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.