Explanation

In simple linear regression, the normality assumption specifically applies to the regression residuals (also called error terms), not to the dependent or independent variables themselves.

Key Points:

1. Normality of Residuals: The classical linear regression model assumes that the error terms (ε) are normally distributed with mean zero and constant variance σ².
2. Why Residuals, Not Variables:
   • The dependent variable (Y) can have any distribution; it's the conditional distribution of Y given X that matters
   • The independent variable (X) can be fixed or random; no normality assumption is required for X
   • The residuals represent the unexplained variation after accounting for the linear relationship
3. Importance: This assumption is crucial for:
   • Valid hypothesis testing (t-tests, F-tests)
   • Constructing confidence intervals
   • Making predictions with proper uncertainty quantification
4. Central Limit Theorem: Even if residuals aren't perfectly normal, with large sample sizes, inference is often robust due to the Central Limit Theorem.

Mathematical Representation:

The regression model: Y = β₀ + β₁X + ε Where ε ~ N(0, σ²)

This means the residuals should be approximately normally distributed around zero.