Chapter 7.8, Problem 31E

a.

To determine

To solve: the system and write the least square regression parabola for the data using Cramer’s Rule.

Answer to Problem 31E

Solution of the system,

  $\begin{array}{l}c\approx \mathrm{84.414.}\\ b\approx \mathrm{4.068.}\\ a\approx -\mathrm{0.143.}\end{array}$

The equation of the least squares regression parabola is,

  $y=-0.143t^2+4.068t+\mathrm{84.414.}$

Explanation of Solution

Given information: The retail sales y (in billions of dollars) of family clothing stores in

the U.S from 2013 through 2017 can be expressed by ordered pairs of the form ( t, y(t)),

where t =3 represents 2013. Ordered pairs are; (3, 95.4),

(4, 98.2),

(5,101.4),

(6,103.6),

(7,105.9).

The coefficients of the least squares regression parabola $y=a{t}^2+bt+c$ can be found by solving the system.

  $\begin{array}{c}5c+25b+135a=504.5\\ 25c+135b+775a=2548.9\\ 135c+775b+4659a=13,883.5\end{array}$

Calculation:

The given system is,

  $\begin{array}{c}5c+25b+135a=504.5\\ 25c+135b+775a=2548.9\\ 135c+775b+4659a=13,883.5\end{array}$

Matrix of Coefficients:

  $\left[\begin{array}{ccc}5& 25& 135\\ 25& 135& 775\\ 135& 775& 4659\end{array}\right]$

Answer column:

  $\left[\begin{array}{c}504.5\\ 2548.9\\ 13883.5\end{array}\right]$

Replace the c −column values with the answer column values:

  $M_c=\left[\begin{array}{ccc}504.5& 25& 135\\ 2548.9& 135& 775\\ 13883.5& 775& 4659\end{array}\right]$

Replace the b −column values with the answer column values

  $M_b=\left[\begin{array}{ccc}5& 504.5& 135\\ 25& 2548.9& 775\\ 135& 13883.5& 4659\end{array}\right]$

Replace the a −column values with the answer column values

  $M_a=\left[\begin{array}{ccc}5& 25& 504.5\\ 25& 135& 2548.9\\ 135& 775& 13883.5\end{array}\right]$

  $\begin{array}{l}D=700.\\ D_c=59090.\\ D_b=2848.\\ D_a=-100.\end{array}$

D denotes the determinant, solve by using Cramer Rule

  $\begin{array}{l}c=\frac{D_c}{D}=\frac{59090}{700}\approx \mathrm{84.414.}\\ b=\frac{D_b}{D}=\frac{2848}{700}\approx \mathrm{4.068.}\\ a=\frac{D_a}{D}=\frac{-100}{700}\approx -0.143\\ \end{array}$

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

  $y=-0.143t^2+4.068t+\mathrm{84.414.}$

b.

To determine

To graph: the parabola with the data and find how does the model fit the data.

Answer to Problem 31E

The model fits the data well.

Explanation of Solution

Given information: The equation of the least squares regression parabola is,

  $y=-0.143t^2+4.068t+84.414$ and ordered pairs are;

(3, 95.4),

(4, 98.2),

(5,101.4),

(6,103.6),

(7,105.9).

Calculation:

The graph of the parabola $y=-0.143t^2+4.068t+84.414$ with the data shown below.

  

So, the model fits the data well.