The variance of two variables ' x ' and ' y ' are 16 and 25 and covariance between ' x ' and ' y ' is 18.5 . Another two variables ' u ' and ' v ' are defined as u = (x-3)/2 and v = (y-2)/3 , then coefficient of correlation between ' u ' and ' v ' is

Option d: 0.925. Solution: Correlation is invariant under linear transformation: Given: - 2 = 16 ⟹ = 4 - 2 = 25 ⟹ = 5 - (x,y) = 18.5 Key property: Correlation coefficient is unchanged by change of origin and scale. For u = -3/2 and v = -2/3 : ruv = rxy Computing rxy : rxy = (x,y) × = 18.5/4 × 5 = 18.5/20 = 0.925 Since u = -3/2 (positive scale) and v = -2/3 (positive scale), both scaling factors are positive, so the sign of r is unchanged. ruv = 0.925 Hence, Option D is the correct answer.

The variance of two variables '$x$' and '$y$' are $16$ and $25$ and covariance between '$x$' and '$y$' is $18.5$. Another two variables '$u$' and '$v$' are defined as $u = (x-3)/2$ and $v = (y-2)/3$, then coefficient of correlation between '$u$' and '$v$' is

The correct answer is Option d: 0.925. **Correlation is invariant under linear transformation:** Given: - $\\sigma_x^2 = 16 \\Rightarrow \\sigma_x = 4$ - $\\sigma_y^2 = 25 \\Rightarrow \\sigma_y = 5$ - $\\text{Cov}(x,y) = 18.5$ **Key property:** Correlation coefficient is unchanged by change of origin and scale. For $u = \\frac{x-3}{2}$ and $v = \\frac{y-2}{3}$: $$r_{uv} = r_{xy}$$ **Computing $r_{xy}$:** $$r_{xy} = \\frac{\\text{Cov}(x,y)}{\\sigma_x \\cdot \\sigma_y} = \\frac{18.5}{4 \\times 5} = \\frac{18.5}{20} = 0.925$$ Since $u = \\frac{x-3}{2}$ (positive scale) and $v = \\frac{y-2}{3}$ (positive scale), both scaling factors are positive, so the sign of $r$ is unchanged. $$r_{uv} = 0.925$$ Hence, **Option D** is the correct answer.

Correlation and RegressionMCQPYQ Sep 24Question 3650 of 188

All Questions

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

Options

A0.800

B0.85

C0.90

D0.925

For any discrepancies in this question, email contact@cadada.in

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: -

\displaystyle \\sigma_x^2 = 16 \\Rightarrow \\sigma_x = 4

\displaystyle \\sigma_y^2 = 25 \\Rightarrow \\sigma_y = 5

\displaystyle \\text{Cov}(x,y) = 18.5

**Key property:** Correlation coefficient is unchanged by change of origin and scale. For

\displaystyle u = \\frac{x-3}{2}

and

\displaystyle v = \\frac{y-2}{3}

r_{uv} = r_{xy}

**Computing

\displaystyle r_{xy}

:**

r_{xy} = \\frac{\\text{Cov}(x,y)}{\\sigma_x \\cdot \\sigma_y} = \\frac{18.5}{4 \\times 5} = \\frac{18.5}{20} = 0.925

Since

\displaystyle u = \\frac{x-3}{2}

(positive scale) and

\displaystyle v = \\frac{y-2}{3}

(positive scale), both scaling factors are positive, so the sign of

\displaystyle r

is unchanged.

r_{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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