Cauchy's Inequality and the Least Squares Line

Abstract

Many basic statistics texts give the impression that there is no simple and straightforward way to obtain the equation defining the least squares line (or regression line). Even though elementary derivations exist (see for instance [3], pp. 231-233), very often the equation is just stated, without proof, after a remark something like, It can be shown by the calculus that... , It is beyond the scope of this text. . . , details are too cumbersome.... Moreover, it does not seem to be common knowledge that there is a connection between the coefficient of linear correlation and Cauchy's inequality, which leads to a deeper understanding of why the correlation coefficient gives a measure of collinearity. The purpose of the present note is to give a pre-calculus approach to the least squares problem, parts of which parallel a procedure in the more advanced text [1]. We also discuss the connection with Cauchy's inequality and give an unexpected by-product of our approach.