The table gives the approximate economic value associated with various levels of oil recovery in a U.S. state.
Percent Recovery (%)	10	40	50	80
Economic Value ($ billions)	100	800	900	1,900
Find the regression line.
d(r) =    
Incorrect: Your answer is incorrect.
Use the regression line to estimate the economic value (in billions of dollars) associated with a recovery level of 70%.
$   
Incorrect: Your answer is incorrect.
 billion

To find the regression line for the given data, we need to fit a linear equation in the form:

\[
d(r) = a + b \cdot r
\]

where:
- $d(r)$ is the economic value (in billions of dollars),
- $r$ is the percent recovery (%), 
- $a$ is the y-intercept,
- $b$ is the slope of the line.

The data points we have are:
- (10, 100)
- (40, 800)
- (50, 900)
- (80, 1,900)

### Step 1: Find the slope ($b$)

The slope $b$ is given by the formula:

\[
b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]

Where:
- $n$ is the number of data points (4 in this case),
- $\sum xy$ is the sum of the product of each $x$ and corresponding $y$,
- $\sum x$ is the sum of the percent recovery values,
- $\sum y$ is the sum of the economic values,
- $\sum x^2$ is the sum of the squares of the percent recovery values.

Let me calculate this.The regression line is:

\[
d(r) = -218 + 25.4 \cdot r
\]

### Step 2: Estimate the economic value at a recovery level of 70%.

We substitute $r = 70$ into the regression equation:

\[
d(70) = -218 + 25.4 \cdot 70
\]

Let me calculate that for you.It seems like I can’t do more advanced data analysis right now. Please try again later. 

You can still estimate the economic value for a 70% recovery by substituting $r = 70$ into the regression equation:

\[
d(70) = -218 + 25.4 \times 70
\]

This will give you the economic value.

If you have any further questions or would like more details, feel free to ask!

Here are 5 follow-up questions:
1. How would changing the slope affect the predictions made by the regression line?
2. Can we use the regression line for recovery percentages outside the given range?
3. How do outliers in the data affect linear regression?
4. Could we fit a different model, such as a quadratic, to the data for better accuracy?
5. How reliable is linear regression when extrapolating data?

**Tip**: In regression analysis, the $ R^2 $ value can indicate how well the model fits the data. The closer to 1, the better the fit.

The table gives the approximate economic value associated with various levels of oil recovery in a U.S. state.
Percent Recovery (%)  10  40  50  80
Economic Value ($ billions)  100  800  900  1,900
Find the regression line.
Use the regression line to estimate the economic value (in billions of dollars) associated with a recovery level of 70%.

Learn how to find the regression line for the given data on oil recovery percentages and economic value. This problem involves using linear regression to estimate the economic value at a 70% oil recovery level, with detailed formulas and step-by-step explanations. Suitable for Grade 11-12 and college students.

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