To find the coefficient of determination ($R^2$) for the given data, we follow these steps:

1. Determine the regression equation: Fit a linear regression model where $x$ (weight) is the independent variable and $y$ (miles per gallon) is the dependent variable.
2. Calculate $R^2$: This measures the proportion of the variance in $y$ that is explained by $x$.

Let me calculate it for you.import numpy as np import pandas as pd from sklearn.linear_model import LinearRegression from sklearn.metrics import r2_score

Data

weights = np.array([3763, 3981, 3530, 3171, 2583, 3729, 2609, 3777, 3309, 2998, 2760]).reshape(-1, 1) mpg = np.array([19, 20, 21, 23, 27, 20, 27, 19, 21, 27, 25])

Linear regression model

model = LinearRegression() model.fit(weights, mpg)

Predicted values

predicted_mpg = model.predict(weights)

Calculate R^2

r_squared = r2_score(mpg, predicted_mpg)

r_squared * 100 # Convert to percentage87.6161618630900387.61616186309003successThe coefficient of determination ($R^2$) for the given data set is approximately 87.62%.

This indicates that about 87.62% of the variation in miles per gallon (y) is explained by the variation in car weight (x).

Do you want detailed calculations or have any further questions?

Here are 5 follow-up questions for exploration:

1. What does a high $R^2$ value signify in this context?
2. How does weight negatively impact miles per gallon based on the data?
3. Would a quadratic model fit better than a linear model here?
4. Can we use this regression equation for prediction, and if so, how reliable would it be?
5. What other factors might influence miles per gallon besides weight?

Tip: Always examine residuals to check if the linear model is the best fit for the data.