import matplotlib.pyplot as plt import numpy as np import pandas as pd from sklearn.linear_model import LinearRegression from sklearn.preprocessing import PolynomialFeatures from sklearn.metrics import r2_score

Data

x = np.array([5, 15, 40, 50, 60, 80]).reshape(-1, 1) y = np.array([12, 15, 24, 46, 76, 112])

Scatter Plot

plt.scatter(x, y, color='blue') plt.title('Scatter Plot of the Data') plt.xlabel('Independent Variable (x)') plt.ylabel('Dependent Variable (y)') plt.show()

Linear Regression

linear_model = LinearRegression() linear_model.fit(x, y) y_pred_linear = linear_model.predict(x)

Linear regression equation

b0_linear = linear_model.intercept_ b1_linear = linear_model.coef_[0]

Significance Test (R^2)

r2_linear = r2_score(y, y_pred_linear)

Polynomial Regression (Quadratic)

poly = PolynomialFeatures(degree=2) x_poly = poly.fit_transform(x)

poly_model = LinearRegression() poly_model.fit(x_poly, y) y_pred_poly = poly_model.predict(x_poly)

Polynomial regression coefficients

b0_poly = poly_model.intercept_ b1_poly, b2_poly = poly_model.coef_[1], poly_model.coef_[2]

R^2 for polynomial model

r2_poly = r2_score(y, y_pred_poly)

Plot the linear regression line

plt.scatter(x, y, color='blue', label='Original Data') plt.plot(x, y_pred_linear, color='red', label='Linear Fit') plt.plot(x, y_pred_poly, color='green', label='Quadratic Fit') plt.title('Scatter Plot with Regression Lines') plt.xlabel('Independent Variable (x)') plt.ylabel('Dependent Variable (y)') plt.legend() plt.show()

Results

{ "Linear Regression Equation": f"y = {b0_linear:.3f} + {b1_linear:.3f}x", "Linear R^2": r2_linear, "Quadratic Regression Equation": f"y = {b0_poly:.3f} + {b1_poly:.3f}x + {b2_poly:.3f}x^2", "Quadratic R^2": r2_poly } <<ImageDisplayed>> <<ImageDisplayed>>{'Linear Regression Equation': 'y = -7.479 + 1.319x', 'Linear R^2': 0.8731001487791359, 'Quadratic Regression Equation': 'y = 13.086 + -0.325x + 0.020x^2', 'Quadratic R^2': 0.97588677203291}{'Linear Regression Equation': 'y = -7.479 + 1.319x', 'Linear R^2': 0.8731001487791359, 'Quadratic Regression Equation': 'y = 13.086 + -0.325x + 0.020x^2', 'Quadratic R^2': 0.97588677203291}success### Analysis of the Data:

a. Scatter Plot:
The scatter plot demonstrates the relationship between the independent variable $x$ and the dependent variable $y$. It appears nonlinear because the points deviate from a straight line and suggest a curve.

b. Linear Regression Equation:
The estimated linear regression equation is: $\hat{y} = -7.479 + 1.319x$

• R² for linear model: $0.873$, indicating a good but not perfect fit.

c. Quadratic Regression Equation:
The estimated quadratic regression equation is: $\hat{y} = 13.086 - 0.325x + 0.020x^2$

• R² for quadratic model: $0.976$, showing a significantly better fit compared to the linear model.

Conclusion:

The quadratic model provides a much better fit to the data than the linear model. It captures the curvature in the data that the linear model misses.

Do you have any specific questions or need further details? Here are some relevant questions:

1. Would you like to understand the significance testing for the linear model's coefficients?
2. Do you want to compare residual plots between the two models?
3. Should I explain how to compute R² values?
4. Would you like to analyze the quadratic model further (e.g., interpretation of coefficients)?
5. Do you need predictions for specific values of $x$?

Tip: A higher R² value indicates that the model explains a greater proportion of the variance in the dependent variable, making it a better fit.