Question

Briefly explain why it is important to consider the value of \(s\) in addition to the value of \(r^{2}\) when evaluating the usefulness of the least squares regression line.

Short answer

The value of \( r^{2} \) only provides a part of the picture about how well the least squares regression line fits the data. Although it tells the proportion of the variation in the dependent variable that can be explained by the model, it doesn't provide any information about the scatter of the points around the regression line. That's where \( s \), the standard error of the estimate, comes in -- it measures the spread of the observations around the fitted regression line.

Step-by-step solution

Understanding the Role of \( r^{2} \)

\( r^{2} \) or the square of the correlation coefficient is a measure of the proportion of the total variation in the dependent variable that's predictable from the independent variable. It has a range between 0 and 1. A value of 1 represents a perfect correlation. Hence, the larger this value, the better our model is at predicting the data. However, this value alone doesn't give us the complete picture.

Role of Standard Error \( s \) in the Model

The standard error \( s \) is essentially the standard deviation of data points as they are distributed around the regression line. A smaller \( s \) signifies that our data points are closer to the line, meaning our model has less unexplained variance and hence, a more reliable predictor. However, if \( s \) is large, it means our data points are spread further from the line, implying that the model might not be reliable.

Balancing Both Parameters

Paying attention to both \( r^{2} \) and \( s \) is crucial. For instance, despite having a high \( r^{2} \), a model could still have a large standard error \( s \). This might be due to outliers or other data disturbances. Hence, while \( r^{2} \) helps to understand the quality of the model, \( s \) helps to understand the robustness.