To determine what it means for a data point to have a residual of -1, we need to understand the concept of residuals in the context of linear regression.
A residual is the difference between the observed value (the actual data point) and the predicted value (the value estimated by the regression line) for a given data point. Mathematically, it can be expressed as:
[tex]\[ \text{Residual} = \text{Observed Value} - \text{Predicted Value} \][/tex]
Given a residual of -1, we can interpret this as follows:
1. Definition: A residual of -1 means that the observed value is 1 unit less than the predicted value.
[tex]\[ \text{Observed Value} = \text{Predicted Value} - 1 \][/tex]
2. Interpretation:
- If the residual were zero, the data point would lie exactly on the regression line.
- A positive residual would indicate that the observed value is above the regression line.
- A negative residual (such as -1) indicates that the observed value is below the regression line.
Therefore, if a data point has a residual of -1, it means that the actual observed value of this data point is 1 unit below the value predicted by the regression line.
Conclusion:
Given the options:
A. The predicted value for that point is -1.
B. The point lies directly on the regression line.
C. The point lies 1 unit above the regression line.
None of these options accurately describe a residual of -1, except when considered the following option:
It is clear that the most accurate representation is:
The point lies 1 unit below the regression line.
This fits with our detailed understanding of what a residual is and how it indicates the position of the observed value in relation to the predicted value given by the regression line.
