The residual is the difference between the actual $Y$-value of a data point and the predicted $Y$-value from the regression line for the corresponding $X$-value. It is given by the formula:

$\text{Residual} = Y_{\text{actual}} - Y_{\text{predicted}}$

Step 1: Identify the given data

• Equation of the regression line: $Y = 5 - 2X$
• Point in the dataset: $(X, Y) = (1, 2)$

Step 2: Calculate the predicted $Y$-value

Substitute $X = 1$ into the regression equation: $Y_{\text{predicted}} = 5 - 2(1) = 5 - 2 = 3$

Step 3: Find the residual

$\text{Residual} = Y_{\text{actual}} - Y_{\text{predicted}} = 2 - 3 = -1$

Final Answer:

The residual at the point $(1, 2)$ is: $\boxed{-1}$

Would you like to see more details or related concepts? Here are some possible questions to expand your understanding:

1. What does a residual signify in the context of regression analysis?
2. How is the sum of residuals related to the regression line?
3. Can the residual be positive? If yes, under what circumstances?
4. How would we compute the residuals for all points in a dataset?
5. What is the role of least squares in determining the regression line?

Tip: Residuals measure how well the regression model fits the data; a smaller residual indicates a better fit for that data point.