Math Problem Statement

Determine the cubic regression function that best approximates the given data.

Round all values to 2 decimal places

y

To determine the cubic regression function $y = ax^3 + bx^2 + cx + d$ , we need to fit the given data using regression analysis. Here's the process:

x: & \; 0, 2, 4, 6, 8, 10, 12 \\ y: & \; 23, 32, 20, 13, 19, 26, 38 \end{aligned}$$ We will calculate the coefficients $$a$$, $$b$$, $$c$$, and $$d$$ using a least-squares fitting approach. Let me compute that for you. numpy as np from numpy.polynomial.polynomial import Polynomial # Input data x = np.array([0, 2, 4, 6, 8, 10, 12]) y = np.array([23, 32, 20, 13, 19, 26, 38]) # Fit a cubic polynomial (degree=3) coefficients = np.polyfit(x, y, 3) # Round coefficients to 2 decimal places coefficients_rounded = np.round(coefficients, 2) # Construct the cubic regression equation coefficients_roundedThe cubic regression function that best approximates the given data is: $$y = 0.08x^3 - 0.97x^2 + 1.83x + 25.43$$ Would you like me to explain the process of deriving this or how to use it? ### Related Questions: 1. What is the significance of each coefficient in the cubic equation? 2. How is the least-squares fitting method used to derive the regression equation? 3. What are other methods to check the accuracy of a regression model? 4. How do you calculate and interpret the residuals in regression analysis? 5. Can the cubic regression model be used for extrapolation beyond the given data? ### Tip: Always plot the data points and the regression curve to visually verify the fit of the model.

Drop file here or Click Here to upload

Cubic Regression
Polynomial Fitting
Least-Squares Method

Cubic regression equation: y = ax^3 + bx^2 + cx + d
Least-squares fitting: minimizes the sum of squared residuals

Least-Squares Regression Theorem

Grades 11-12 and above

Cubic Regression to Find Polynomial for Given Points

Cubic and Quartic Polynomial Regression for Curve Fitting

Best Fit Quadratic Function for Data Set Using Least-Squares Regression

Cubic and Quartic Polynomial Models: Data Analysis Example

Cubic and Linear Function Models for Data Points x = [1, 2, 3, 4, 5, 6] and y = [-2, -1, 0, 4, 8, 16]