Chapter 4, Problem 3P

a.

To determine

Does the scatter plot shown suggest an exponential model?

Answer to Problem 3P

  $\boxed{\text{exponential model}}$

Explanation of Solution

Given information:

The U.S. health-care expenditures for $1970-2001$ are given in the table below, and a scatter plot of the data is shown in the ?gure.

  

Does the scatter plot shown suggest an exponential model?

Calculation:

The U.S. health-care expenditures for $1970-2001$ are shown in the scatter plot.

  

The scatter plot appears to be an upward growth, increasing rapidly.

This is an $\boxed{\text{exponential model}}$ .

Hence, the solution is, $\boxed{\text{exponential model}}$

b.

To determine

Make a table of the values $\left(\text{t,lnE}\right)$ and a scatter plot. Does the scatter plot appear to be linear?

Answer to Problem 3P

The scatter plot appears to be linear.

Explanation of Solution

Given information:

The U.S. health-care expenditures for $1970-2001$ are given in the table below, and a scatter plot of the data is shown in the ?gure.

  

Make a table of the values $\left(\text{t,lnE}\right)$ and a scatter plot. Does the scatter plot appear to be linear?

Calculation:

Make a table of the values $\left(\text{t,lnE}\right)$ .

  $\begin{array}{cccccccccccc}t& 0& 10& 15& 17& 20& 22& 24& 26& 28& 30& 31\\ E& 74.3& 251.1& 434.5& 506.2& 696.6& 820.3& 937.2& 1039.4& 1150.0& 1310.0& 1424.5\\ \ln E& 4.30& 5.52& 6.07& 6.22& 6.54& 6.70& 6.84& 6.94& 7.04& 7.17& 7.26\\ t,\ln E& 0,4.30& 10,5.52& 15,6.07& 17,6.22& 20,6.54& 22,6.70& 24,6.84& 26,6.94& 28,7.04& 30,7.17& 31,7.26\end{array}$

Press $\boxed{STAT}$ and select EDIT1, enter the values from the table.

  

Press $2nd\boxed{Y=}$ enter into the plot $1$

  

Press the button window and set the range of $x,y$ as $\left[-1,32\right]\times \left[4,7.5\right]$ then press the GRAPH button to plot the scatter plot.

  

The scatter plot appears to be linear.

Hence, The scatter plot appears to be linear .

c.

To determine

Find the regression line for the data in part (b).

Answer to Problem 3P

  $\boxed{\ln E=0.092t+4.55}$

Explanation of Solution

Given information:

The U.S. health-care expenditures for $1970-2001$ are given in the table below, and a scatter plot of the data is shown in the ?gure.

  

Find the regression line for the data in part (b).

Calculation:

To find the regression line for the data Press $\boxed{STAT}$ and go to the $CALC$ menu and select.

  

Press $\boxed{enter}$ the regression line for the data.

  

Hence, the regression line for the data is, $\boxed{\ln E=0.092t+4.55}$

d.

To determine

To find an exponential model for the growth of health-care expenditures.

Answer to Problem 3P

  $\boxed{E=94.63e^{at}}$

Explanation of Solution

Given information:

The U.S. health-care expenditures for $1970-2001$ are given in the table below, and a scatter plot of the data is shown in the ?gure.

  

Use the results of part (c) to find an exponential model for the growth of health-care expenditures.

Calculation:

Find an exponential model for the growth of health-care expenditures.

Now,

  $\begin{array}{l}\ln E=0.092t+4.55\\ E=e^{0.092t+4.55}\\ =e^{0.092t}\times e^{4.55}\\ =e^{0.092t}\times 94.63\left[\therefore a=0.092\right]\\ \boxed{E=94.63e^{at}}\end{array}$

Hence, an exponential model for the growth is, $\boxed{E=94.63e^{at}}$

e.

To determine

To find an exponential model for the growth of health-care expenditures.

Answer to Problem 3P

  $\boxed{3421.91million}$

Explanation of Solution

Given information:

The U.S. health-care expenditures for $1970-2001$ are given in the table below, and a scatter plot of the data is shown in the ?gure.

  

Use your model to predict the total health-care expenditures in $2009$ .

Calculation:

Exponential model for the growth of health-care expenditures.

  $E=94.63e^{0.092t}$

Put $t=39\left[\therefore 2009-1970=39\right]$

  $\begin{array}{l}E=\left(94.63\right)e^{0.092\left(39\right)}\\ =\left(94.63\right)\left(36.161\right)\\ =3421.91\end{array}$

Hence, the total health-care expenditures in $2009$ is, $\boxed{3421.91million}$