Chapter 1.7, Problem 22E

(a)

To determine

To graph: The scatter plot of the data with $t=5$ corresponding to $2005$ .

Explanation of Solution

Given information:

The mean salaries $S$ (in thousands of dollars) of public school teachers in United States from $2005$ through $2011$ are shown in the table.

Year	Mean salary, $S$ (in thousand of dollars)
$2005$	$47.5$
$2006$	$49.1$
$2007$	$51.1$
$2008$	$52.8$
$2009$	$54.3$
$2010$	$55.5$
$2011$	$57.2$

Graph:

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 $\boxed{\text{ZOOM}}\boxed{9}$ key and $\boxed{\text{TRACE}}$ key to produce the graph in standard window as shown in Figure (1).

  

Figure (1)

(b)

To determine

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

Answer to Problem 22E

The linear model for the data is equal to $S=1.61t+39.61$ .

Explanation of Solution

Given information:

The mean salaries $S$ (in thousands of dollars) of public school teachers in United States from $2005$ through $2011$ are shown in the table.

Year	Mean salary, $S$ (in thousand of dollars)
$2005$	$47.5$
$2006$	$49.1$
$2007$	$51.1$
$2008$	$52.8$
$2009$	$54.3$
$2010$	$55.5$
$2011$	$57.2$

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(2)

Therefore, the linear model for the data is equal to $S=1.61t+39.61$ .

(c)

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 22E

The model is best fit for the data as all the points of the data lies on the graph of the model.

Explanation of Solution

Given information:

The mean salaries $S$ (in thousands of dollars) of public school teachers in United States from $2005$ through $2011$ are shown in the table.

Year	Mean salary, $S$ (in thousand of dollars)
$2005$	$47.5$
$2006$	$49.1$
$2007$	$51.1$
$2008$	$52.8$
$2009$	$54.3$
$2010$	$55.5$
$2011$	$57.2$

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 $y=1.61t+39.61$ . Enter the keystrokes $\boxed{1.61}\boxed{\text{X}}\boxed{+}\boxed{39.61}$ .

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 (3)

(d)

To determine

To find: The mean salaries in $2020$ and $2022$ and check whether the answer is reasonable or not.

Answer to Problem 22E

The mean salaries in $2020$ and $2022$ is equal to $71.81$ and $75.03$ thousand dollars respectively.

Explanation of Solution

Given information:

The mean salaries $S$ (in thousands of dollars) of public school teachers in United States from $2005$ through $2011$ are shown in the table.

Year	Mean salary, $S$ (in thousand of dollars)
$2005$	$47.5$
$2006$	$49.1$
$2007$	$51.1$
$2008$	$52.8$
$2009$	$54.3$
$2010$	$55.5$
$2011$	$57.2$

Calculation:

As found in part(b), the model for the best fit of data is equal to $S=1.61t+39.61$ and the year $2020$ is represented by $t=20$ and the year $2022$ is represented by $t=22$ .

Substitute $20$ for $t$ in the model.

  $\begin{array}{c}S=1.61t+39.61\\ =1.61\left(20\right)+39.61\\ =32.2+39.61\\ =71.81\end{array}$

Substitute $22$ for $t$ in the model.

  $\begin{array}{c}S=1.61t+39.61\\ =1.61\left(22\right)+39.61\\ =35.42+39.61\\ =75.03\end{array}$

Therefore, the mean salaries in $2020$ and $2022$ is equal to $71.81$ and $75.03$ thousand dollars respectively.

The answer to the mean salaries look reasonable as the value is increasing on a constant rate or linearly.