle u = \x−32\displaystyle \frac{x-3}{2} and v=fracy−23\displaystyle v = \y−23\displaystyle \frac{y-2}{3}: ruv=rxyr_{uv} = r_{xy} **Computing rxy\displaystyle r_{xy}:** rxy=fractextCov(x,y)sigmaxcdotsigmay=frac18.54times5=frac18.520=0.925r_{xy} = \textCov(x,y)sigmaxcdotsigmay\displaystyle \frac{\\text{Cov}(x,y)}{\\sigma_x \\cdot \\sigma_y} = \18.54times5\displaystyle \frac{18.5}{4 \\times 5} = \18.520=0\displaystyle \frac{18.5}{20} = 0.925 Since u=fracx−32\displaystyle u = \x−32\displaystyle \frac{x-3}{2} (positive scale) and v=fracy−23\displaystyle v = \y−23\displaystyle \frac{y-2}{3} (positive scale), both scaling factors are positive, so the sign of r\displaystyle r is unchanged. ruv=0.925r_{uv} = 0.925 Hence, **Option D** 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.

Variance

A measure of dispersion equal to the average of the squared deviations of each data point from the mean. Formula: σ² = Σ(xᵢ − x̄)² / n. Standard Deviation = √Variance.

Related Comparison Tables

Correlation vs Regression: Statistics for CA Foundation

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