(Medicare). The annual expenditures for Medicare (in billions of dollars) by the US Government for selected years since 1980 are shown in Table 4. Let x represent the years since 1980. (Use 4 decimal places in all answers.)
Year	Billion $
1980	
37
1985	
72
1990	
111
1995	
181
2000	
197
2005	
330
Table 4: Medicare Expenditures

A. Find the exponential regression model for the data.
M(x) = 
  
Incorrect: Your answer is incorrect.

B. What is the r2 value for this model?
r2 = 
  
Incorrect: Your answer is incorrect.

C. Extimate (to the nearest billion) the expenditures in 2008?
The Medicare expenditures in 2008 were about $ 
  
Incorrect: Your answer is incorrect.
 billion.

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.

The annual expenditures for Medicare (in billions of dollars) by the US Government for selected years since 1980 are shown in Table 4. Let x represent the years since 1980. Find the exponential regression model for the data, the r2 value for the model, and estimate the expenditures in 2008.

Find the exponential regression model for Medicare expenditures (in billions) from 1980 to 2005. Learn how to calculate the r² value and estimate the expenditures for 2008 using regression analysis. This problem involves key statistical concepts and exponential growth modeling, suitable for advanced students.

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