Regression Line Characteristics and Implications

A

Since there is no error in the regression, the R-square equals 1 (100%). The slope coefficient can be any real number, not necessarily 1. The residual error will be zero.

If all the data points in a regression lie exactly on a straight line, it means that the relationship between the independent variable(s) and the dependent variable is perfectly linear. In this scenario, the following statements are true:

I. The observed values of the dependent variable will equal the predicted values. When the data points lie exactly on a straight line, it means that the regression model perfectly fits the data. Therefore, the predicted values of the dependent variable will be equal to the observed values. This statement is true.

II. The R-square will equal 100%. The R-square (coefficient of determination) measures the proportion of the variation in the dependent variable that is explained by the independent variable(s). In this case, where the data points lie exactly on a straight line, it implies that the regression model explains all the variation in the dependent variable. Thus, the R-square will be equal to 100%. This statement is true.

III. The slope coefficient will be 1. The slope coefficient represents the change in the dependent variable for a unit change in the independent variable. When all the data points lie exactly on a straight line, it indicates a perfect positive relationship between the variables. In such a case, the slope coefficient will be equal to 1. This statement is true.