Chapter 13, Problem 38RE

To determine

Finding a Least Squares Regression Parabola.

Answer to Problem 38RE

Least Squares Regression Parabola is $y=\frac{8}{7}{x}^2+\frac{233}{35}x+\frac{48}{5}.$

Explanation of Solution

Given information: Find the least squares regression parabola for the points,

(-5,5),(-4,1),(-3,0),(-2,1),(-1,4).

Calculation:

Formula:

The table shows the calculation involved in finding the least squares regression using $n=5.$

  

The least squares regression parabola or quadratic given by ,

  $f(x)=a{x}^2+bx+c.$ where :

  $\begin{array}{l}nc+({\displaystyle {\sum }_{i=1}^n{x}_i})b+({\displaystyle {\sum }_{i=1}^n{x}_i{}^2})a={\displaystyle {\sum }_{i=1}^n{y}_i}.\\ 55a-15b+5c=11\mathrm{..............}(1).\\ ({\displaystyle {\sum }_{i=1}^n{x}_i})c+({\displaystyle {\sum }_{i=1}^n{x}_i{}^2})b+({\displaystyle {\sum }_{i=1}^n{x}^3{}_i})a={\displaystyle {\sum }_{i=1}^n{x}_i}{y}_i.\\ -225a+55b-15c=-35.\\ -45a+11b-3c=-7\mathrm{...............}(2).\\ ({\displaystyle {\sum }_{i=1}^n{x}_i{}^2})c+({\displaystyle {\sum }_{i=1}^n{x}_i{}^3})b+({\displaystyle {\sum }_{i=1}^n{x}_i{}^4})a={\displaystyle {\sum }_{i=1}^n{x}_i{}^2}{y}_i.\\ 979a-225b+55c=149\mathrm{...............}(3).\\ \end{array}$

Solving for the system of three variables ,we have : $a=\frac{8}{7},b=\frac{233}{35},c=\frac{48}{5}.$

Therefore ,the least squares regression parabola is $y=\frac{8}{7}{x}^2+\frac{233}{35}x+\frac{48}{5}.$