## What is Regression Analysis? Regression analysis is a statistical method used to model the relationship between a dependent variable \( Y \) and one or more independent variables \( X \). ### **1. Simple Linear Regression** In a simple linear regression model, the relationship between \( X \) and \( Y \) is given by: \[ Y = \beta_0 + \beta_1 X + \epsilon \] where: - \( Y \) is the dependent variable (response), - \( X \) is the independent variable (predictor), - \( \beta_0 \) is the intercept, - \( \beta_1 \) is the slope (coefficient), - \( \epsilon \) is the error term. ### **2. Example Dataset** Assume we have a dataset: | X (Hours Studied) | Y (Exam Score) | |------------------|--------------| | 1 | 50 | | 2 | 55 | | 3 | 65 | | 4 | 70 | | 5 | 75 | ### **3. Fitting the Model** We estimate \( \beta_0 \) and \( \beta_1 \) using the **least squares method**: \[ \hat{\beta}_1 = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2} \] \[ \hat{\beta}_0 = \bar{Y} - \hat{\beta}_1 \bar{X} \] Using these equations, we can fit a line to the data. ### **4. Visualization** Below is a plot showing the regression line:  ### **5. Conclusion** Regression is a powerful tool in statistics and machine learning for predicting numerical outcomes. More advanced models include **multiple regression**, **logistic regression**, and **nonlinear regression**. --- 