The table shows the national expenditures for health care in the United States for selected years, with projections to 2020.
Year	National Expenditures
for Health Care (billions)
1960	$28
1970	75
1980	254
1990	714
1995	1017
2000	1353
2005	1981
2010	2954
2015	3308
2020	4487
(a) Use a scatter plot with x as the number of years past 1950 and y as the total expenditures for health care (in billions) to identify what type (or types) of function(s) would make a good model for these data. 
(b) Find a power model and a quadratic model for the data. (Round all numerical values to four decimal places.)
power model    	y(x)	=	
 
quadratic model    	y(x)	=	
 

(c) Which model from part (b) approximates more accurately the data point for 2020?
power model
quadratic model    

(d) Use the quadratic model to predict the 2025 expenditures for national health care. (Round your answer to the nearest billion dollars.)

 billion dollars

Let's go through each part of this question step by step.

### (a) Scatter Plot & Function Identification

You are asked to create a scatter plot where $ x $ is the number of years past 1950 and $ y $ is the national expenditures in billions. This step is crucial for identifying the type of function (power, quadratic, or others) that best models the data.

The table values would be:

\[
\begin{array}{|c|c|}
\hline
\text{Year (x = years past 1950)} & \text{Expenditure (y in billions)} \\
\hline
10 & 28 \\
20 & 75 \\
30 & 254 \\
40 & 714 \\
45 & 1017 \\
50 & 1353 \\
55 & 1981 \\
60 & 2954 \\
65 & 3308 \\
70 & 4487 \\
\hline
\end{array}
\]

From the shape of the scatter plot (which would have to be visualized on a graph), it's likely that the data follows a curve that resembles either a **power model** (of the form $ y = ax^b $) or a **quadratic model** (of the form $ y = ax^2 + bx + c $).

### (b) Power and Quadratic Models

To find the power and quadratic models, you would typically apply regression techniques to fit the data. I'll outline what each model would look like and how we determine the parameters.

#### Power Model

The general form of a power model is:

\[
y(x) = ax^b
\]

Using a logarithmic transformation of the data, you can estimate the parameters $ a $ and $ b $ through regression analysis. However, for brevity, I will compute the coefficients directly in the next step.

#### Quadratic Model

The general form of a quadratic model is:

\[
y(x) = ax^2 + bx + c
\]

We will similarly use quadratic regression to fit the coefficients $ a $, $ b $, and $ c $.

I will compute both models now.### (b) Power and Quadratic Models

Here are the models based on the regression analysis:

- **Power model**:  
  \[
  y(x) = 0.0034x^{3.3175}
  \]

- **Quadratic model**:  
  \[
  y(x) = 1.7606x^2 - 71.0061x + 679.1671
  \]

### (c) Which model approximates 2020 more accurately?

For the year 2020, corresponding to $ x = 70 $ (years past 1950):

- **Power model prediction**: 4463.12 billion
- **Quadratic model prediction**: 4335.49 billion

The actual value in 2020 is 4487 billion. The **power model** is closer to the actual value (4463.12 vs. 4487), making it the more accurate model for 2020.

### (d) Prediction for 2025 using the Quadratic Model

Using the quadratic model to predict the expenditures for 2025 ($ x = 75 $ years past 1950), we get:

\[
\text{2025 prediction} \approx 5257 \text{ billion dollars}
\]

### Summary:
- The power model is a closer fit for the year 2020.
- The predicted expenditure for 2025 using the quadratic model is approximately **5257 billion dollars**.

Let me know if you want further details or have any questions.

Here are five questions to expand on this:
1. How is regression used to fit models to data points?
2. What are the assumptions behind using power and quadratic models?
3. How could you visually compare these models with a plot of residuals?
4. What impact does extrapolating data have on prediction accuracy?
5. Why is the power model more accurate for predicting 2020?

**Tip**: When predicting future data points, it's essential to consider the limits of the models and be aware of potential overfitting.

The table shows the national expenditures for health care in the United States for selected years, with projections to 2020.
Year National Expenditures for Health Care (billions)
1960 $28
1970 75
1980 254
1990 714
1995 1017
2000 1353
2005 1981
2010 2954
2015 3308
2020 4487
(a) Use a scatter plot with x as the number of years past 1950 and y as the total expenditures for health care (in billions) to identify what type (or types) of function(s) would make a good model for these data.
(b) Find a power model and a quadratic model for the data. (Round all numerical values to four decimal places.)
power model y(x) =
quadratic model y(x) =
(c) Which model from part (b) approximates more accurately the data point for 2020?
power model
quadratic model
(d) Use the quadratic model to predict the 2025 expenditures for national health care. (Round your answer to the nearest billion dollars.)

Analyze U.S. national health care expenditures from 1960 to 2020 using power and quadratic models. Learn how to model real-world data with regression techniques, fit power and quadratic functions, and predict future expenditures. Suitable for Grades 11-12.

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