To solve this problem, we need to determine the standard deviations of [tex]\( x \)[/tex] and [tex]\( y \)[/tex], as well as the covariance between [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
Here are the given values:
- [tex]\( b_{x y} = 0.6 \)[/tex] (regression coefficient of [tex]\( y \)[/tex] on [tex]\( x \)[/tex])
- [tex]\( r = 0.8 \)[/tex] (correlation coefficient between [tex]\( x \)[/tex] and [tex]\( y \)[/tex])
- [tex]\( \text{variance of } x = 9 \)[/tex]
First, we will find the standard deviation of [tex]\( x \)[/tex]. Since the variance is provided:
1. Calculate the standard deviation of [tex]\( x \)[/tex]:
[tex]\[
\text{std\_dev}_x = \sqrt{\text{variance}_x} = \sqrt{9} = 3.0
\][/tex]
Next, using the regression coefficient [tex]\( b_{x y} = 0.6 \)[/tex], we can find the standard deviation of [tex]\( y \)[/tex]:
2. Using the relation [tex]\( b_{x y} = \frac{\text{std\_dev}_x}{\text{std\_dev}_y} \)[/tex], solve for [tex]\( \text{std\_dev}_y \)[/tex]:
[tex]\[
\text{std\_dev}_y = \frac{\text{std\_dev}_x}{b_{x y}} = \frac{3.0}{0.6} = 5.0
\][/tex]
Finally, we use the correlation coefficient [tex]\( r = 0.8 \)[/tex] to find the covariance between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
3. The covariance formula is given by:
[tex]\[
\text{cov}(x, y) = r \times \text{std\_dev}_x \times \text{std\_dev}_y
\][/tex]
Substituting the values:
[tex]\[
\text{cov}(x, y) = 0.8 \times 3.0 \times 5.0 = 12.0
\][/tex]
So, the standard deviations and covariance are:
- [tex]\(\text{std\_dev}_x = 3.0\)[/tex]
- [tex]\(\text{std\_dev}_y = 5.0\)[/tex]
- [tex]\(\text{cov}(x, y) = 12.0\)[/tex]
Given this, the correct values do not match any of the choices from the provided options exactly, so the answer is:
(d) None of these
