Find the formula that gives the least squares error for the points (-1, 5), (2, 2), (5, -1).
A) $( A - B + 5 ) ^ { 2 }$ + $( 2 A + B - 2 ) ^ { 2 }$ + $( A + B + 1 ) ^ { 2 }$
B) $( - 2 A + B - 5 ) ^ { 2 }$ + $( A + B - 2 ) ^ { 2 }$ + $( 5 \mathrm {~A} + \mathrm { B } + 1 ) ^ { 2 }$
C) $( - A + B - 5 ) ^ { 2 }$ + $( 2 A + B - 2 ) ^ { 2 }$ + $( 5 \mathrm {~A} + \mathrm { B } + 1 ) ^ { 2 }$
D) $( - A + B - 5 ) ^ { 2 }$ + $( 2 A + B - 2 ) ^ { 2 }$ + $( A + B + 5 ) ^ { 2 }$

Question

Quizplus · Accepted Answer

The least squares error formula for a set of points and a line $y = Ax + B$ involves the sum of the squares of the differences between the y-values of the points and the y-values predicted by the line equation. For each point $(x, y)$, the error term is $(y - (Ax + B))^2$. Substituting the given points into this formula:1. For $(-1, 5)$, the error term is $(5 - (A(-1) + B))^2 = (-A + B - 5)^2$.2. For $(2, 2)$, the error term is $(2 - (A(2) + B))^2 = (2A + B - 2)^2$.3. For $(5, -1)$, the error term is $(-1 - (A(5) + B))^2 = (5A + B + 1)^2$.Adding these terms together gives the formula in choice C.

Find the Formula That Gives the Least Squares Error for the Points