Question

The coefficient of correlation between two variables $X$ and $Y$ is 0.6. Their covariance is 4.8. The variance of $X$ is 9. Then the standard deviation of $Y$ is

• A. $\frac{4.8}{3 \times 0.6}$
• B. $\frac{3}{4.8 \times 0.6}$
• C. $\frac{9}{4.8 \times 0.6}$
• D. $\frac{4.8}{9 \times 0.6}$

Hint:

Always calculate the standard deviation from the given variance first before plugging it into the correlation coefficient equation.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The correlation coefficient ($r$) is a standardized measure of the linear relationship between two variables, expressed as the ratio of their covariance to the product of their individual standard deviations.
Key Formula or Approach:
The formula for the Pearson correlation coefficient is:
\[ r = \frac{\text{Cov}(X, Y)}{\sigma_X \cdot \sigma_Y} \]

Step 2: Detailed Explanation:
We are given the following values:
Correlation coefficient, $r = 0.6$
Covariance, $\text{Cov}(X, Y) = 4.8$
Variance of $X$, $\text{Var}(X) = \sigma_X^2 = 9 \implies \sigma_X = \sqrt{9} = 3$
Substitute these parameters into the correlation formula:
\[ 0.6 = \frac{4.8}{3 \cdot \sigma_Y} \]
Rearrange the equation to solve for the standard deviation of $Y$, $\sigma_Y$:
\[ 3 \cdot \sigma_Y = \frac{4.8}{0.6} \]
\[ \sigma_Y = \frac{4.8}{3 \cdot 0.6} \]
Therefore, the standard deviation of $Y$ is given by the expression $\frac{4.8}{3 \times 0.6}$.

Step 3: Final Answer
The correct option is (A).