Chapter 7.2, Problem 86E

(a)

To determine

To find: the least regression line.

Answer to Problem 86E

The least squares regression line is $y=-44.2x+89.51$ .

Explanation of Solution

Given information:

Since, $y$ represents demand of a candy bar and $x$ represents price of the products.

The result can be arranged in table format as shown below:

$x$	$y$
$\$1.00$	$45$
$\$1.20$	$37$
$\$1.50$	$23$

Calculation:

The system of linear equations for getting values of $a$ and $b$ :

  $3b+3.7a=105$  ...(1)

  $3.7b+4.69a=123.9$  ...(2)

Multiply $3.7$ and $3$ in equation (1) and (2) respectively:

  $11.1b+13.69a=388.5$  ...(3)

  $11.1b+14.07a=371.7$  ...(4)

Subtracting equation (3) from equation (4):

  $\begin{array}{c}11.1b+14.07a-11.1b-13.69a=371.7-388.5\\ 0.38a=-16.8\\ a=-44.2\end{array}$

Put the value of $a$ in equation (1) to get the value of $b$ :

  $\begin{array}{c}3b+3.7\times \left(-44.2\right)=105\\ 3b=105+163.54\\ 3b=268.54\\ b=89.51\end{array}$

Therefore, the value of $a$ and $b$ by elimination method are $-44.2$ and $89.51$ respectively.

Calculate the least squares regression line.

  $\begin{array}{l}y=ax+b\\ y=-44.2x+89.51\end{array}$

(b)

To determine

To validate: the result by using a regression feature of graphing utility.

Answer to Problem 86E

The least squares regression line by graphing utility is $y=-44.2x+89.53$ .

Explanation of Solution

Given information:

Since, $y$ represents demand of a candy bar and $x$ represents price of the products.

The result can be arranged in table format as shown below:

$x$	$y$
$\$1.00$	$45$
$\$1.20$	$37$
$\$1.50$	$23$

Calculation:

Use regression feature of a graphing utility to confirm the result.

  

Therefore, the least squares regression line by graphing utility is $y=-44.2x+89.53$ .

(c)

To determine

To plot : the given data and graph the linear model .

Answer to Problem 86E

Plotted the given data and graphed the linear model.

Explanation of Solution

Given information:

Since, $y$ represents demand of a candy bar and $x$ represents price of the products.

The result can be arranged in table format as shown below:

$x$	$y$
$\$1.00$	$45$
$\$1.20$	$37$
$\$1.50$	$23$

Plot the data on graph as shown below.

  

(d)

To determine

To predict: the demand when the price is $\$1.75$

Answer to Problem 86E

The demand of product when the price $\$1.75$ is $12$ .

Explanation of Solution

Given information:

Since, $y$ represents demand of a candy bar and $x$ represents price of the products.

The result can be arranged in table format as shown below:

$x$	$y$
$\$1.00$	$45$
$\$1.20$	$37$
$\$1.50$	$23$

Price of the product $=\$1.75$

Calculation:

To predict the demand of product when the price is $\$1.75$ , use the least square regression line that is

  $y=-44.2x+89.51$  ...(1)

Here, $x=1.75$

Putting the value of $x$ in equation (1):

  $\begin{array}{l}y=\left(-44.2\times 1.75\right)+89.51\\ y=-77.35+89.51\\ y=12.16\\ y\approx 12\end{array}$

Therefore, the demand of product when the price $\$1.75$ is $12$ .