What does a larger value of r2 indicate about the observations in relation to the least squares line?

A larger value of ( r^2 )—the coefficient of determination—indicates that a greater proportion of the variance in the dependent variable can be explained by the independent variable through the least squares regression line. This means that the observations are more closely grouped around the least squares line, suggesting a stronger linear relationship between the variables.

When ( r^2 ) is high, it reflects that the fitted model does a good job of capturing the underlying relationship in the data, leading to observations that deviate minimally from the predicted values. Therefore, a strong clustering of points around the line is observed, demonstrating a clear and predictable pattern.

On the contrary, a lower value of ( r^2 ) would imply that the observations are more spread out from the least squares line, indicating a weaker relationship between the variables. The observations would appear more random with greater variance not explained by the model.