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The image presents a problem where a cubic regression model is sought for a company's annual profits, measured in millions of dollars, over several years. The table provides profits from 2006 to 2010, with profits and years listed as follows:

- Years (relabelled as \( x \) starting from 0 for 2006):  
  \[
  x = 0 \ (2006), \ 1 \ (2007), \ 2 \ (2008), \ 3 \ (2009), \ 4 \ (2010)
  \]
  
- Profits (in millions of dollars):  
  \[
  P = 87.6, \ 85.4, \ 83.1, \ 81.5, \ 80.3
  \]

The goal is to find the cubic regression model and determine the coefficient of the \( x^2 \) term in the model. The available options are:

- A) \(-0.15\)
- B) \(-0.2320\)
- C) \(-0.2697\)
- D) \(-0.40\)
- E) None of the above

I will now fit a cubic regression model to this data and calculate the coefficient for the \(x^2\) term. Let's proceed.The coefficient of the \(x^2\) term in the cubic regression model is approximately \(-0.0571\). Since none of the provided options match this value, the correct answer is:

**E) None of the above.**

Would you like more details or have any questions? Here are five related questions you might want to explore:

1. How is a cubic regression model calculated from data points?
2. Can we use other types of regression (e.g., linear, quadratic) for this data set?
3. How do you interpret the coefficients of a cubic regression model in real-world scenarios?
4. What is the role of higher-order terms (like \(x^3\), \(x^2\)) in a polynomial model?
5. How would the predictions change if we extended the model to predict future profits?

**Tip:** Always ensure your model fits the trend in the data before trusting its predictions—using a polynomial of too high a degree can lead to overfitting.

The following table of values gives a company's annual profits in millions of dollars. Reindex the data so that the year 2006 corresponds to x = 0. Find the cubic regression model for the data. Which of these is the coefficient of the x^2 term of the cubic regression model?

Explore how to find the cubic regression model for a company's annual profits over several years. Learn to calculate the coefficient of the x^2 term in the model using real data points. Suitable for high school students in Grades 10-12.

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