To find the variance of $Y$ when given that one of the regression coefficients (likely $\beta_1$) is 1.3 and the standard deviation of $X$ is 2, we can use the following relationship:

$\beta_1 = \frac{\text{Cov}(X, Y)}{\text{Var}(X)}$

Given:

• Regression coefficient $\beta_1 = 1.3$
• Standard deviation of $X$, $\sigma_X = 2$ (so $\text{Var}(X) = \sigma_X^2 = 4$)

First, we calculate the covariance $\text{Cov}(X, Y)$ using the formula for the regression coefficient:

$\text{Cov}(X, Y) = \beta_1 \times \text{Var}(X)$

$\text{Cov}(X, Y) = 1.3 \times 4 = 5.2$

The variance of $Y$ can be found using the formula:

$\text{Var}(Y) = \beta_1^2 \times \text{Var}(X) + \text{Var}(\epsilon)$

However, the problem does not mention the variance of the error term ($\text{Var}(\epsilon)$). If we assume that there is no error or that the error term has a variance of zero, then:

$\text{Var}(Y) = \beta_1^2 \times \text{Var}(X)$

$\text{Var}(Y) = (1.3)^2 \times 4 = 1.69 \times 4 = 6.76$

Thus, the variance of $Y$ is $\text{Var}(Y) = 6.76$.

Do you want more details or have any questions?

Related Questions:

1. How would the variance of $Y$ change if the variance of the error term is not zero?
2. What is the covariance between $X$ and $Y$ if the regression coefficient is different?
3. How does changing the standard deviation of $X$ affect the variance of $Y$?
4. What assumptions are necessary for the linear regression model to hold?
5. How would the interpretation change if $\beta_1$ was negative instead of positive?

Tip: When dealing with regression analysis, understanding the role of variance and covariance is crucial, as they are foundational to determining how changes in one variable affect another.