The covariance between two variables x and y is 72 . The variances of x and y are 144 and 81 . The coefficient of correlation is

Option c: 2/3. Solution: Given Data: - Covariance between x and y : (x, y) = 72 - Variance of x : (x) = 2 = 144 - Variance of y : (y) = 2 = 81 Step 1: Calculate the Standard Deviations The standard deviation of x ( ) is: = ()(x) = (144) = 12 The standard deviation of y ( ) is: = ()(y) = (81) = 9 Step 2: Calculate the Correlation Coefficient ( r ) The formula for the Karl Pearson correlation coefficient is: r = (x, y) × Substitute the given values into the formula: r = 72/12 × 9 r = 72/108 Simplifying the fraction: r = 2/3 Note on Typo: The source file lists Option A ( 1/3 ) as the correct option, which is a typographical error. The mathematically correct value is 2/3 , which corresponds to Option C. Hence, Option C is the correct answer.

The covariance between two variables $x$ and $y$ is $72$. The variances of $x$ and $y$ are $144$ and $81$. The coefficient of correlation is

The correct answer is Option c: $\\frac{2}{3}$. **Given Data:** - Covariance between $x$ and $y$: $\\text{Cov}(x, y) = 72$ - Variance of $x$: $\\text{Var}(x) = \\sigma_x^2 = 144$ - Variance of $y$: $\\text{Var}(y) = \\sigma_y^2 = 81$ **Step 1: Calculate the Standard Deviations** The standard deviation of $x$ ($\\sigma_x$) is: $$\\sigma_x = \\sqrt{\\text{Var}(x)} = \\sqrt{144} = 12$$ The standard deviation of $y$ ($\\sigma_y$) is: $$\\sigma_y = \\sqrt{\\text{Var}(y)} = \\sqrt{81} = 9$$ **Step 2: Calculate the Correlation Coefficient ($r$)** The formula for the Karl Pearson correlation coefficient is: $$r = \\frac{\\text{Cov}(x, y)}{\\sigma_x \\times \\sigma_y}$$ Substitute the given values into the formula: $$r = \\frac{72}{12 \\times 9}$$ $$r = \\frac{72}{108}$$ Simplifying the fraction: $$r = \\frac{2}{3}$$ **Note on Typo:** The source file lists Option A ($\\frac{1}{3}$) as the correct option, which is a typographical error. The mathematically correct value is $\\frac{2}{3}$, which corresponds to Option C. Hence, **Option C** is the correct answer.

Correlation and RegressionMCQMTP Nov 19Question 3735 of 188

All Questions

The covariance between two variables $\displaystyle x$ and $\displaystyle y$ is $\displaystyle 72$ . The variances of $\displaystyle x$ and $\displaystyle y$ are $\displaystyle 144$ and $\displaystyle 81$ . The coefficient of correlation is

Options

\displaystyle \\frac{1}{3}

\displaystyle \\frac{4}{5}

\displaystyle \\frac{2}{3}

\displaystyle \\frac{3}{5}

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

Correct Answer

✅ Option c — $\displaystyle \\frac{2}{3}$

All Options:

A $\displaystyle \\frac{1}{3}$
B $\displaystyle \\frac{4}{5}$
C $\displaystyle \\frac{2}{3}$
D $\displaystyle \\frac{3}{5}$

Detailed Solution & Explanation

**Given Data:** - Covariance between

\displaystyle x

and

\displaystyle y

\displaystyle \\text{Cov}(x, y) = 72

- Variance of

\displaystyle x

\displaystyle \\text{Var}(x) = \\sigma_x^2 = 144

- Variance of

\displaystyle y

\displaystyle \\text{Var}(y) = \\sigma_y^2 = 81

**Step 1: Calculate the Standard Deviations** The standard deviation of

\displaystyle x

(

\displaystyle \\sigma_x

) is:

\\sigma_x = \\sqrt{\\text{Var}(x)} = \\sqrt{144} = 12

The standard deviation of

\displaystyle y

(

\displaystyle \\sigma_y

) is:

\\sigma_y = \\sqrt{\\text{Var}(y)} = \\sqrt{81} = 9

**Step 2: Calculate the Correlation Coefficient (

\displaystyle r

)** The formula for the Karl Pearson correlation coefficient is:

r = \\frac{\\text{Cov}(x, y)}{\\sigma_x \\times \\sigma_y}

Substitute the given values into the formula:

r = \\frac{72}{12 \\times 9}

r = \\frac{72}{108}

Simplifying the fraction:

r = \\frac{2}{3}

**Note on Typo:** The source file lists Option A (

\displaystyle \\frac{1}{3}

) as the correct option, which is a typographical error. The mathematically correct value is

\displaystyle \\frac{2}{3}

, which corresponds to Option C. 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

Correlation

A statistical measure that expresses the extent to which two variables are linearly related.

Related Comparison Tables

Correlation vs Regression: Statistics for CA Foundation

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The covariance between two variables x\displaystyle xx and y\displaystyle yy is 72\displaystyle 7272. The variances of x\displaystyle xx and y\displaystyle yy are 144\displaystyle 144144 and 81\displaystyle 8181. The coefficient of correlation is