Theory

Linear Regression is one of the most fundamental and widely used techniques in statistical modeling and machine learning. It helps in understanding the relationship between two continuous variables — typically one independent (predictor) variable and one dependent (target) variable. In the context of this experiment, we aim to explore how the number of hours a student studies influences their final exam score.

1. Concept of Linear Regression

Linear regression attempts to model the relationship between two variables by fitting a linear equation (a straight line) to the observed data. The general form of the linear regression equation is:

            𝑌=𝑎+𝑏𝑋+𝜀

Where:

• Y: Dependent variable (Final Exam Score)
• X: Independent variable (Study Hours)
• a: Intercept of the regression line (value of Y when X = 0)
• b: Slope of the regression line (indicates how much Y changes for a unit change in X)
• ε: Error term (represents the variability in Y not explained by X)

2. Data Collection and Preprocessing

Before fitting a regression model, it is essential to gather accurate and representative data. In this simulation, the dataset consists of study hours (X) and corresponding final exam scores (Y) for a number of students. Data preprocessing includes:

• Handling missing or null values
• Identifying and removing outliers
• Normalizing or standardizing values if needed

3. Data Exploration

Exploratory Data Analysis (EDA) helps in visualizing patterns, trends, and distributions in the data. Common techniques include:

• Scatter plots to view the relationship between X and Y
• Summary statistics (mean, median, standard deviation)
• Correlation analysis to quantify the strength of association

4. Model Building

The simple linear regression model is trained using the training dataset. The model learns the best-fitting line by minimizing the difference between predicted values and actual outcomes — typically by minimizing the Mean Squared Error (MSE).

        MSE = (1/n) ∑_i=1ⁿ (Y_i - Ŷ_i)²

Where:

• Y_i: Actual value
• Ŷ_i: Predicted value
• n: Number of observations

A lower MSE indicates a better fit of the model to the data.

5. Model Evaluation

After training, the model is evaluated on a testing dataset that was not used during training. This helps in assessing how well the model generalizes to new, unseen data. Performance metrics used include:

• Mean Squared Error (MSE)
• Root Mean Squared Error (RMSE)
• Mean Squared Error (MSE)

6. Interpretation

Once the model is evaluated, it can be used to make predictions. For example, if a student studies for a certain number of hours, the model can predict their likely exam score. However, it is important to remember that linear regression assumes a linear relationship and may not capture complex patterns.