import pandas as pd import statsmodels.api as sm

Define the data

data = { "x1": [30, 47, 25, 51, 40, 51, 74, 36, 59, 76], "y": [94, 108, 112, 178, 94, 175, 170, 117, 142, 211] }

Create a DataFrame

df = pd.DataFrame(data)

Add a constant to the independent variable (for intercept)

X = sm.add_constant(df["x1"]) y = df["y"]

Fit the regression model

model = sm.OLS(y, X).fit()

Extract the regression equation coefficients

coefficients = model.params

coefficientsconst 45.059369 x1 1.943571 dtype: float64const 45.059369 x1 1.943571 dtype: float64successThe estimated regression equation relating $y$ to $x_1$ is:

$\hat{y} = 45.1 + 1.9x_1$

Do you want further details on this calculation or have any questions? Here are some related questions for further exploration:

1. How is the intercept ($45.1$) interpreted in this context?
2. What does the slope ($1.9$) indicate about the relationship between $x_1$ and $y$?
3. How can we test the significance of this regression model?
4. Can we plot the data points and the regression line for visualization?
5. How would adding $x_2$ to the model potentially improve its predictive power?

Tip: Always examine residuals to check the validity of a regression model.