Chapter 13.3, Problem 29E

To determine

To find:The least squares regression parabola for the given points and verify the equation using a graphing utility.

Answer to Problem 29E

The equation is $y=2x^2-3x+4$ and the graph is shown in Figure 1.

Explanation of Solution

Given:

The points are $\left(-2,18\right)$ , $\left(-1,9\right)$ , $\left(0,4\right)$ , $\left(1,3\right)$ and $\left(2,6\right)$ .

Calculation:

Let the general equation of the least squares regression parabola be $f\left(x\right)=a{x}^2+bx+c$ where the following equations are used to determine the unknowns a, b and c:

  $\begin{array}{c}cn+b{\displaystyle \sum _{i=1}^n{x}_i}+a{\displaystyle \sum _{i=1}^n{x}_i{}^2}={\displaystyle \sum _{i=1}^n{y}_i}\\ c{\displaystyle \sum _{i=1}^n{x}_i}+b{\displaystyle \sum _{i=1}^n{x}_i{}^2}+a{\displaystyle \sum _{i=1}^n{x}_i{}^3}={\displaystyle \sum _{i=1}^n{x}_i{y}_i}\\ c{\displaystyle \sum _{i=1}^n{x}_i{}^2}+b{\displaystyle \sum _{i=1}^n{x}_i{}^3}+a{\displaystyle \sum _{i=1}^n{x}_i{}^4}={\displaystyle \sum _{i=1}^n{x}_i{}^2{y}_i}\end{array}$

…(1)

In order to determine value of the coefficients of a, b and c in the set of equations (1), construct a table as shown below:

x	y	$x^2$	$x^3$	$x^4$	$xy$	$x^{2}y$
-2	18	4	-8	16	-36	72
-1	9	1	-1	1	-9	9
0	4	0	0	0	0	0
1	3	1	1	1	3	3
2	6	4	8	16	12	24
${\displaystyle \sum _{i=1}^3{x}_i}=0$	${\displaystyle \sum _{i=1}^3{y}_i}=40$	${\displaystyle \sum _{i=1}^3{x}_i{}^2}=10$	${\displaystyle \sum _{i=1}^3{x}_i{}^3}=0$	${\displaystyle \sum _{i=1}^3{x}_i{}^4}=34$	${\displaystyle \sum _{i=1}^3{x}_i{y}_i}=-30$	${\displaystyle \sum _{i=1}^3{x}_i{}^2{y}_i}=108$

Table 1

Substitute the values from the above table into the set of equations (1).

  $\begin{array}{c}5c+0b+10a=40\\ 0c+10b+0a=-30\\ 10c+0b+34a=108\end{array}$

Simplify the set of equations to determine the values of a, b and c.

  $\begin{array}{c}a=2\\ b=-3\\ c=4\end{array}$

So, the equation of the least squares regression parabola becomes,

  $\begin{array}{c}f\left(x\right)=\left(2\right)x^2+\left(-3\right)x+4\\ y=2x^2-3x+4\end{array}$

In order to verify the equation $y=2x^2-3x+4$ , plot the points and the equation in a graphing utility. The graph is shown below:

  

Figure 1