Question

In some cases, the best-fitting multiple regression equation is of the form y^ =b₀ + b₁x + b₂x² + b₃x³. This equation represents a cubic model. Using the provided data and regression techniques, demonstrate the step-by-step process of determining the coefficients b₀, b₁, b₂ and b₃ for this cubic regression equation. Show your work comprehensively.

Answer 1

To determine the coefficients for the cubic regression equation, you would need to use regression techniques such as the method of least squares.

Step-by-step explanation:

To determine the coefficients for the cubic regression equation, you would need to use regression techniques such as the method of least squares. This involves minimizing the sum of squared errors (SSE) between the predicted values and the actual values of the dependent variable.

1. First, you need to gather data on the independent variable (x) and the dependent variable (y).
2. Next, calculate the squared values of x and x^3.
3. Construct a matrix equation using the data and squared values of x and x^3. This equation will be of the form [1 x x^2 x^3] [b₀ b₁ b₂ b₃]ᵀ = [y], where [b₀ b₁ b₂ b₃]ᵀ represents the coefficients.
4. Use matrix algebra to solve for the coefficients. You can do this by multiplying both sides of the equation by the transpose of the matrix [1 x x^2 x^3] and then solving for [b₀ b₁ b₂ b₃]ᵀ.

The resulting values of b₀, b₁, b₂, and b₃ represent the coefficients for the cubic regression equation.