Chapter 13.3, Problem 25E

To determine

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

Answer to Problem 25E

The equation is $y=-0.75x^2+3.5x$ and the graph is shown in Figure 1.

Explanation of Solution

Given:

The points are $\left(0,0\right)$ , $\left(2,4\right)$ and $\left(4,2\right)$ .

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	$x^2$	$x^3$	$x^4$	$xy$	$x^{2}y$
0	0	0	0	0	0	0
2	4	4	8	16	8	16
4	2	16	64	256	8	32
${\displaystyle \sum _{i=1}^3{x}_i}=6$	${\displaystyle \sum _{i=1}^3{y}_i}=6$	${\displaystyle \sum _{i=1}^3{x}_i{}^2}=20$	${\displaystyle \sum _{i=1}^3{x}_i{}^3}=72$	${\displaystyle \sum _{i=1}^3{x}_i{}^4}=272$	${\displaystyle \sum _{i=1}^3{x}_i{y}_i}=16$	${\displaystyle \sum _{i=1}^3{x}_i{}^2{y}_i}=48$

Table 1

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

  $\begin{array}{c}3c+6b+20a=6\\ 6c+20b+72a=16\\ 20c+72b+272a=48\end{array}$

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

  $\begin{array}{c}a=-0.75\\ b=3.5\\ c=0\end{array}$

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

  $\begin{array}{c}f\left(x\right)=\left(-0.75\right)x^2+3.5x+0\\ y=-0.75x^2+3.5x\end{array}$

In order to verify the equation $y=-0.75x^2+3.5x$ , plot the points and the equation in a graphing utility. The graph is shown below:

  

Figure 1