Chapter 7.3, Problem 69E

a.

To determine

To find: The least squares regression parabola for the given points.

Answer to Problem 69E

The least squares regression parabola is $y=0.0514286x^2+0.877143x+1.88563$ .

Explanation of Solution

Given information:

In testing a new automobile breaking system, engineers recorded the speed $x$ (in miles per hour) and the stopping distance $y$ (in feet). The table shows the results.

  

For the above data points, the sums have been evaluated and that results to the following system of linear equations.

  $\{\begin{array}{c}5c+250b+13500a=923\\ 250c+13500b+775000a=52170\\ 13500c+775000b+46590000a=3101300\end{array}$

Concept used:

One way to find the least squares regression parabola $y=a{x}^2+bx+c$ for a set of points $\left(x_1,y_1\right),\left(x_2,y_2\right),\mathrm{...},\left(x_n,y_n\right)$ is by solving the system below for $a,b,\text{and }c.$

  $\{\begin{array}{c}nc+\left({\displaystyle \sum _{i=1}^n{x}_i}\right)b+\left({\displaystyle \sum _{i=1}^n{x}_i{}^2}\right)a={\displaystyle \sum _{i=1}^n{y}_i}\\ \left({\displaystyle \sum _{i=1}^n{x}_i}\right)c+\left({\displaystyle \sum _{i=1}^n{x}_i{}^2}\right)b+\left({\displaystyle \sum _{i=1}^n{x}_i{}^3}\right)a={\displaystyle \sum _{i=1}^n{x}_i{y}_i}\\ \left({\displaystyle \sum _{i=1}^n{x}_i{}^2}\right)c+\left({\displaystyle \sum _{i=1}^n{x}_i{}^3}\right)b+\left({\displaystyle \sum _{i=1}^n{x}_i{}^4}\right)a={\displaystyle \sum _{i=1}^n{x}_i{}^2{y}_i}\end{array}$

Calculations:

The steps of solving the system of equations using Row-Echelon Form and Back substitutions are as follows:

Subtracting 50 times the first equation from the second equation to produce a new second equation, and subtracting 2700 times the first equation from the third equation to produce a new third equation, we get

  $\{\begin{array}{c}5c+250b+13500a=923\\ 1000b+100000a=6020\\ 100000b+10140000a=609200\end{array}$

Subtracting 100 times the second equation from the third equation to produce a new third equation, we get

  $\{\begin{array}{c}5c+250b+13500a=923\\ 1000b+100000a=6020\\ 140000a=7200\end{array}$

Now, solving $140000a=7200$ for $a$ , we obtain $a=0.0514286$ .

Now, substituting the above finding value in the second equation, we get

  $\begin{array}{l}1000b+100000(0.0514286)=6020\\ \Rightarrow 1000b=877.14\\ \Rightarrow b=0.877143\end{array}$

Substituting the above finding values in the first equation, we get

  $\begin{array}{l}5c+250(0.877143)+13500(0.0514286)=923\\ \Rightarrow 5c+219.28575+694.2861=923\\ \Rightarrow 5c=923-913.57185\\ \Rightarrow 5c=9.42815\\ \Rightarrow c=1.88563\end{array}$

Therefore, the values of $a,b,\text{and c}$ in the given figure are $0.0514286,0.877143,\text{and 1}\text{.88563}$ , respectively.

b.

To determine

To verify: The least squares regression parabola using the graphing utility to verify the obtained result in the previous part.

Explanation of Solution

Given information:

In testing a new automobile breaking system, engineers recorded the speed $x$ (in miles per hour) and the stopping distance $y$ (in feet). The table shows the results.

  

For the above data points, the sums have been evaluated and that results to the following system of linear equations.

  $\{\begin{array}{c}5c+250b+13500a=923\\ 250c+13500b+775000a=52170\\ 13500c+775000b+46590000a=3101300\end{array}$

Concept used:

One way to find the least squares regression parabola $y=a{x}^2+bx+c$ for a set of points $\left(x_1,y_1\right),\left(x_2,y_2\right),\mathrm{...},\left(x_n,y_n\right)$ is by solving the system below for $a,b,\text{and }c.$

  $\{\begin{array}{c}nc+\left({\displaystyle \sum _{i=1}^n{x}_i}\right)b+\left({\displaystyle \sum _{i=1}^n{x}_i{}^2}\right)a={\displaystyle \sum _{i=1}^n{y}_i}\\ \left({\displaystyle \sum _{i=1}^n{x}_i}\right)c+\left({\displaystyle \sum _{i=1}^n{x}_i{}^2}\right)b+\left({\displaystyle \sum _{i=1}^n{x}_i{}^3}\right)a={\displaystyle \sum _{i=1}^n{x}_i{y}_i}\\ \left({\displaystyle \sum _{i=1}^n{x}_i{}^2}\right)c+\left({\displaystyle \sum _{i=1}^n{x}_i{}^3}\right)b+\left({\displaystyle \sum _{i=1}^n{x}_i{}^4}\right)a={\displaystyle \sum _{i=1}^n{x}_i{}^2{y}_i}\end{array}$

Calculations:

Using Desmos graphing utility, the model that fit the data is obtained as follows:

  

The values of $a,b,\text{and c}$ found in the previous partare $0.0514286,0.877143,\text{and 1}\text{.88563}$ , respectively.

Conclusion:

Comparing the values of $a,b,\text{and c}$ with the values obtained using the graphing utility, the values are matching.

Therefore, the model is a best fit.

c.

To determine

To verify: The least squares regression parabola using the graphing utility to verify the obtained result in the previous part.

Answer to Problem 69E

The estimated stopping distance is $357ft$ when the speed is 75 miles per hour.

Explanation of Solution

Given information:

In testing a new automobile breaking system, engineers recorded the speed $x$ (in miles per hour) and the stopping distance $y$ (in feet). The table shows the results.

  

For the above data points, the sums have been evaluated and that results to the following system of linear equations.

  $\{\begin{array}{c}5c+250b+13500a=923\\ 250c+13500b+775000a=52170\\ 13500c+775000b+46590000a=3101300\end{array}$

Concept used:

One way to find the least squares regression parabola $y=a{x}^2+bx+c$ for a set of points $\left(x_1,y_1\right),\left(x_2,y_2\right),\mathrm{...},\left(x_n,y_n\right)$ is by solving the system below for $a,b,\text{and }c.$

  $\{\begin{array}{c}nc+\left({\displaystyle \sum _{i=1}^n{x}_i}\right)b+\left({\displaystyle \sum _{i=1}^n{x}_i{}^2}\right)a={\displaystyle \sum _{i=1}^n{y}_i}\\ \left({\displaystyle \sum _{i=1}^n{x}_i}\right)c+\left({\displaystyle \sum _{i=1}^n{x}_i{}^2}\right)b+\left({\displaystyle \sum _{i=1}^n{x}_i{}^3}\right)a={\displaystyle \sum _{i=1}^n{x}_i{y}_i}\\ \left({\displaystyle \sum _{i=1}^n{x}_i{}^2}\right)c+\left({\displaystyle \sum _{i=1}^n{x}_i{}^3}\right)b+\left({\displaystyle \sum _{i=1}^n{x}_i{}^4}\right)a={\displaystyle \sum _{i=1}^n{x}_i{}^2{y}_i}\end{array}$

Calculations:

The model we got is $y=0.0514286x^2+0.877143x+1.88563$ .

Substituting, 75 for $x$ in the equation, we get

  $\begin{array}{l}$ y=0.0514286{(75)}^2+0.877143(75)+1.88563$\Rightarrow y=356.95731\end{array}$