What is linear regression? Provide at least one example using math and one using Python.

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Linear regression is a fundamental statistical method used to model relationships between variables. It finds the best-fitting straight line through data points, allowing us to make predictions. The line is described by the equation y equals beta zero plus beta one times x, where beta zero is the y-intercept and beta one is the slope. Let's work through a mathematical example. Given three data points: one comma three, two comma five, and three comma seven, we need to find the best-fitting line. First, we calculate the slope beta one by finding the change in y over change in x, which equals two. Next, we find the y-intercept beta zero by substituting a known point into our equation, giving us beta zero equals one. Therefore, our linear regression equation is y equals one plus two x. We can use this to predict that when x equals four, y equals nine. Now let's implement this in Python using the scikit-learn library. First, we import numpy and LinearRegression. We prepare our data by reshaping x into a two-dimensional array as required by scikit-learn. We create a LinearRegression model and fit it to our data. The model automatically calculates the intercept and slope coefficients. Finally, we can make predictions for new values. Running this code gives us an intercept of one point zero, a slope of two point zero, and predicts nine point zero for x equals four, confirming our mathematical calculation. Linear regression uses the least squares method to find the best-fitting line. The goal is to minimize the sum of squared errors, where each error is the difference between the actual y value and the predicted y value. The blue dots represent actual data points, while the red dots show predicted values on our regression line. The green lines represent the errors or residuals. By minimizing the sum of these squared errors, we find the optimal slope and intercept that best describe the relationship in our data. This method is called Ordinary Least Squares or OLS. Linear regression has numerous real-world applications across many fields. In economics, it's used for price prediction. In medicine, it helps model drug dosage effects. Engineers use it for performance modeling, and marketers apply it for sales forecasting. The key points to remember are that linear regression models linear relationships using the least squares method, follows the equation y equals beta zero plus beta one x, is easy to implement in Python, and serves as a foundation for more advanced machine learning techniques. This example shows predicting salary based on years of experience, demonstrating how linear regression can provide valuable insights for decision-making.