Question

Briefly explain why a small value of \(s_{e}\) is desirable in a regression setting.

Short answer

A small \(s_{e}\), or standard error of estimate, is desirable in a regression setting because it suggests a high level of precision in the model's predictions, with a minimal level of unexplained variability in the dependent variable. It indicates that the observed data points closely fit the estimated regression line, implying that the model is reliable and accurate.

Step-by-step solution

Understand the Role of \(s_{e}\) in Regression

In regression analysis, the standard error of the estimate, or \(s_{e}\), indicates how close the data are to the fitted regression line. It's the standard deviation of the residuals.

Explain the Desirability of Small \(s_{e}\)

A small \(s_{e}\) is desirable as it indicates that the observed values or data points are closely packed around the estimated regression line. This means that there's a high degree of precision in the predictions made by the model, and a minimal level of unaccounted variability in the response variable, hence maximizing the explanatory power of the independent variables.

Conclude

Overall, a small \(s_{e}\) indicates that the regression model predicts the dependent variable well, with very little inaccuracy or variability. Therefore, in a regression setting, one would ideally want the \(s_{e}\) to be as small as possible to ensure the accuracy and reliability of the model's predictions.