The table below shows the median weekly earnings of a certain population of workers. If this trend​ continues, predict the median weekly earnings in 201616. For this​ exercise, find a linear and a quadratic regression equation and use both to predict the earnings in 201616. 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)
570570
580580
613613
625625
614614
Question content area bottom
Part 1
create A linear regression equation y=?

Let's solve the problem step by step, beginning with the **linear regression equation**.

### Step 1: Set up the data points.
Since we are told to let \( x \) be the number of years since 2000 and \( y \) be the weekly earnings, we will convert the year values into \( x \)-values:

- 2002 → \( x = 2 \)
- 2004 → \( x = 4 \)
- 2006 → \( x = 6 \)
- 2008 → \( x = 8 \)
- 2010 → \( x = 10 \)

Thus, the table looks like this:

| \( x \) (years since 2000) | \( y \) (weekly earnings in dollars) |
|----------------------------|--------------------------------------|
| 2                          | 570                                  |
| 4                          | 580                                  |
| 6                          | 613                                  |
| 8                          | 625                                  |
| 10                         | 614                                  |

### Step 2: Use the formula for linear regression.
The general form of the linear regression equation is:

\[
y = mx + b
\]

Where:
- \( m \) is the slope,
- \( b \) is the y-intercept.

#### Formula for \( m \) (slope):
\[
m = \frac{n \sum{xy} - (\sum{x})(\sum{y})}{n \sum{x^2} - (\sum{x})^2}
\]

#### Formula for \( b \) (y-intercept):
\[
b = \frac{\sum{y} - m \sum{x}}{n}
\]

Where \( n \) is the number of data points. Let's calculate each term.

### Step 3: Calculate the necessary sums.
We need to compute:
- \( \sum{x} \)
- \( \sum{y} \)
- \( \sum{xy} \)
- \( \sum{x^2} \)

Let's calculate these values and then solve for \( m \) and \( b \).

I'll do the computations now.The linear regression equation is:

\[
y = 6.65x + 560.5
\]

This represents the trend of median weekly earnings based on the number of years since 2000.

Would you like to proceed with predicting the earnings for 2016 using this equation, or should I first provide the quadratic regression? Let me know!

Here are some related questions for further exploration:
1. How do we compute the slope in linear regression?
2. Why is the y-intercept important in regression models?
3. What are the limitations of linear regression for long-term predictions?
4. How would the model change if outliers existed in the data?
5. What real-world factors could cause deviations from this model?

**Tip**: In a linear regression, always check the residuals to see how well the model fits the data.

The table below shows the median weekly earnings of a certain population of workers. If this trend continues, predict the median weekly earnings in 2016. For this exercise, find a linear and a quadratic regression equation and use both to predict the earnings in 2016. 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) 570, 580, 613, 625, 614. Create a linear regression equation y=?

Predict the median weekly earnings for 2016 using both linear and quadratic regression. This problem uses historical earnings data from 2002 to 2010 and involves calculating both regression equations. Key concepts include the slope formula, y-intercept, and quadratic regression model. Suitable for college-level students studying statistics and algebra.

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