Which of the following best explains why the slope m that minimizes must satisfy ?

MATHEMATICS · Single correct · AP · AP Statistics · Exploring Two-Variable Data · Exploring Two-Variable Data

Problem

Which of the following best explains why the slope m that minimizes E(m)=∑i=1n(yi−mxi)2E(m)=\sum_{i=1}^n\bigl(y_i-mx_i\bigr)^2 must satisfy ∑i=1nxi(yi−mxi)=0\sum_{i=1}^n x_i\bigl(y_i-mx_i\bigr)=0? (a) Because squaring the errors makes the largest deviation dominate, the largest individual error must be zero at the minimum. (b) Minimizing forces the unweighted sum of residuals to vanish, so ∑i=1n(yi−mxi)=0\sum_{i=1}^n (y_i-mx_i)=0 and hence ∑xi(yi−mxi)=0\sum x_i(y_i-mx_i)=0. (c) Setting the derivative of EE with respect to mm equal to zero makes the rate of change of total squared error vanish, so an extremum occurs when ∑i=1nxi(yi−mxi)=0\sum_{i=1}^n x_i(y_i-mx_i)=0. (d) At the minimum each residual is individually 00, implying their xix_i–weighted sum is also 00.

Related theory

Browse the problem library