Let's tackle this problem by going through the steps for exponential regression. We are given the following dataLet's tackle this problem by going through the steps for exponential regression. We are given the following

Year	Billion $ ($y$)
1980	37
1985	72
1990	111
1995	181
2000	197
2005	330

Step 1: Define $x$ Values

First, we need to assign $x$ as the number of years since 1980. So:

$x = \text{Year} - 1980$

Thus, we get the following $x$ values:

Year	Billion $ ($y$)	$x$
1980	37	0
1985	72	5
1990	111	10
1995	181	15
2000	197	20
2005	330	25

Step 2: Exponential Regression Model

The exponential model has the form:

$M(x) = a \cdot e^{bx}$

To find $a$ and $b$, we can use the data to perform an exponential regression. I'll calculate the best-fit model now.### A. Exponential Regression Model

The best-fit exponential regression model is:

$M(x) = 50.6554 \cdot e^{0.0742x}$

B. $r^2$ Value

The $r^2$ value for this model is approximately:

$r^2 = 0.9701$

This indicates a strong fit between the model and the data.

C. Estimation of Expenditures in 2008

To estimate the Medicare expenditures in 2008, first, calculate $x = 2008 - 1980 = 28$. Now, substitute $x = 28$ into the model:

$M(28) = 50.6554 \cdot e^{0.0742 \cdot 28}$

Let me calculate that for you.### C. Estimate for Expenditures in 2008

The estimated Medicare expenditures in 2008 are approximately $404 billion.

Would you like more details on any step? Here are some follow-up questions you might explore:

1. How do we interpret the $r^2$ value in regression analysis?
2. How can you calculate residuals to evaluate model accuracy?
3. Can this model be applied to future years, and what assumptions are involved?
4. What are the limitations of exponential models in economic data?
5. How does Medicare expenditure growth compare to inflation or GDP growth?

Tip: In regression analysis, always verify the model fit using the $r^2$ value and residual analysis to ensure accurate predictions.