Chapter 2.7, Problem 82E

a.

To determine

To create a scatter plot of the data by using a graphing utility.

Answer to Problem 82E

  

Explanation of Solution

Given information:

The mean salaries (in thousands of dollars) of public school classroom teachers in the United States from $2000$ through $2011$ are shown in the table.

  $\begin{array}{ll}Year\hfill & Salary,S\hfill \\ 2000\hfill & 42.2\hfill \\ 2001\hfill & 43.7\hfill \\ 2002\hfill & 43.8\hfill \\ 2003\hfill & 45.0\hfill \\ 2004\hfill & 45.6\hfill \\ 2005\hfill & 45.9\hfill \\ 2006\hfill & 48.2\hfill \\ 2007\hfill & 49.3\hfill \\ 2008\hfill & 51.3\hfill \\ 2009\hfill & 52.9\hfill \\ 2010\hfill & 54.4\hfill \\ 2011\hfill & 54.2\hfill \end{array}$

A model that approximates these data is given by:

  $S=\frac{42.6-0.236t}{1-0.026t},0\le t\le 11$

Where represents the year, with corresponding to $2000$ .

Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window.

Calculation:

Here, we will consider the following table:

  $\begin{array}{ll}Year\hfill & Salary,S\hfill \\ 2000\hfill & 42.2\hfill \\ 2001\hfill & 43.7\hfill \\ 2002\hfill & 43.8\hfill \\ 2003\hfill & 45.0\hfill \\ 2004\hfill & 45.6\hfill \\ 2005\hfill & 45.9\hfill \\ 2006\hfill & 48.2\hfill \\ 2007\hfill & 49.3\hfill \\ 2008\hfill & 51.3\hfill \\ 2009\hfill & 52.9\hfill \\ 2010\hfill & 54.4\hfill \\ 2011\hfill & 54.2\hfill \end{array}$

Now, by using a graphical utility the scatter plot of the above data and the graph of the model together are as follows:

  

Hence, the result is shown in the graph.

b.

To determine

To find, how well does the model fit the data?

Answer to Problem 82E

The model fits the data very closely.

Explanation of Solution

Given information:

The mean salaries (in thousands of dollars) of public school classroom teachers in the United States from $2000$ through $2011$ are shown in the table.

  $\begin{array}{ll}Year\hfill & Salary,S\hfill \\ 2000\hfill & 42.2\hfill \\ 2001\hfill & 43.7\hfill \\ 2002\hfill & 43.8\hfill \\ 2003\hfill & 45.0\hfill \\ 2004\hfill & 45.6\hfill \\ 2005\hfill & 45.9\hfill \\ 2006\hfill & 48.2\hfill \\ 2007\hfill & 49.3\hfill \\ 2008\hfill & 51.3\hfill \\ 2009\hfill & 52.9\hfill \\ 2010\hfill & 54.4\hfill \\ 2011\hfill & 54.2\hfill \end{array}$

A model that approximates these data is given by:

  $S=\frac{42.6-0.236t}{1-0.026t},0\le t\le 11$

Where represents the year, with corresponding to $2000$ .

How well does the model fit the data? Explain.

Calculation:

Here, we have observed from the above graph that the model fits the data very closely. That is, the mean salaries have been on the rise gradually every year.

Hence, the result is shown in the above graph.

c.

To determine

By using the model to predict when the salary for classroom teachers will exceed $\$60,000$ .

Answer to Problem 82E

The model fits the data very closely.

Explanation of Solution

Given information:

The mean salaries (in thousands of dollars) of public school classroom teachers in the United States from $2000$ through $2011$ are shown in the table.

  $\begin{array}{ll}Year\hfill & Salary,S\hfill \\ 2000\hfill & 42.2\hfill \\ 2001\hfill & 43.7\hfill \\ 2002\hfill & 43.8\hfill \\ 2003\hfill & 45.0\hfill \\ 2004\hfill & 45.6\hfill \\ 2005\hfill & 45.9\hfill \\ 2006\hfill & 48.2\hfill \\ 2007\hfill & 49.3\hfill \\ 2008\hfill & 51.3\hfill \\ 2009\hfill & 52.9\hfill \\ 2010\hfill & 54.4\hfill \\ 2011\hfill & 54.2\hfill \end{array}$

A model that approximates these data is given by:

  $S=\frac{42.6-0.236t}{1-0.026t},0\le t\le 11$

Where represents the year, with corresponding to $2000$ .

Use the model to predict when the salary for classroom teachers will exceed $\$60,000$ .

Calculation:

By using the zoom and trace features, the salary for classroom teachers will exceed $\$60,000$ in the year $\boxed{2013}$ .

d.

To determine

To find if the model valid for long-term predictions of classroom teacher salaries.

Answer to Problem 82E

No, the model is not valid.

Explanation of Solution

Given information:

The mean salaries (in thousands of dollars) of public school classroom teachers in the United States from $2000$ through $2011$ are shown in the table.

  $\begin{array}{ll}Year\hfill & Salary,S\hfill \\ 2000\hfill & 42.2\hfill \\ 2001\hfill & 43.7\hfill \\ 2002\hfill & 43.8\hfill \\ 2003\hfill & 45.0\hfill \\ 2004\hfill & 45.6\hfill \\ 2005\hfill & 45.9\hfill \\ 2006\hfill & 48.2\hfill \\ 2007\hfill & 49.3\hfill \\ 2008\hfill & 51.3\hfill \\ 2009\hfill & 52.9\hfill \\ 2010\hfill & 54.4\hfill \\ 2011\hfill & 54.2\hfill \end{array}$

A model that approximates these data is given by:

  $S=\frac{42.6-0.236t}{1-0.026t},0\le t\le 11$

Where represents the year, with corresponding to $2000$ .

Is the model valid for long-term predictions of classroom teacher salaries? Explain

Calculation:

No, the model is not valid for long-term predictions of classroom teacher salaries. As, the salary predicted by the model increases very fast and reaches approximately $\$10,000$ in the year $\boxed{2024}$ .