To solve this problem, we will perform linear regression on the data provided in the table. The aim is to determine the linear equation that best fits the data and then use that equation to estimate the profit for 2006. Here's the step-by-step process:

Step 1: Understanding the table

The table gives us the following points (x, y), where x is the number of years since 1998, and y is the profit in thousands of dollars:

• (0, 72)
• (1, 84)
• (2, 104)
• (3, 156)

We are tasked with finding the equation of the line $y = mx + b$ (the line of best fit) that approximates this data.

Step 2: Calculating the slope (m)

The slope formula for a linear regression line is:

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

Where:

• $n$ is the number of data points (in this case, 4).
• $\sum xy$ is the sum of the products of each x and 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 the x-values.

Let’s calculate these values:

• $\sum x = 0 + 1 + 2 + 3 = 6$
• $\sum y = 72 + 84 + 104 + 156 = 416$
• $\sum xy = (0 \times 72) + (1 \times 84) + (2 \times 104) + (3 \times 156) = 0 + 84 + 208 + 468 = 760$
• $\sum x^2 = (0^2) + (1^2) + (2^2) + (3^2) = 0 + 1 + 4 + 9 = 14$

Now substitute into the slope formula:

$m = \frac{4(760) - (6)(416)}{4(14) - (6)^2} = \frac{3040 - 2496}{56 - 36} = \frac{544}{20} = 27.2$

Step 3: Calculating the y-intercept (b)

The y-intercept $b$ can be found using the formula:

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

Substitute the known values:

$b = \frac{416 - 27.2(6)}{4} = \frac{416 - 163.2}{4} = \frac{252.8}{4} = 63.2$

Step 4: Writing the equation of the line

Now that we have $m = 27.2$ and $b = 63.2$, the equation of the line of best fit is:

$y = 27.2x + 63.2$

Step 5: Finding the profit for 2006

To find the profit for 2006, we need to plug in $x = 8$ (since 2006 is 8 years after 1998) into the equation:

$y = 27.2(8) + 63.2 = 217.6 + 63.2 = 280.8$

Therefore, the projected profit for 2006 is approximately $281,000.

Would you like further details or explanations on any of the steps?

Here are 5 related questions you might want to explore:

1. How can we use this method to project profits for other years?
2. How does the slope affect the projected profits?
3. What are the limitations of using linear regression for this data?
4. How can we determine if a linear model is the best fit for the data?
5. How does rounding affect the accuracy of the projections?

Tip: When performing linear regression, it is important to check how well the data fits the linear model by analyzing residuals or using statistical tools like the coefficient of determination ($R^2$).