Chapter 13.3, Problem 26E

To determine

  $y=1.5x^2+0.5x+1$

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

Answer to Problem 26E

The equation is and the graph is shown in Figure 1.

Explanation of Solution

Given:

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

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	$x^2$	$x^3$	$x^4$	$xy$	.
-2	6	4	-8	16	-12	24
-1	2	1	-1	1	-2	2
1	3	1	1	1	3	3
${\displaystyle \sum _{i=1}^3{x}_i}=-2$	${\displaystyle \sum _{i=1}^3{y}_i}=11$	${\displaystyle \sum _{i=1}^3{x}_i{}^2}=6$	${\displaystyle \sum _{i=1}^3{x}_i{}^3}=-8$	${\displaystyle \sum _{i=1}^3{x}_i{}^4}=18$	${\displaystyle \sum _{i=1}^3{x}_i{y}_i}=-11$	${\displaystyle \sum _{i=1}^3{x}_i{}^2{y}_i}=29$

Table 1

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

  $\begin{array}{c}3c-2b+6a=11\\ -2c+6b-8a=-11\\ 6c-8b+18a=29\end{array}$

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

  $\begin{array}{c}a=1.5\\ b=0.5\\ c=1\end{array}$

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

  $\begin{array}{c}f\left(x\right)=\left(1.5\right)x^2+\left(0.5\right)x+1\\ y=1.5x^2+0.5x+1\end{array}$

In order to verify the equation $y=1.5x^2+0.5x+1$ , plot the points and the equation in a graphing utility. The graph is shown below:

  

Figure 1