• Simple Linear Regression is a method to model the relationship between two continuous variables.
• It predicts the dependent variable \(Y\) based on the independent variable \(X\).
• The linear regression equation is: \[ Y = \beta_0 + \beta_1 X + \epsilon \] where:
  • \(\beta_0\) is the intercept,
  • \(\beta_1\) is the slope,
  • \(\epsilon\) is the error term.

• Determine the strength and direction of the relationship between two variables.
• Estimate the value of the dependent variable based on the independent variable.
• Example applications:
  • Predicting a student’s score based on study hours.
  • Estimating a car’s fuel efficiency based on engine size.

• This scatter plot shows the relationship between \(X\) and \(Y\).

• Simple Linear Regression uses the formula:

  \[ \hat{Y} = \beta_0 + \beta_1 X \]

• To estimate \(\beta_0\) (intercept) and \(\beta_1\) (slope), we minimize the sum of squared errors: \[ \text{SSE} = \sum_{i=1}^n (Y_i - \hat{Y}_i)^2 \]

Summary:

• Linear regression predicts \(Y\) based on \(X\).
• The regression line minimizes the sum of squared residuals.

Applications:

• Predictive modeling in finance, healthcare, and more.

# Scatter plot code
library(ggplot2)
ggplot(data, aes(x = x, y = y)) +
  geom_point(color = "blue") +
  labs(title = "Scatter Plot of X vs Y", x = "X", y = "Y")