Chapter 13.3, Problem 31E

To determine

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

Answer to Problem 31E

The equation is $y=-\frac{5}{4}{x}^2+\frac{9}{20}x+\frac{199}{20}$ and the graph is shown in Figure 1.

Explanation of Solution

Given:

The points are $\left(0,10\right)$ , $\left(1,9\right)$ , $\left(2,6\right)$ and $\left(3,0\right)$ .

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	10	0	0	0	0	0
1	9	1	1	1	9	9
2	6	4	8	16	12	24
3	0	9	27	81	0	0
${\displaystyle \sum _{i=1}^3{x}_i}=6$	${\displaystyle \sum _{i=1}^3{y}_i}=25$	${\displaystyle \sum _{i=1}^3{x}_i{}^2}=14$	${\displaystyle \sum _{i=1}^3{x}_i{}^3}=36$	${\displaystyle \sum _{i=1}^3{x}_i{}^4}=98$	${\displaystyle \sum _{i=1}^3{x}_i{y}_i}=21$	${\displaystyle \sum _{i=1}^3{x}_i{}^2{y}_i}=33$

Table 1

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

  $\begin{array}{c}4c+6b+14a=25\\ 6c+14b+36a=21\\ 14c+36b+98a=33\end{array}$

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

  $\begin{array}{c}a=-\frac{5}{4}\\ b=\frac{9}{20}\\ c=\frac{199}{20}\end{array}$

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

  $\begin{array}{c}f\left(x\right)=\left(-\frac{5}{4}\right)x^2+\left(\frac{9}{20}\right)x+\frac{199}{20}\\ y=-\frac{5}{4}{x}^2+\frac{9}{20}x+\frac{199}{20}\end{array}$

In order to verify the equation $y=-\frac{5}{4}{x}^2+\frac{9}{20}x+\frac{199}{20}$ , plot the points and the equation in a graphing utility. The graph is shown below:

  

Figure 1