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Part 1
The table below shows the median weekly earnings of a certain population of workers. If this trend​ continues, predict the median weekly earnings in 201515. For this​ exercise, find a linear and a quadratic regression equation and use both to predict the earnings in 201515. Let x be the number of years since the year 2000 and y be the weekly earnings in dollars.
Year
2002
2004
2006
2008
2010
 
Weekly Earnings
​(in dollars)
566
582
619
628
621


We are tasked with predicting the median weekly earnings for the year 201515, using both a linear and quadratic regression model. Let's break down the process into steps:

### Step 1: Defining Variables
Let $ x $ represent the number of years since the year 2000, and $ y $ represent the weekly earnings in dollars. The data table can be transformed accordingly:

| Year  | $ x $ (Years since 2000) | $ y $ (Weekly Earnings) |
|-------|-----------------------------|---------------------------|
| 2002  | 2                           | 566                       |
| 2004  | 4                           | 582                       |
| 2006  | 6                           | 619                       |
| 2008  | 8                           | 628                       |
| 2010  | 10                          | 621                       |

### Step 2: Linear Regression Model
A linear regression equation has the form:
\[
y = mx + b
\]
Where $ m $ is the slope, and $ b $ is the y-intercept.

We can calculate the slope and intercept using the least squares method based on the data points. After obtaining the linear regression model, we will use it to predict the value of $ y $ when $ x = 201515 - 2000 = 201515 $.

### Step 3: Quadratic Regression Model
A quadratic regression equation has the form:
\[
y = ax^2 + bx + c
\]
Here, $ a $, $ b $, and $ c $ are constants that will be determined using the least squares method based on the data points. We will then use this model to predict $ y $ for $ x = 201515 $.

I will now compute both the linear and quadratic regression equations and make predictions for the year 201515.### Regression Equations

1. **Linear Regression Equation**:
   \[
   y = 7.8x + 556.4
   \]
   This equation predicts a consistent increase of approximately \$7.80 in weekly earnings for every additional year.

2. **Quadratic Regression Equation**:
   \[
   y = -1.32x^2 + 23.66x + 519.4
   \]
   This equation includes both a linear and quadratic term, capturing more complex trends over time.

### Predictions for the year 201515:
- **Linear Prediction**: Using the linear model, the predicted earnings in the year 201515 are approximately **\$1,572,373.40**.
- **Quadratic Prediction**: Using the quadratic model, the predicted earnings are approximately **-\$53,656,193,758.78**, which is clearly unreasonable due to the large value of $x$.

The quadratic prediction is highly unrealistic because quadratic models tend to exaggerate results for large inputs when fitted over limited data.

### Explanation:
- The linear model produces a much more reasonable output for large values of $x$.
- The quadratic model, while fitting the data well for smaller values of $x$, generates unrealistic values for extremely large $x$, showing its limitations for extrapolation.

Do you have any further questions or need clarification on these models?

#### Related questions:
1. Why does quadratic regression perform poorly for large future values?
2. How does the slope of a linear regression affect predictions?
3. Can regression models be improved with more data points?
4. What are the practical limits of using regression for predictions in the far future?
5. How do you determine if a regression model fits the data well?

**Tip**: Always check the behavior of regression models when extrapolating far from the data range, as higher-degree models may behave unpredictably.

The table below shows the median weekly earnings of a certain population of workers. If this trend​ continues, predict the median weekly earnings in 201515. For this​ exercise, find a linear and a quadratic regression equation and use both to predict the earnings in 201515. Let x be the number of years since the year 2000 and y be the weekly earnings in dollars. Year: 2002, 2004, 2006, 2008, 2010. Weekly Earnings (in dollars): 566, 582, 619, 628, 621.

Learn how to predict median weekly earnings using linear and quadratic regression models. This problem involves data extrapolation and analysis with formulas y = mx + b and y = ax^2 + bx + c. Explore the limitations of both models for future predictions. Suitable for students in Grades 10-12.

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