A random sample of ten professional athletes produced the following data where x is the number of endorsements the player has and y is the amount of money made (in millions of dollars).
x	y	x	y
0	2	5	11
3	8	4	9
2	6	3	8
1	3	0	3
6	14	5	10
Use regression to find the equation for the line of best fit. (Round your answers to three decimal places.)

To find the equation for the line of best fit using simple linear regression, we need to determine the slope \( m \) and the y-intercept \( b \) of the equation of the line:

\[
y = mx + b
\]

We can calculate the slope \( m \) and intercept \( b \) using the following formulas:

\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]

\[
b = \frac{\sum y - m(\sum x)}{n}
\]

Where:
- \( n \) is the number of data points,
- \( \sum x \) is the sum of the \( x \)-values,
- \( \sum y \) is the sum of the \( y \)-values,
- \( \sum xy \) is the sum of the products of corresponding \( x \) and \( y \) values,
- \( \sum x^2 \) is the sum of the squares of the \( x \)-values.

Let's compute these values based on the given dataTo find the equation for the line of best fit using simple linear regression, we need to determine the slope \( m \) and the y-intercept \( b \) of the equation of the line:

\[
y = mx + b
\]

We can calculate the slope \( m \) and intercept \( b \) using the following formulas:

\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]

\[
b = \frac{\sum y - m(\sum x)}{n}
\]

Where:
- \( n \) is the number of data points,
- \( \sum x \) is the sum of the \( x \)-values,
- \( \sum y \) is the sum of the \( y \)-values,
- \( \sum xy \) is the sum of the products of corresponding \( x \) and \( y \) values,
- \( \sum x^2 \) is the sum of the squares of the \( x \)-values.

Let's compute these values based on the given 

**Given Data:**
\[
(0, 2), (3, 8), (2, 6), (1, 3), (6, 14), (5, 11), (4, 9), (3, 8), (0, 3), (5, 10)
\]

I'll calculate the slope and intercept now.The equation of the line of best fit is:

\[
y = 1.795x + 2.196
\]

This means for every additional endorsement, the player's earnings increase by approximately 1.795 million dollars.

Would you like further details or explanations?

Here are 5 related questions to explore:
1. How do you interpret the slope of the regression line?
2. What does the y-intercept represent in the context of this problem?
3. How can you use this equation to predict earnings for a player with 7 endorsements?
4. What are the assumptions behind linear regression, and are they met in this data?
5. How would you assess the accuracy of this regression model?

**Tip:** Always check the correlation coefficient to see how well the data fits the regression line.

A random sample of ten professional athletes produced the following data where x is the number of endorsements the player has and y is the amount of money made (in millions of dollars). Use regression to find the equation for the line of best fit. (Round your answers to three decimal places.)

Use linear regression to determine the line of best fit for a sample of 10 athletes' endorsements and earnings. This problem involves calculating the slope and intercept using the least squares method and analyzing the relationship between endorsements and earnings. Suitable for advanced high school or college-level students.

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