Chapter 1.7, Problem 25E

(a)

To determine

To find: The linear model for the data with the help of regression of graphing utility.

Answer to Problem 25E

The linear model for the data is equal to $y=45.7x+108.04$ .

Explanation of Solution

Given information:

The table shows the advertising expenditures $x$ and sales volumes $y$ for a company for nine randomly selected months. Both are measured in Thousands of dollars.

Month	Advertising expenditures, $x$	Sales volume, $y$
$1$	$2.4$	$215$
$2$	$1.6$	$182$
$3$	$1.4$	$172$
$4$	$2.3$	$214$
$5$	$2.0$	$201$
$6$	$2.6$	$230$
$7$	$1.4$	$176$
$8$	$1.6$	$177$
$9$	$2.0$	$196$

Calculation:

For the regression linear model of data follow the steps below:

First press the $\boxed{2\text{nd}}$ key and then $\boxed{\text{Y}=}$ and make sure that Plot $1$ is ON and choose the scatter plot icon in type.

Go to ${\text{Y}}_1$ and then press the key $\boxed{\text{STAT}}$ and then in edit enter the data in ${\text{L}}_1$ and ${\text{L}}_2$ .

Now enter the keystrokes $\boxed{\text{STAT}}\boxed{\to }\boxed{4}$ and press the $\boxed{\text{ENTER}}$ key for the regression.

  

Figure(1)

Therefore, the linear model for the data is equal to $y=45.7x+108.04$ .

(b)

To determine

To find: The graph of model and the scatter plot of data in same viewing window and check the model is best fit or not.

Answer to Problem 25E

The model is not the best fit for the data as all the points of the data doesn’t lie on the graph of the model.

Explanation of Solution

Given information:

The table shows the advertising expenditures $x$ and sales volumes $y$ for a company for nine randomly selected months. Both are measured in Thousands of dollars.

Month	Advertising expenditures, $x$	Sales volume, $y$
$1$	$2.4$	$215$
$2$	$1.6$	$182$
$3$	$1.4$	$172$
$4$	$2.3$	$214$
$5$	$2.0$	$201$
$6$	$2.6$	$230$
$7$	$1.4$	$176$
$8$	$1.6$	$177$
$9$	$2.0$	$196$

Calculation:

To graph the points on scatter plot, follow the steps using graphing calculator.

First press the $\boxed{2\text{nd}}$ key and then $\boxed{\text{Y}=}$ and make sure that Plot $1$ is ON and choose the scatter plot icon in type.

Go to ${\text{Y}}_1$ and then press the key $\boxed{\text{STAT}}$ and then in edit enter the data in ${\text{L}}_1$ and ${\text{L}}_2$ .

Now press the key $\boxed{\text{Y}=}$ and go to ${\text{Y}}_2$ and enter the right hand side of the function $y=45.7x+108.04$ . Enter the keystrokes $\boxed{45.7}\boxed{\text{X}}\boxed{+}\boxed{108.04}$ .

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph in standard window as shown in Figure (1).

  

Figure (2)

(c)

To determine

To find: Interpretation of slope of the model in context of the problem.

Answer to Problem 25E

The slope of the model is equal to $45.7$ and it represents the average increase in sales with increase in advertising.

Explanation of Solution

Given information:

The table shows the advertising expenditures $x$ and sales volumes $y$ for a company for nine randomly selected months. Both are measured in Thousands of dollars.

Month	Advertising expenditures, $x$	Sales volume, $y$
$1$	$2.4$	$215$
$2$	$1.6$	$182$
$3$	$1.4$	$172$
$4$	$2.3$	$214$
$5$	$2.0$	$201$
$6$	$2.6$	$230$
$7$	$1.4$	$176$
$8$	$1.6$	$177$
$9$	$2.0$	$196$

Calculation:

As found in part(a), the model for the best fit of data is equal to $y=45.7x+108.04$ .

Compare it with general equation of line $y=mx+c$ .

Therefore, the slope of model as $45.7$ and it represents the average increase in sales with increase in advertising.

(d)

To determine

To find: The sales for the advertising expenditures of $\$1500$ .

Answer to Problem 25E

The sales for the advertising expenditures of $\$1500$ is equal to $\$68658.04$ .

Explanation of Solution

Given information:

The table shows the advertising expenditures $x$ and sales volumes $y$ for a company for nine randomly selected months. Both are measured in Thousands of dollars.

Month	Advertising expenditures, $x$	Sales volume, $y$
$1$	$2.4$	$215$
$2$	$1.6$	$182$
$3$	$1.4$	$172$
$4$	$2.3$	$214$
$5$	$2.0$	$201$
$6$	$2.6$	$230$
$7$	$1.4$	$176$
$8$	$1.6$	$177$
$9$	$2.0$	$196$

Calculation:

As found in part(a), the model for the best fit of data is equal to $y=45.7x+108.04$ .

Substitute $1500$ for $x$ in the model.

  $\begin{array}{c}y=45.7x+108.04\\ =45.7\left(1500\right)+108.04\\ =68658.04\end{array}$

Therefore, the sales for the advertising expenditures of $\$1500$ is equal to $\$68658.04$ .