Question

If the variances of two variables $x$ and $y$ are respectively $9$ and $16$ and their covariance is $8$, then their coefficient of correlation is

• A. $\frac{2}{3}$
• B. $\frac{8}{3\sqrt{2}}$
• C. $\frac{9}{8\sqrt{2}}$
• D. $\frac{2}{9}$

Answer 1

The correct option is A

$\frac{2}{3}$

Explanation for the correct option.

Step 1. Find the coefficient of correlation.

The coefficient of correlation is given as: $r=\frac{\mathrm{cov}\left(x,y\right)}{\sqrt{\mathrm{var}\left(x\right)·\mathrm{var}\left(y\right)}}$.

The covariance of two variables $x$ and $y$ is $\mathrm{cov}\left(x,y\right)=8$.

The variance of $x$ is $9$ and the variance of $y$ is $16$.

So the coefficient of correlation is:

$$\begin{gathered} r=\frac{8}{\sqrt{9·16}} \\ r=\frac{8}{\sqrt{144}} \\ r=\frac{8}{12} \\ r=\frac{2}{3} \end{gathered}$$

Hence, the correct option is A.