If an observation has a residual of 0, it means that there is no difference between the observed value and the predicted value for that observation. Let's evaluate each of the given statements in detail:
OA. An error was made in the calculation as a residual cannot be zero.
This statement is false. A residual of 0 indicates that the predicted value perfectly matches the observed value for that particular observation. Thus, it is possible for a residual to be zero without any errors in calculation.
OB. The R-square will be 1.
This statement is false. The R-square value of 1 indicates that the regression model perfectly predicts all observations, meaning all residuals are zero for the entire dataset. However, a single observation with a residual of 0 does not imply that the R-square for the entire model is 1.
C. Its predicted value is the same as its observed value.
This statement is true. A residual of 0 means no difference between the observed value and the predicted value; hence they are the same.
OD. The correlation coefficient will be 0.
This statement is false. The correlation coefficient measures the strength and direction of a linear relationship between two variables. A single residual of 0 does not determine the overall correlation coefficient for the dataset.
OE. It is an outlier.
This statement is false. An outlier is typically defined as an observation that deviates significantly from other observations. An observation with a residual of 0 indicates it perfectly fits the regression line, which is the opposite of being an outlier.
Therefore, the correct answer is:
C. Its predicted value is the same as its observed value.
