Step 1: Identify the points

We are given the following data points:

$x$ (Years since 2009)	$y$ (Profits in thousands of dollars)
0	4747
11	6464
22	6666
33	8989
44	9898

Step 2: Apply linear regression

The linear regression equation is of the form: $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept.

We will calculate the slope $m$ and the intercept $b$ using the formulas for linear regression: $m = \frac{n\sum{xy} - \sum{x}\sum{y}}{n\sum{x^2} - (\sum{x})^2}$ $b = \frac{\sum{y} - m\sum{x}}{n}$ Where $n$ is the number of data points.

Step 3: Calculate the necessary sums

Let's calculate the necessary sums for this dataset. There are 5 data points ($n = 5$):

$\sum{x} = 0 + 11 + 22 + 33 + 44 = 110$ $\sum{y} = 4747 + 6464 + 6666 + 8989 + 9898 = 36764$ $\sum{xy} = (0 \times 4747) + (11 \times 6464) + (22 \times 6666) + (33 \times 8989) + (44 \times 9898) = 561967$ $\sum{x^2} = 0^2 + 11^2 + 22^2 + 33^2 + 44^2 = 3796$

Step 4: Calculate the slope ($m$) and intercept ($b$)

Now, we can use these sums to calculate the slope $m$ and the intercept $b$.

$m = \frac{5(561967) - (110)(36764)}{5(3796) - (110)^2}$ $m = \frac{2809835 - 4044040}{18980 - 12100} = \frac{-1230205}{6880} \approx 178.78$

Now calculate the intercept $b$: $b = \frac{36764 - (178.78)(110)}{5}$ $b = \frac{36764 - 19665.8}{5} = \frac{17098.2}{5} \approx 3419.64$

Step 5: Write the linear regression equation

Thus, the linear regression equation is: $y = 178.78x + 3419.64$

Step 6: Project the profit for 2021

Since 2021 is 12 years since 2009, $x = 12$. We substitute this into the linear regression equation: $y = 178.78(12) + 3419.64 = 2145.36 + 3419.64 = 5565$

Thus, the projected profit for 2021 is approximately 5565 thousand dollars.

Final Answer:

The projected profit for 2021 is 5565 thousand dollars.

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Would you like more details or have any questions?

Here are 5 follow-up questions:

1. How would the equation change if another data point were added?
2. Can you explain how the slope of the regression line influences the predicted values?
3. What does the y-intercept of the regression line represent in the context of this problem?
4. How would changing the time frame affect the projected profit for future years?
5. What are the assumptions behind using linear regression for this data?

Tip: When working with real-world data, always check if a linear model is appropriate by plotting the data points to see if they roughly follow a straight-line trend.