Question

Explain why it can be dangerous to use the leastsquares line to obtain predictions for \(x\) values that are substantially larger or smaller than those contained in the sample.

Short answer

Using the least squares line helps predict one variable based on another, but extrapolation, i.e., predicting for values outside the sample range, is dangerous. That's because the linearity assumption may not hold, all influencing factors may not be captured, and prediction errors are larger outside the sample range.

Step-by-step solution

Understanding the Role of Least Squares Line

The least squares line, also known as the line of best fit, is used in regression analysis to predict the value of one variable (dependent variable) based on the value of another variable (independent variable). It works under the assumption that there is a linear relationship between these variables.

Conceptualizing Out-of-Sample Predictions

An out-of-sample prediction refers to using the regression line (least squares line) to predict the dependent variable for values of the independent variable that are not within the range of the sample data. If \(x\) values are substantially larger or smaller than those of in the sample, it means the prediction is extrapolated far beyond the range of observed data.

Explaining the Dangers of Extrapolation

Predicting values for a variable outside the range of sample data is known as extrapolation. It can be dangerous for several reasons: 1) The assumption of linearity might not hold outside the sample range. Real-world relationships can be non-linear, and trends may change. 2) The sample data may not incorporate all factors affecting the relationship, which may be more prevalent outside the sample range. 3) Errors of estimates are larger and predictions are less reliable outside the sample range. Therefore, using the least squares line to obtain predictions for \(x\) values substantially larger or smaller than those in the sample can be risky and misleading.