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Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable (Y) and one or more independent variables (X). The goal is to find a linear equation that provides the best fit to the observed data.

Form of the linear equation:

The general form of simple linear regression is: \[ Y = \beta_0 + \beta_1X + \varepsilon \]

where \( \beta_0 \) is the y-intercept, \( \beta_1 \) is the regression coefficient (slope), and \( \varepsilon \) is the error term.

Regression Coefficient (Slope):

The regression coefficient (\( \beta_1 \)) indicates the change in the dependent variable for a one-unit increase in the independent variable. A positive coefficient signifies a positive correlation, while a negative coefficient suggests a negative correlation.

Additional Information:

• Coefficient of Determination (R²): Indicates the proportion of variation in the dependent variable explained by the independent variable.
• P-Value: Indicates the significance of the regression coefficient. A low p-value suggests that the coefficient is statistically significant.
• Residuals: The difference between observed values and predicted values. Residuals should be randomly and evenly distributed.

Example:

Suppose we are examining the relationship between the number of hours a student studies (X) and their grades in a subject (Y). Linear regression could help us find an equation modeling this relationship.