Chapter 13.3, Problem 23E

(a)

To determine

To find: The least squares regression line for the given data.

Answer to Problem 23E

The equation is $y=-1.78x+127.6$ .

Explanation of Solution

Given:

The monthly sales for four different prices of the tool:

Price, x	Demand, y
$25	82
$30	75
$35	67
$40	55

Calculation:

Let the general equation of the least squares regression line be $f\left(x\right)=ax+b$ where the following equations are used to determine the unknowns a and b:

  $\begin{array}{c}a=\frac{n{\displaystyle \sum _{i=1}^n{x}_i{y}_i}-{\displaystyle \sum _{i=1}^n{x}_i}{\displaystyle \sum _{i=1}^n{y}_i}}{n{\displaystyle \sum _{i=1}^n{x}_i{}^2}-{\left({\displaystyle \sum _{i=1}^n{x}_i}\right)}^2}\\ b=\frac{1}{n}\left({\displaystyle \sum _{i=1}^n{y}_i}-a{\displaystyle \sum _{i=1}^n{x}_i}\right)\end{array}$.......(1)

In order to determine value of the coefficients of a and b in the set of equations (1), construct a table as shown below:

x	y	$xy$	$x^2$
25	82	2050	625
30	75	2250	900
35	67	2345	1225
40	55	2200	1600
${\displaystyle \sum _{i=1}^4{x}_i}=130$	${\displaystyle \sum _{i=1}^4{y}_i}=279$	${\displaystyle \sum _{i=1}^4{x}_i{y}_i}=8845$	${\displaystyle \sum _{i=1}^4{x}_i{}^2}=4350$

Table 1

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

  $\begin{array}{l}a=\frac{4\left(8845\right)-\left(130\right)\left(279\right)}{4\left(4350\right)-{\left(130\right)}^2}\\ b=\frac{1}{4}\left(279-a\left(130\right)\right)\end{array}$

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

  $\begin{array}{c}a=-1.78\\ b=127.6\end{array}$

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

  $\begin{array}{c}f\left(x\right)=\left(-1.78\right)x+\left(127.6\right)\\ y=-1.78x+127.6\end{array}$

(b)

To determine

To estimate: The demand when the price is $32.95.

Answer to Problem 23E

The demand is 69 tools.

Explanation of Solution

Given:

The lease squares regression line is $y=-1.78x+127.6$ .

Calculation:

To determine the demand, substitute 32.95 for x in the least squares regression line and simplify as shown below:

  $\begin{array}{c}y=-1.78\left(32.95\right)+127.6\\ \approx 69\text{ tools}\end{array}$

(c)

To determine

To estimate: The price that creates a demand of 83 tools.

Answer to Problem 23E

The price is $25.

Explanation of Solution

Given:

The lease squares regression line is $y=-1.78x+127.6$ .

Calculation:

To determine the price, substitute 83 for y in the least squares regression line and simplify as shown below:

  $\begin{array}{c}83=-1.78x+127.6\\ 1.78x=127.6-83\\ \approx \$25\end{array}$