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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)

566566

582582

619619

628628

621621

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Part 1

A linear regression equation is

yequals=7.8 x plus 556.47.8x+556.4.

​(Use integers or decimals for any numbers in the equation. Round to three decimal places as​ needed.)

Part 2

What is the median weekly earnings for

201515

predicted by the linear regression​ equation?

​$673673

​(Round to the nearest dollar as​ needed.)

Part 3

A quadratic regression equation is

yequals=enter your response here.

​(Use integers or decimals for any numbers in the equation. Round to three decimal places as​ needed.)

### Part 1: Linear Regression Equation

The linear regression equation is given as:
\[
y = 7.8x + 556.4
\]
where:
- $ y $ is the median weekly earnings (in dollars).
- $ x $ is the number of years since 2000.

For the year 2015, $ x = 2015 - 2000 = 15 $.

### Part 2: Predicting Median Weekly Earnings in 2015 Using Linear Regression

Substitute $ x = 15 $ into the linear regression equation:
\[
y = 7.8(15) + 556.4
\]
\[
y = 117 + 556.4 = 673.4
\]
So, the median weekly earnings predicted by the linear regression equation for 2015 is approximately:
\[
y \approx 673 \, \text{dollars}
\]
This matches the provided value.

### Part 3: Finding the Quadratic Regression Equation

To find the quadratic regression equation, we need to fit the following form:
\[
y = ax^2 + bx + c
\]
Using the data points from the given years and earnings:

- $ x_1 = 2 $, $ y_1 = 566 $
- $ x_2 = 4 $, $ y_2 = 582 $
- $ x_3 = 6 $, $ y_3 = 619 $
- $ x_4 = 8 $, $ y_4 = 628 $
- $ x_5 = 10 $, $ y_5 = 621 $

I will now calculate the quadratic regression equation based on this data.The quadratic regression equation based on the given data is:

\[
y = -1.321x^2 + 23.657x + 519.4
\]

### Part 3: Predicting Median Weekly Earnings for 2015 Using Quadratic Regression

Now, using this equation, you can substitute $ x = 15 $ to predict the median weekly earnings for the year 2015. Would you like me to calculate that prediction for you?

---
Here are 5 related questions:
1. How do linear and quadratic regression equations differ in predicting trends?
2. Why might a quadratic model be more accurate for certain datasets?
3. How can residuals help determine the best-fit model for data?
4. What happens when you apply cubic or higher-order polynomials for regression?
5. How is overfitting detected when using regression models?

**Tip:** Quadratic regressions can better fit data with curves, whereas linear regressions work best with constant trends.

The table below shows the median weekly earnings of a certain population of workers. If this trend​ continues, predict the median weekly earnings in 2015. For this​ exercise, find a linear and a quadratic regression equation and use both to predict the earnings in 2015. Let x be the number of years since the year 2000 and y be the weekly earnings in dollars.

This exercise shows how to predict the median weekly earnings for 2015 using both linear and quadratic regression equations based on historical data from 2002 to 2010. Learn how to apply the linear regression equation y = 7.8x + 556.4 and the quadratic regression equation to make accurate predictions. Suitable for Grades 10-12 students interested in algebra and data analysis.

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