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The task is to find both a **linear** and a **quadratic regression equation** based on the table of weekly earnings, and then predict the median weekly earnings in **2017**. Here’s the table:

| Year  | 2002 | 2004 | 2006 | 2008 | 2010 |
|-------|------|------|------|------|------|
| Earnings ($) | 566  | 584  | 613  | 622  | 615  |

The instructions say to let $ x $ represent the number of years since 2000, meaning:
- 2002 → $ x = 2 $
- 2004 → $ x = 4 $
- 2006 → $ x = 6 $
- 2008 → $ x = 8 $
- 2010 → $ x = 10 $

The weekly earnings $ y $ correspond to the earnings in dollars for each year.

### Steps for Linear Regression:
We can now find the equation of the linear regression $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept.

### Steps for Quadratic Regression:
For quadratic regression, the equation will have the form $ y = ax^2 + bx + c $, where $ a $, $ b $, and $ c $ are coefficients to be determined.

I'll compute both the linear and quadratic regression equations and predict the earnings for the year **2017** (which corresponds to $ x = 17 $). Let's do this now.### Linear Regression Equation:
The linear regression equation is:
\[
y = 6.800x + 559.200
\]
Using this equation, the predicted weekly earnings for 2017 ($x = 17$) are:
\[
y = 6.800(17) + 559.200 = 674.800
\]

### Quadratic Regression Equation:
The quadratic regression equation is:
\[
y = -1.250x^2 + 21.800x + 524.200
\]
Using this equation, the predicted weekly earnings for 2017 ($x = 17$) are:
\[
y = -1.250(17)^2 + 21.800(17) + 524.200 = 533.550
\]

### Summary of Predictions:
- **Linear prediction for 2017**: $674.80
- **Quadratic prediction for 2017**: $533.55

Would you like more details or have any further questions?

Here are some related questions:
1. How does the linear regression method differ from quadratic regression?
2. Can we rely more on the linear or quadratic model based on the data?
3. How do outliers affect the regression model predictions?
4. What other types of regression models exist besides linear and quadratic?
5. How would the prediction change if more data points were added?

**Tip:** Always check the residuals (errors) in regression models to ensure your model fits the data well.

The table shows the median weekly earnings of a certain population of workers. Find a linear and quadratic regression equation to predict the weekly earnings in 2017. The table includes data from 2002 to 2010.

Learn how to apply linear and quadratic regression equations to predict weekly earnings based on historical data. This problem covers the use of regression models for data forecasting, suitable for high school students in Grades 9-12.

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