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)	500	1,200	1,300	2,300
Find the regression line.
d(r) = 
  

To find the regression line for the given data, we will calculate the linear relationship between the percent recovery $ r $ and the economic value $ d(r) $.

### Step 1: Data points
The data points provided are:

- $ r_1 = 10\% $, $ d(r_1) = 500 $ billion dollars
- $ r_2 = 40\% $, $ d(r_2) = 1,200 $ billion dollars
- $ r_3 = 50\% $, $ d(r_3) = 1,300 $ billion dollars
- $ r_4 = 80\% $, $ d(r_4) = 2,300 $ billion dollars

These are points in the form $ (r, d(r)) $.

### Step 2: Formula for the regression line
The equation of the regression line is:

\[
d(r) = mr + b
\]

where $ m $ is the slope and $ b $ is the y-intercept.

#### The slope $ m $ is calculated by:

\[
m = \frac{n(\sum r d(r)) - (\sum r)(\sum d(r))}{n(\sum r^2) - (\sum r)^2}
\]

Where:
- $ n $ is the number of data points (4 in this case),
- $ r $ is the percent recovery,
- $ d(r) $ is the economic value.

#### The intercept $ b $ is calculated by:

\[
b = \frac{\sum d(r) - m(\sum r)}{n}
\]

Let’s calculate the regression line using this data.The regression line for the given data is:

\[
d(r) = 25.4r + 182
\]

This means that for each 1% increase in oil recovery, the economic value increases by approximately 25.4 billion dollars, and the base economic value starts at 182 billion dollars when the recovery is 0%.

Would you like further details on how this was calculated or how to interpret it? Here are five related questions:

1. How does linear regression work in general?
2. Can the regression line be used for predictions beyond the given data points?
3. What are some limitations of using a linear regression for this type of data?
4. How would the regression change if more data points were added?
5. Could a different type of regression (e.g., quadratic) fit the data better?

**Tip:** Always check the residuals (the differences between actual and predicted values) to ensure the linear model fits the data well.

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) 500 1,200 1,300 2,300 Find the regression line.

Find the regression line for oil recovery percent and economic value data. This problem covers concepts of linear regression, slope and intercept formulas, and how to apply the least squares method. Suitable for students in Grades 10-12.

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