Chapter 2.8, Problem 22E

a

To determine

To graph: Graph and find the linear model and quadratic model for the data.

Explanation of Solution

Given information:

  

Graph : linear regression graph

  

Quadratic regression graph

  

Interpretation : A quadratic model finds the equation of the parabola that best fits the set of data and its equation is given by $y=a{x}^2+bx+c$ .

Linear regression is used to predict the relationship between two variables and is given by the formula $y=mx+b$

We will define the value of R square to get the best fit line from both the graph.

b

To determine

To calculate: calculate which graph best fit the data.

Answer to Problem 22E

Since the value of R square is more nearer to one in quadratic model that is $R^2=0.9825$ , therefore quadratic model will best suit for the best fit line.

Explanation of Solution

Given information:

Formula used: A quadratic model finds the equation of the parabola that best fits the set of data and its equation is given by $y=a{x}^2+bx+c$ .

  $\left[\begin{array}{ccc}{\displaystyle \sum _{i=1}^n{X}_i^4}& {\displaystyle \sum _{i=1}^n{X}_i^3}& {\displaystyle \sum _{i=1}^n{X}_i^2}\\ {\displaystyle \sum _{i=1}^n{X}_i^3}& {\displaystyle \sum _{i=1}^n{X}_i^2}& {\displaystyle \sum _{i=1}^n{X}_i}\\ {\displaystyle \sum _{i=1}^n{X}_i^2}& {\displaystyle \sum _{i=1}^n{X}_i}& n\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\end{array}\right]=\left[\begin{array}{c}{\displaystyle \sum _{i=1}^n{X}^2{}_i{Y}_i}\\ {\displaystyle \sum _{i=1}^n{X}_i{Y}_i}\\ {\displaystyle \sum _{i=1}^n{Y}_i}\\ \end{array}\right]$ and $\begin{array}{l}{R}^2=1-\frac{SSE}{SST}\\ SSE={\displaystyle \sum _{i=1}^n{(y_i-a{x}_i^2-b{x}_i-c)}^2}\\ SST={\displaystyle \sum _{i=1}^n{(y_i-\overline{y})}^2}\end{array}$

Linear regression is used to predict the relationship between two variables and is given by the formula $y=mx+b$

  $b=\frac{{\displaystyle \sum _{}^{}y}{\displaystyle \sum _{}^{}{x}^2}-{\displaystyle \sum _{}^{}xy}{\displaystyle \sum _{}^{}x}}{{\displaystyle \sum _{}^{}{x}^2}-{\displaystyle \sum _{}^{}{(x)}^2}}$ , $m=\frac{n{\displaystyle \sum _{}^{}xy}-{\displaystyle \sum _{}^{}y}{\displaystyle \sum _{}^{}x}}{n{\displaystyle \sum _{}^{}{x}^2}-{\displaystyle \sum _{}^{}{(x)}^2}}$

Calculation:

From above graph the respective values of quadratic model are $\begin{array}{l}a=5.196,\\ b=-20898.4,\\ c=2.101\times {10}^7\end{array}$ and $R^2=0.9825$

Therefore the equation of the quadratic model is given by,

  $y=5.196x^2-20898.4x+2.101\times {10}^7$

Now the respective values of linear model are $\begin{array}{l}m=27.657\\ b=-55475.7\end{array}$ and $R^2=0.9137$

Therefore the equation of the quadratic model is given by,

  $y=27.657x-55475.7$

Since the value of R square is more nearer to one in quadratic model , therefore quadratic model will best suit for the best fit line.