Agenda

<aside> 📌 Reminders

• Final Tutorial today

• Final Exam December 12, 2024 </aside>

• Regression is a statistical method used to model the relationship between two variables:

  1. Independent variable (predictor): $X$
  2. Dependent variable (response): $Y$

• Goal: Find an equation that best describes how $Y$ changes with $X$.

The Least Squares Regression Line

• The least squares regression line is the line that minimizes the sum of squared differences (errors) between the observed data points and the predicted values on the line.
• Equation of the line:

$$ Y = b_0 + b_1X $$

Where:

• $b_0$ = y-intercept,
• $b_1$ = slope.

Calculating the Slope ($b_1$) and Intercept ($b_0$)

• Slope ($b_1$):

$$ b_1 = \frac{\sum{(X_i - \bar{X})(Y_i - \bar{Y})}}{\sum{(X_i - \bar{X})^2}} $$

• Intercept ($b_0$):

  $$ b_0 = \bar{Y} - b_1\bar{X} $$

Where:

• $X_i, Y_i$: Data points,
• $\bar{X}, \bar{Y}$: Means of $X$ and $Y$.

Key Concepts

1. Residuals:

   • The residual for a data point is the difference between the observed and predicted values:

     $$ \text{Residual} = Y_i - \hat{Y}_i $$

   • A good regression line minimizes the sum of squared residuals.

2. Minimizing Error:

   • The least squares method ensures that the total error, measured as:

     $$ \text{Sum of Squared Errors (SSE)} = \sum{(y_i - \hat{y}_i)^2}  $$

     is as small as possible.

3. Assumptions:

   • The relationship between $X$ and $Y$ is linear.
   • Errors (residuals) are normally distributed with mean 0 and constant variance.