Let’s solve this step-by-step:
First, we need to find the standard deviations of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] using their variances.
1. Given the variance of [tex]\(x\)[/tex], [tex]\(\text{var}(x) = 6.25\)[/tex], we calculate the standard deviation of [tex]\(x\)[/tex]:
[tex]\[
\sigma_x = \sqrt{\text{var}(x)} = \sqrt{6.25} = 2.5
\][/tex]
2. Given the variance of [tex]\(y\)[/tex], [tex]\(\text{var}(y) = 4\)[/tex], we calculate the standard deviation of [tex]\(y\)[/tex]:
[tex]\[
\sigma_y = \sqrt{\text{var}(y)} = \sqrt{4} = 2.0
\][/tex]
Next, we use the given covariance of [tex]\(x\)[/tex] and [tex]\(y\)[/tex], [tex]\(\text{cov}(x, y) = 0.9\)[/tex], to find the coefficient of correlation, [tex]\(r\)[/tex], using the formula:
[tex]\[
r = \frac{\text{cov}(x, y)}{\sigma_x \cdot \sigma_y}
\][/tex]
3. Substitute the calculated standard deviations and the given covariance into the formula:
[tex]\[
r = \frac{0.9}{2.5 \cdot 2.0}
\][/tex]
4. Calculate the denominator:
[tex]\[
2.5 \cdot 2.0 = 5.0
\][/tex]
5. Now, perform the division:
[tex]\[
r = \frac{0.9}{5.0} = 0.18
\][/tex]
Therefore, the coefficient of correlation between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is [tex]\(0.18\)[/tex].
The correct answer is:
(B) 0.18
