Chapter 7, Problem 49RE

a.

To determine

To Analysis E-Commerce Data.

Answer to Problem 49RE

  $\boxed{y=-5.95t^2+104.45t-312.9}$

Explanation of Solution

Given information:

The table shows the retail e-commerce sales $y$ (in billions of dollars) in the United States from $2007$ through $2009$ .

  

Find the least squares regression parabola $y=a{t}^2+bt+c$ for the data, where $t$ represents the year with $t=7$ corresponding to $2007$ , by solving the system for $a,b,c$

  $\{\begin{array}{l}3c+24b+19a=413.8\\ 24c+194b+1584a=3328.9\\ 194c-158b+13,058a=27051.1\end{array}$

Calculation:

The given system of equations is,

  $\begin{array}{l}3c+24b+19a=\mathrm{413.8........}\left(1\right)\\ 24c+194b+1584a=\mathrm{3328.9.....}\left(2\right)\\ 194c-158b+13,058a=\mathrm{27051.1.....}\left(3\right)\end{array}$

Multiply equation $\left(1\right)$ by $-8$ and add to equation $\left(2\right)$

  $\begin{array}{l}\underset{\_}{\begin{array}{l}-24c-192b-1552a=-3310.4\\ 24c+194b+1584a=3328.9\end{array}}\\ 2b+32a=18.5\end{array}$

Multiply equation $\left(1\right)$ by $-\frac{194}{3}$ and add to equation $\left(3\right)$

  $\begin{array}{l}\underset{\_}{\begin{array}{l}-194c-155b+\frac{37636}{3}a=-\frac{80277.2}{3}\\ 194c+1584b+13058a=27051.1\end{array}}\\ 32b+\frac{1538}{3}a=\frac{8761}{30}\end{array}$

The new system of equations is,

  $\begin{array}{l}3c+24b+194a=\mathrm{413.8........}\left(1\right)\\ 2b+32a=\mathrm{18.5.....}\left(2\right)\\ 32b+\frac{1538}{3}a=\frac{8761}{30}\mathrm{.....}\left(3\right)\end{array}$

Multiply equation $\left(2\right)$ by $-16$ and add to equation $\left(3\right)$

  $\begin{array}{l}\underset{\_}{\begin{array}{l}32b+\frac{1538}{3}a=\frac{8761}{30}\\ -32b-512a=-296\end{array}}\\ \frac{2}{3}a=-\frac{119}{30}\end{array}$

The new system of equations is,

  $\begin{array}{l}3c+24b+194a=\mathrm{413.8........}\left(1\right)\\ 2b+32a=\mathrm{18.5.....}\left(2\right)\\ \frac{2}{3}a=-\frac{119}{30}\mathrm{.....}\left(3\right)\end{array}$

Divide equation $\left(3\right)$ $\frac{2}{3}$

  $a=-5.95$

Substitute $a=-5.95$ in equation $\left(2\right)$

  $\begin{array}{l}2b+32\times \left(-5.95\right)=18.5\\ 2b=208.9\\ b=104.45\end{array}$

Substitute $a=-5.95$ and $b=104.45$ in equation $\left(1\right)$

  $\begin{array}{l}3c+24\left(104.45\right)+194\left(-5.95\right)=413.8\\ 3c+1352.5=413.8\\ 3c=-938.7\\ c=-312.9\end{array}$

Hence, $a=-5.95,b=104.45,c=-312.9$

Hence, the equation of parabola is $\boxed{y=-5.95t^2+104.45t-312.9}$

b.

To determine

To Analysis E-Commerce Data.

Answer to Problem 49RE

The model fit the data exactly.

Explanation of Solution

Given information:

The table shows the retail e-commerce sales $y$ (in billions of dollars) in the United States from $2007$ through $2009$ .

  

Use a graphing utility to graph the parabola and the data in the same viewing window. How well does the model fit the data?

Calculation:

Considering the data and model in the same window,

The model fit the data exactly.

  

Hence, the model fit the data exactly.

c.

To determine

To Analysis E-Commerce Data.

Answer to Problem 49RE

The answer is not reasonable.

Explanation of Solution

Given information:

The table shows the retail e-commerce sales $y$ (in billions of dollars) in the United States from $2007$ through $2009$ .

  

Use the model to estimate the retail e-commerce sales in the United States in $2015$ . Does your answer seem reasonable?

Calculation:

Consider the equation.

  $y=-5.95t^2+104.45t+-312.9$

Put $t=15$ in above equation.

  $\begin{array}{l}y=-5.95{\left(15\right)}^2+104.45\left(15\right)-312.9\\ y=-84.9\end{array}$

Hence, the sales in $2015$ is $y=\$\left(-84.9\right)billion$ .

The answer is not reasonable.