• Linear regression models the relationship between two variables
  
• One variable explains or predicts another
  
• This method is widely used in many fields including science and engineering

• Suppose we want to understand how studying affects exam scores
  
• We collect data on study hours and exam performance
  
• We then try to model this relationship mathematically

hours <- c(1, 2, 3, 4, 5, 6, 7, 8)
scores <- c(50, 55, 65, 70, 75, 78, 85, 90)

data <- data.frame(hours = hours, scores = scores)

• (1, 50)
  
• (2, 55)
  
• (3, 65)
  
• (4, 70)
  
• (5, 75)
  
• (6, 78)
  
• (7, 85)
  
• (8, 90)

• The data shows a clear upward trend
  
• As study hours increase, exam scores also increase
  
• This suggests a positive linear relationship
  
• Linear regression helps quantify this pattern

In simple linear regression, we model the relationship as:

\[ Y = \beta_0 + \beta_1 X + \epsilon \]

• \(Y\) is the response (exam score)
  
• \(X\) is the predictor (study hours)
  
• \(\beta_0\) is the intercept
  
• \(\beta_1\) is the slope
  
• \(\epsilon\) represents random error

• The intercept is the predicted value when \(X = 0\)
  
• The slope tells us how much \(Y\) changes for each unit increase in \(X\)
  
• The goal is to find the best-fitting line through the data

model <- lm(scores ~ hours, data = data)
summary(model)

• The lm() function fits a linear regression model
  
• It estimates the best line relating study hours to exam scores
  
• The detailed output is hidden to keep the slide clean

Residuals measure prediction error:

\[ e_i = y_i - \hat{y}_i \]

• If residuals are small, the model fits well
  
• Random patterns suggest a good linear fit

• The slope represents the change in score per additional hour studied
  
• A positive slope indicates improvement with more study time
  
• This allows us to make predictions for new values

The regression line minimizes the total squared error:

\[ \sum_{i=1}^{n}(y_i - \hat{y}_i)^2 \]

• This ensures the best possible fit to the data
  
• It penalizes large errors more than small ones

• Linear regression is simple but powerful
  
• It models relationships and supports prediction
  
• It is easy to implement using R
  
• Visualization helps interpret results clearly

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