To find the linear regression equation $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept, we need to calculate both $m$ and $b$ using the given data points:

• $(0, 127)$
• $(1, 126)$
• $(2, 151)$
• $(3, 169)$

Step 1: Calculate the slope $m$

The formula for the slope $m$ in linear regression 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
• $x$ and $y$ represent the data points
• $\sum xy$ is the sum of the product of each $x$ and $y$
• $\sum x^2$ is the sum of the squares of the $x$ values

Step 2: Calculate the y-intercept $b$

The formula for the y-intercept $b$ is: $b = \frac{\sum y - m \sum x}{n}$

Step 3: Plug in the data and calculate

Let's calculate the necessary sums:

• $\sum x = 0 + 1 + 2 + 3 = 6$
• $\sum y = 127 + 126 + 151 + 169 = 573$
• $\sum xy = (0 \times 127) + (1 \times 126) + (2 \times 151) + (3 \times 169) = 0 + 126 + 302 + 507 = 935$
• $\sum x^2 = (0^2) + (1^2) + (2^2) + (3^2) = 0 + 1 + 4 + 9 = 14$
• $n = 4$ (since there are 4 data points)

Now calculate the slope $m$: $m = \frac{4(935) - (6)(573)}{4(14) - (6)^2} = \frac{3740 - 3438}{56 - 36} = \frac{302}{20} = 15.1$

Now calculate the y-intercept $b$: $b = \frac{573 - (15.1)(6)}{4} = \frac{573 - 90.6}{4} = \frac{482.4}{4} = 120.6$

Step 4: Write the regression equation

The linear regression equation is: $y = 15.1x + 120.6$

Step 5: Find the projected profit for 2021

In 2021, the number of years since 2012 is $x = 9$. Substituting $x = 9$ into the equation: $y = 15.1(9) + 120.6 = 135.9 + 120.6 = 256.5$

Thus, the projected profit for 2021 is 256,500 dollars, or 257 thousand dollars when rounded to the nearest thousand.

Would you like more details or have any questions?

Here are five related questions to expand on this:

1. How would the regression equation change if the company's profits were nonlinear?
2. What does the slope of 15.1 mean in this context?
3. How could this model be used to project profits further into the future?
4. What are the limitations of using a linear model in predicting profits?
5. How can the accuracy of the linear regression model be evaluated?

Tip: Always check the residuals (differences between actual and predicted values) to ensure that a linear model is appropriate for the data.