The question in the image asks for the regression residual when $X_2 = 3$. We already know that the fitted value is $\hat{Y}_2 = 11$, which is derived from the regression equation $\hat{Y} = 2 + 3X$.

To find the regression residual, we can use the formula for residuals:

$\text{Residual} = Y_{\text{actual}} - \hat{Y}$

From the table, the actual value of $Y_2$ when $X_2 = 3$ is $Y_2 = 5$. So, the residual is:

$\text{Residual} = Y_2 - \hat{Y}_2 = 5 - 11 = -6$

Thus, the regression residual when $X_2 = 3$ is $-6$.

Would you like more details or have further questions?

Here are some related questions to explore:

1. How is the slope of a regression line determined?
2. What does a negative residual indicate in regression?
3. How can we assess the accuracy of a regression model?
4. What are the assumptions behind simple linear regression?
5. How can you interpret the y-intercept in a regression equation?

Tip: Always check for patterns in residuals, as they can indicate whether the linear model is appropriate for the data.