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The annual profits for a company are given in the following table, where x represents the number of years since 2001, and y represents the profit in thousands of dollars. Write the linear regression equation that represents this set of data, rounding all coefficients *to the nearest tenth*. Using this equation, find the projected profit (in thousands of dollars) for 2012, rounded *to the nearest thousand dollars*.

To find the linear regression equation (the line of best fit) for the given data, we will need to calculate the slope and intercept of the line.

The formula for the line of best fit is:

\[
y = mx + b
\]

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

We can calculate the slope \(m\) using the formula:

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

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

After calculating \(m\) and \(b\), we will substitute these values into the equation \(y = mx + b\).

To find the projected profit for 2012, we will use \(x = 11\) (since 2012 is 11 years after 2001) and substitute it into the equation to calculate the profit.

I will now perform these calculations to find the linear regression equation and the projected profit.The linear regression equation (line of best fit) for the given data is:

\[
y = 14.5x + 145.5
\]

Using this equation, the projected profit for the year 2012 (11 years after 2001) is approximately \(305\) thousand dollars.

Would you like more details or have any questions?

Here are some related questions to consider:
1. How is the slope of the line related to the rate of change in profit over the years?
2. Can this linear model be used to predict profits far beyond 2012? Why or why not?
3. What other factors could influence the accuracy of this projection in real-world scenarios?
4. How can we assess the quality of this linear regression model?
5. What would the profit be projected for the year 2020 based on this model?

**Tip**: Linear regression works well for trends over time, but it may not account for unexpected changes or external factors that can influence real-world data.

The annual profits for a company are given in the following table, where x represents the number of years since 2001, and y represents the profit in thousands of dollars. Write the linear regression equation that represents this set of data, rounding all coefficients to the nearest tenth. Using this equation, find the projected profit (in thousands of dollars) for 2012, rounded to the nearest thousand dollars.

Learn how to write the linear regression equation for a company's annual profits since 2001 and use it to project profits for 2012. This problem involves calculating the slope and intercept to form the best-fit line equation, perfect for students in Grades 10-12.

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