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

Amazon revenue data

data = { "Year": [2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019], "Years_since_2012": [0, 1, 2, 3, 4, 5, 6, 7], "Revenue": [61.09, 74.45, 88.99, 107.01, 135.99, 177.87, 232.89, 280.52], }

df = pd.DataFrame(data)

Independent and dependent variables

X = np.array(df["Years_since_2012"]).reshape(-1, 1) # Years since 2012 y = np.array(df["Revenue"]) # Revenue in billions

Linear regression

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

Coefficients

slope = model.coef_[0] intercept = model.intercept_

Correlation coefficient (r)

correlation_matrix = np.corrcoef(df["Years_since_2012"], df["Revenue"]) correlation_coefficient = correlation_matrix[0, 1]

R-squared value

r_squared = r2_score(y, model.predict(X))

Scatter plot and regression line

plt.figure(figsize=(10, 6)) plt.scatter(df["Years_since_2012"], df["Revenue"], color="blue", label="Data points") plt.plot(df["Years_since_2012"], model.predict(X), color="red", label="Regression line") plt.title("Amazon Revenue Over Years Since 2012") plt.xlabel("Years since 2012") plt.ylabel("Revenue (in billions of $)") plt.legend() plt.grid(True) plt.show()

Displaying results

slope, intercept, correlation_coefficient, r_squared <<ImageDisplayed>>(31.23607142857144, 35.52499999999996, 0.9667066412895324, 0.9345217303132884)(31.23607142857144, 35.52499999999996, 0.9667066412895324, 0.9345217303132884)success### Results:

1. Correlation Coefficient (r): 0.97

   • Meaning: The correlation coefficient of 0.97 indicates a very strong positive linear relationship between the number of years since 2012 and Amazon's revenue.

2. Least Squares Regression Line:
   $\hat{y} = 31.24x + 35.52$

   • Here, $\hat{y}$ is the predicted revenue (in billions), and $x$ is the number of years since 2012.

3. Slope Meaning:
   The slope (31.24) indicates that for each additional year since 2012, Amazon's revenue increases by approximately $31.24 billion on average.

4. Y-Intercept Meaning:
   The y-intercept (35.52) represents the estimated revenue of Amazon in the year 2012 (when $x = 0$). It is meaningful in this context as it aligns with the starting point of the data.

5. R-squared (R²): 0.93

   • Meaning: 93.45% of the variability in Amazon's revenue can be explained by the number of years since 2012. This suggests that the model fits the data very well.

Let me know if you'd like further details on any part!

Related Questions:

1. How can the regression model be used to forecast Amazon's revenue for future years?
2. What factors might account for any deviations between predicted and actual revenues in the dataset?
3. What does a high correlation coefficient imply about the dependence of revenue growth on time?
4. How would adding more data from before 2012 affect the model and its predictions?
5. Are there other variables besides time that could improve the accuracy of the prediction model?

Tip:

Always examine residuals (differences between predicted and observed values) to ensure there are no patterns, which might indicate non-linearity or missing variables.