The problem in the image reads as follows:

18. Coefficient of correlation between variables $X$ and $Y$ is 0.3, and their covariance is 12. If the variance of $X$ is 9, find the standard deviation of $Y$.

We are given the following information:

• Correlation coefficient, $r = 0.3$
• Covariance, $\text{Cov}(X, Y) = 12$
• Variance of $X$, $\sigma_X^2 = 9$

We need to find the standard deviation of $Y$, denoted by $\sigma_Y$.

Formula to use:

The formula for the correlation coefficient $r$ in terms of covariance and standard deviations is:

$r = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}$

Where:

• $\text{Cov}(X, Y)$ is the covariance,
• $\sigma_X$ is the standard deviation of $X$, and
• $\sigma_Y$ is the standard deviation of $Y$.

Step 1: Calculate the standard deviation of $X$

Since the variance of $X$ is given as $9$, we can find the standard deviation of $X$ by taking the square root of the variance: $\sigma_X = \sqrt{\sigma_X^2} = \sqrt{9} = 3$

Step 2: Rearrange the formula for $\sigma_Y$

Rearranging the formula for the correlation coefficient to solve for $\sigma_Y$:

$\sigma_Y = \frac{\text{Cov}(X, Y)}{r \sigma_X}$

Step 3: Plug in the values

Substituting the given values:

$\sigma_Y = \frac{12}{0.3 \times 3}$

Now, let's calculate the result:

$\sigma_Y = \frac{12}{0.9} = 13.33$

Final Answer:

The standard deviation of $Y$ is approximately 13.33.

Would you like further details on any step, or do you have any other questions?

Here are 5 questions you might find useful:

1. How does covariance relate to correlation?
2. What is the geometric interpretation of the correlation coefficient?
3. How would you find variance if the standard deviation was given?
4. What are other ways to calculate covariance besides the given information?
5. How do outliers affect the correlation coefficient?

Tip: The correlation coefficient $r$ always lies between -1 and 1, where values closer to 1 or -1 indicate a stronger linear relationship between the variables.