Chapter 13.3, Problem 27E

To determine

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

Answer to Problem 27E

The equation is $y=\frac{10}{11}{x}^2-\frac{104}{55}x+\frac{78}{55}$ and the graph is shown in Figure 1.

Explanation of Solution

Given:

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

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	$x^2$	$x^3$	$x^4$	$xy$	$x^{2}y$
-1	4	1	-1	1	-4	4
0	2	0	0	0	0	0
1	0	1	1	1	0	0
3	4	9	27	81	12	36
${\displaystyle \sum _{i=1}^3{x}_i}=3$	${\displaystyle \sum _{i=1}^3{y}_i}=10$	${\displaystyle \sum _{i=1}^3{x}_i{}^2}=11$	${\displaystyle \sum _{i=1}^3{x}_i{}^3}=27$	${\displaystyle \sum _{i=1}^3{x}_i{}^4}=83$	${\displaystyle \sum _{i=1}^3{x}_i{y}_i}=8$	${\displaystyle \sum _{i=1}^3{x}_i{}^2{y}_i}=40$

Table 1

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

  $\begin{array}{c}4c+3b+11a=10\\ 3c+11b+27a=8\\ 11c+27b+83a=40\end{array}$

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

  $\begin{array}{c}a=\frac{10}{11}\\ b=-\frac{104}{55}\\ c=\frac{78}{55}\end{array}$

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

  $\begin{array}{c}f\left(x\right)=\left(\frac{10}{11}\right)x^2+\left(-\frac{104}{55}\right)x+\frac{78}{55}\\ y=\frac{10}{11}{x}^2-\frac{104}{55}x+\frac{78}{55}\end{array}$

In order to verify the equation $y=\frac{10}{11}{x}^2-\frac{104}{55}x+\frac{78}{55}$ , plot the points and the equation in a graphing utility. The graph is shown below:

  

Figure 1