<p>We need to choose the correct regression line because each line is designed to minimise a different type of prediction error. When we use \(x\) to estimate \(y\), the regression line of \(y\) on \(x\) is chosen because it makes the errors in the predicted \(y\)-values as small as possible. <br></p>
<p>On the other hand, when we use \(y\) to estimate \(x\), the regression line of \(x\) on \(y\) is used because it minimises the errors in the predicted \(x\)-values. Therefore, the choice of regression line depends on which variable we are trying to predict. Using the wrong line may not give the smallest possible errors for the variable of interest.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: center; --cell-justify: center">Scenario</th><th style="text-align: center; --cell-justify: center">Given&nbsp;\(x\) to estimate \(y\)</th><th style="text-align: center; --cell-justify: center">Given&nbsp;\(y\) to estimate \(x\)</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(y\) depends on \(x\)</td><td colspan="2" style="text-align: center; --cell-justify: center">Use the \(y\) on \(x\) line equation.</td></tr><tr><td style="text-align: center; --cell-justify: center">\(x\) depends on \(y\)</td><td colspan="2" style="text-align: center; --cell-justify: center">Use the \(x\) on \(y\) line equation.</td></tr><tr><td style="text-align: center; --cell-justify: center">No observable dependence</td><td style="text-align: center; --cell-justify: center">Use the \(y\) on \(x\) line equation.</td><td style="text-align: center; --cell-justify: center">Use the \(x\) on \(y\) line equation.</td></tr></tbody></table></div></div>

We need to choose the correct regression line because each line is designed to minimise a different type of prediction error. When we use \(x\) to estimate \(y\), the regression line of \(y\) on \(x\) is chosen because it makes the errors in the predicted \(y\)-values as small as possible. 


On the other hand, when we use \(y\) to estimate \(x\), the regression line of \(x\) on \(y\) is used because it minimises the errors in the predicted \(x\)-values. Therefore, the choice of regression line depends on which variable we are trying to predict. Using the wrong line may not give the smallest possible errors for the variable of interest.

Correlation & Regressions

International Baccalaureate Higher Level Mathematics

Choice of Regression Lines

Choice of Regression Lines