• Simple Linear Regression analyzes the relationship between a single predictor (independent variable) and a response (dependent variable).
  
• We typically model it as \(y = \beta_0 + \beta_1 x + \varepsilon\).
  
• We will use the built-in cars dataset in R, which measures speed (mph) and stopping distances (ft).

• Why study linear regression?
  • It is one of the most fundamental and widely used statistical techniques.
  • Helps us understand how one variable changes in relation to another.
  • Forms a basis for more complex models like multiple regression, logistic regression, etc.

Here is the simple linear regression model with common assumptions:

\[ y_i = \beta_0 + \beta_1 x_i + \varepsilon_i \]

\[ \varepsilon_i \sim N(0, \sigma^2) \]

independent errors (\(\varepsilon_i\) are independent).

• \(\beta_0\) (intercept) and \(\beta_1\) (slope) are unknown parameters.
• \(\varepsilon_i\) are random errors assumed to have mean 0 and constant variance \(\sigma^2\).

We often estimate \(\beta_0\) and \(\beta_1\) via Least Squares, which minimizes the sum of squared residuals:

\[ \hat{\beta}_1 = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})} {\sum_{i=1}^n (x_i - \bar{x})^2}, \quad \hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \,\bar{x}. \]

\[ \bar{x} = \frac{1}{n}\sum_{i=1}^n x_i, \quad \bar{y} = \frac{1}{n}\sum_{i=1}^n y_i. \]

ggplot(cars, aes(x = speed, y = dist)) +
  geom_point() +
  labs(title = "Scatter Plot of Speed vs. Distance",
       x = "Speed (mph)",
       y = "Stopping Distance (ft)") +
  theme_minimal()

## `geom_smooth()` using formula = 'y ~ x'

• Simple Linear Regression is a straightforward way to quantify and test the relationship between two variables.
• In our cars example, speed is a significant predictor of stopping distance.
• Next steps might include:
  • Checking diagnostic plots (residuals, normality).
  • Trying transformations or polynomial terms if the relationship is non-linear.
  • Extending to multiple regression if more predictors are available.