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If var(x) = 6.25, var(y) = 4, and cov(x, y) = 0.9, then the coefficient of correlation between x and y is:

A. 0.25
B. 0.18
C. 0.32
D. 0.29


Sagot :

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