Question

Both \(r^{2}\) and \(s_{e}\) are used to assess the fit of a line. a. Is it possible that both \(r^{2}\) and \(s_{e}\) could be large for a bivariate data set? Explain. (A picture might be helpful.) b. Is it possible that a bivariate data set could yield values of \(r^{2}\) and \(s_{e}\) that are both small? Explain. (Again, a picture might be helpful.) c. Explain why it is desirable to have \(r^{2}\) large and \(s_{e}\) small if the relationship between two variables \(x\) and \(y\) is to be described using a straight line.

Short answer

Yes, both \(r^{2}\) and \(s_{e}\) could be large for a bivariate data set if there are outliers. Also, \(r^{2}\) and \(s_{e}\) can both be small if data points are close to the line but the line has a poor slope. It is desirable to have a large \(r^{2}\) and small \(s_{e}\) for a statistical model that accurately predicts the dependent variable.

Step-by-step solution

Scenario Analysis for large \(r^{2}\) and large \(s_{e}\)

Yes, it is possible that both \(r^{2}\) and \(s_{e}\) could be large for a bivariate data set. While \(r^{2}\) represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s), \(s_{e}\) is a measure of the differences between the predicted value and what is observed. A large \(r^{2}\) value signifies that there is a lot of variability that can be accounted for by the independent variable(s) in the model. However, a large standard error (\(s_{e}\)) would indicate that the observed values deviate significantly from the line produced by the model. This scenario could occur when there are outliers in the data.

Scenario Analysis for small \(r^{2}\) and small \(s_{e}\)

Also, it's possible to have a situation where a bivariate data set could yield values of \(r^{2}\) and \(s_{e}\) that are both small. A small \(r^{2}\) means that the predictor (independent variables) does not explain the variability in the output (dependent variable) well. A small \(s_{e}\) indicates that the difference between observed and predicted values is small. This could happen if the data points are close to the line, but the line does not have a good slope (possibly nearly horizontal) and hence does not provide a good fit.

Desirability of large \(r^{2}\) and small \(s_{e}\)

It is desirable to have the \(r^{2}\) large and \(s_{e}\) small when describing a relationship between two variables using a straight line. A large \(r^{2}\) implies that the independent variable(s) explain a large proportion of the variance in the dependent variable. A small \(s_{e}\) means the residuals (the differences between the observed and predicted values) are small, implying a good fit of the model. The combination indicates a good statistical model that accurately predicts the dependent variable.