Chapter 7.2, Problem 58E

a.

To determine

To verify defence department outlays.

Answer to Problem 58E

  $\boxed{y=38.363t+299.84}$

Explanation of Solution

Given information:

The table shows the total national outlays $y$ for defence functions (in billions of dollars) for the years $2004$ through $2011$ .

  

Find the least squares regression line $y=at+b$ for the data, where $t$ represents the year with $t=4$ corresponding to $2004$ , by solving the system for $a$ and $b$ .

  $\{\begin{array}{l}8b+60a=4700.5\\ 60b+492a=36,865.0\end{array}$

Calculation:

Consider the given system of equations,

  $\begin{array}{l}8b+60a=\mathrm{4700.5..............}\left(1\right)\\ 60b+492a=36,\mathrm{865.0..........}\left(2\right)\end{array}$

Multiply equation $\left(1\right)$ by $-15$ and $\left(2\right)$ by $2$ add both equations to eliminate ‘ $b$ ’.

  $\frac{\begin{array}{l}-120b-900a=-70507.5\\ 120b+984a=73730\end{array}}{\begin{array}{l}84a=3222.5\\ a=38.363\end{array}}$

Substitute $a=38.363$ in equation $\left(1\right)$

  $\begin{array}{l}8b+60\left(38.363\right)=4700.5\\ 8b=2398.72\\ b=299.84\end{array}$

Hence the result is $\boxed{a=38.363,b=299.84}$

Now the least square regression formula is $\boxed{y=38.363t+299.84}$

b.

To determine

To verify defence department outlays.

Answer to Problem 58E

The results are same as part $\left(a\right)$

Explanation of Solution

Given information:

The table shows the total national outlays $y$ for defence functions (in billions of dollars) for the years $2004$ through $2011$ .

  

Use the regression feature of a graphing utility to confirm the result of part $\left(a\right)$ .

Calculation:

Now the least square regression formula is $\boxed{y=38.363t+299.84}$

Now use calculator to verify the part $\left(a\right)$ results.

  

Hence, the results are same as part $\left(a\right)$

c.

To determine

To verify defence department outlays.

Answer to Problem 58E

  $\begin{array}{l}\begin{array}{ccccccccc}Year& 2004& 2005& 2006& 2007& 2008& 2009& 2010& 2011\\ outlays\text{'}y\text{'}& 453.292& 491.655& 530.018& 568.381& 606.744& 645.107& 683.47& 721.833\end{array}\\ \end{array}$

Explanation of Solution

Given information:

The table shows the total national outlays $y$ for defence functions (in billions of dollars) for the years $2004$ through $2011$ .

  

Use the linear model to create a table of estimated values of Compare the estimated values with the actual data.

Calculation:

  $\begin{array}{l}\begin{array}{ccccccccc}Year& 2004& 2005& 2006& 2007& 2008& 2009& 2010& 2011\\ outlays\text{'}y\text{'}& 453.292& 491.655& 530.018& 568.381& 606.744& 645.107& 683.47& 721.833\end{array}\\ \end{array}$

d.

To determine

To verify defence department outlays.

Answer to Problem 58E

  $\boxed{\$760.196billion}$

Explanation of Solution

Given information:

The table shows the total national outlays $y$ for defence functions (in billions of dollars) for the years $2004$ through $2011$ .

  

Use the linear model to estimate the total national outlay for $2012$ .

Calculation:

Let $t=12$ and $y=38.363t+299.84$

Put $t=12$ in regression equation,

  $\begin{array}{l}y=38.363\left(12\right)+299.84\\ =760.196\end{array}$

Hence the total national outlay in $2012$ is $\boxed{\$760.196billion}$

e.

To determine

To verify defence department outlays.

Answer to Problem 58E

(Calculated value) $\$760.196billion$ >$\$701billion$ (actual value).

Explanation of Solution

Given information:

The table shows the total national outlays $y$ for defence functions (in billions of dollars) for the years $2004$ through $2011$ .

  

Use the Internet, your school’s library, or some other reference source to find the total national outlay for $2012$ . How does this value compare with your answer in part $\left(d\right)$ ?

Calculation:

In $2012$ total national outlay is $\$701billion$

But the calculates value is $\$760.196billion$ which is larger than the actual value.

Hence, (calculated value) $\$760.196billion$ >$\$701billion$ (actual value).

f.

To determine

To verify defence department outlays.

Answer to Problem 58E

This model is not valid for long-term predictions of total national outlays.

Explanation of Solution

Given information:

The table shows the total national outlays $y$ for defence functions (in billions of dollars) for the years $2004$ through $2011$ .

  

Is the linear model valid for long-term predictions of total national outlays? Explain.

Calculation:

This model is not valid for long-term predictions of total national outlays.