This question involves determining the coefficients for a multiple regression equation in the form:

$\hat{y} = b_0 + b_1x + b_2x^2 + b_3x^3$

using the given dataset where $x$, $x^2$, and $x^3$ represent the independent variables. Here's how you solve this:

1. Dataset Analysis: For each $x$ value, calculate $x^2$ and $x^3$. These will be additional columns in the dataset.

2. Set up the Regression Equation: Use the regression equation format to solve for $b_0$, $b_1$, $b_2$, and $b_3$ that minimize the error between $\hat{y}$ and $y$.

3. Matrix Formulation: Set this up as a system of equations using matrices to solve for the coefficients. This involves constructing a design matrix $X$, where each row corresponds to $[1, x, x^2, x^3]$, and using the normal equation:

   $\mathbf{b} = (\mathbf{X}^\top \mathbf{X})^{-1} \mathbf{X}^\top \mathbf{y}$

4. Compute the Coefficients: Perform the matrix multiplication and inversion to solve for the coefficients.

Would you like me to compute the specific coefficients using this dataset? If so, I can walk you through the steps or perform a detailed calculation.

Follow-Up Questions:

1. Would you like to know more about how regression equations are derived?
2. Do you want a step-by-step demonstration of setting up the matrix?
3. Shall we explore how errors are minimized in least-squares regression?
4. Interested in visualizing the data and cubic fit using a graph?
5. Would you like to learn how this process is implemented programmatically (e.g., Python)?

Tip:

For complex regression problems, computational tools like Excel, Python, or R can save time and ensure accuracy when solving for coefficients.