The coefficient of correlation between the two variables X and Y is 0.6. Their covariance is 3.6 and variance of X is 4. The variance of Y is3.55.54.59

1. Break Down the Problem

We need to analyze the relationship between the two variables $X$ and $Y$ using the given statistics: correlation coefficient, covariance, and variances of $X$ and $Y$. The goal is to understand how the correlation coefficient relates to covariance and the variances of the two variables.

2. Relevant Concepts

The correlation coefficient $r$ between two variables can be expressed using the formula: $r = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X) \cdot \text{Var}(Y)}}$ Where:

• $\text{Cov}(X, Y)$ is the covariance between $X$ and $Y$.
• $\text{Var}(X)$ is the variance of $X$.
• $\text{Var}(Y)$ is the variance of $Y$.

3. Analysis and Detail

We are given:

• $r = 0.6$
• $\text{Cov}(X, Y) = 3.6$
• $\text{Var}(X) = 4$
• $\text{Var}(Y) = 3.55$

We can rearrange the correlation formula to solve for $\text{Var}(Y)$: $r \cdot \sqrt{\text{Var}(X) \cdot \text{Var}(Y)} = \text{Cov}(X, Y)$ Substituting in the known values: $0.6 \cdot \sqrt{4 \cdot \text{Var}(Y)} = 3.6$

First, isolate $\sqrt{\text{Var}(Y)}$: $\sqrt{4 \cdot \text{Var}(Y)} = \frac{3.6}{0.6} = 6$

Next, square both sides to eliminate the square root: $4 \cdot \text{Var}(Y) = 6^2 = 36$

Now, solve for $\text{Var}(Y)$: $\text{Var}(Y) = \frac{36}{4} = 9$

4. Verify and Summarize

We computed $\text{Var}(Y)$ under the assumption that all provided values are correct. The calculated variance $\text{Var}(Y)$ is $9$. The information provided suggests an inconsistency in the stated variance options for $Y$. The variance of $Y$ based on our calculations appears to differ from possible given options (3.55, 4.59).

Final Answer

The variance of $Y$ calculated from the given data is $\text{Var}(Y) = 9$.