To calculate the coefficient of determination ($R^2$) for the expanded data set, we need to perform a regression analysis. The coefficient of determination is the square of the correlation coefficient between the dependent variable ($y$) and the independent variable ($x$). It represents the proportion of variance in $y$ that is predictable from $x$.

Let’s compute it step by step.

1. Identify variables: $x$ is the car weight in pounds, and $y$ is the miles per gallon (MPG).

2. Input the data: Perform a regression calculation.

3. Find $R^2$: Calculate $R^2$ as: $R^2 = \text{Correlation coefficient}^2$

I'll calculate $R^2$ now.import numpy as np import pandas as pd from sklearn.linear_model import LinearRegression from sklearn.metrics import r2_score

Data

data = { "Weight (pounds)": [3769, 3987, 3525, 3174, 2583, 3740, 2608, 3774, 3313, 2990, 2753], "Miles per Gallon": [20, 19, 21, 24, 27, 19, 26, 19, 20, 26, 27] }

Convert to DataFrame

df = pd.DataFrame(data)

Extract x and y

x = np.array(df["Weight (pounds)"]).reshape(-1, 1) y = np.array(df["Miles per Gallon"])

Perform Linear Regression

model = LinearRegression() model.fit(x, y)

Predict y values

y_pred = model.predict(x)

Calculate R-squared

r_squared = r2_score(y, y_pred) r_squared_rounded = round(r_squared * 100, 1)

r_squared_rounded89.789.7successThe coefficient of determination ($R^2$) for the expanded data set is 89.7%.

This indicates that approximately 89.7% of the variation in miles per gallon (MPG) can be explained by the weight of the car.

Would you like me to explain the calculation process in more detail or explore further related questions? Here are some related questions:

1. What does $R^2$ signify in a regression analysis?
2. How does the weight of a car typically impact its MPG based on this analysis?
3. What are the potential limitations of relying solely on $R^2$ for assessing model fit?
4. Can we visualize this data with a regression line plot for better understanding?
5. How would the result change if we removed one or more data points from the analysis?

Tip: Always inspect residuals when analyzing regression to verify the model's assumptions!