【Discussion on Normality Verification】: Verification Methods for Normality Assumption in Linear Regression

# 1. Introduction to the Normality Assumption in Linear Regression

In conducting linear regression analysis, the normality assumption is one of the crucial prerequisites. In simple terms, the normality assumption posits that the dependent variable follows a normal distribution at each value of the independent variables. The validity of this assumption is vital for the parameter estimation and significance testing of the linear regression model. If the normality assumption does not hold, it may lead to inaccuracies in the regression analysis results, impacting the reliability and effectiveness of the model. Therefore, it is essential to verify the normality assumption in practice by checking if the residuals conform to a normal distribution.

# 2.1 Analysis of the Normal Distribution Concept
The normal distribution, also known as the Gaussian distribution, is one of the most common continuous probability distributions in statistics. Data from the natural world and various fields often exhibit a normal distribution pattern. Understanding the concept of the normal distribution is crucial for grasping subsequent statistical knowledge and the normality assumption in linear regression.

### 2.1.1 Definition of the Normal Distribution
The normal distribution is named after the mathematician Gauss and is described by the following probability density function:

$$ f(x | \mu, \sigma) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$

Here, $\mu$ represents the mean, and $\sigma$ is the standard deviation. The shape of the normal distribution is determined by these two parameters, with the mean dictating the position of the distribution, and the standard deviation determining its spread.

### 2.1.2 Characteristics of the Normal Distribution
Characteristics of the normal distribution include:
- Bell-shaped curve, symmetric about the center;
- Mean, median, and mode are equal;
- 68% of the data falls within one standard deviation of the mean, and 95% within two standard deviations;
- The intervals divided by three σ are known as the rule of thumb golden triangle.

### 2.1.3 Applications of the Normal Distribution
The normal distribution is widely applied in statistical analysis, hypothesis testing, quality control, and other fields. Its significance lies in the fact that many natural phenomena, social phenomena, as well as some physical and mathematical models exhibit the properties of a normal distribution.

In the next section, we will continue to explore the relationship between the normal distribution and hypothesis testing.

# 3. Linear Regression Model

### 3.1 Basic Concepts of Linear Regression

Linear regression is a statistical model used to study the relationship between independent variables (or explanatory variables) and a dependent variable. In linear regression, it is assumed that the relationship between the independent variables and the dependent variable can be described by a linear equation, which can be used to predict the values of the dependent variable. In practical applications, linear regression is typically divided into simple linear regression and multiple linear regression.

#### 3.1.1 Simple Linear Regression and Multiple Linear Regression

- **Simple Linear Regression**: When only one independent variable and one dependent variable are involved, simple linear regression is used. The equation of a simple linear regression model is expressed as: $Y = β0 + β1*X + ε$, where $Y$ is the dependent variable, $X$ is the independent variable, $β0$ and $β1$ are regression coefficients, and $ε$ represents error.
- **Multiple Linear Regression**: When multiple independent variables influence the dependent variable, multiple linear regression is used. The equation of a multiple linear regression model can be expressed as: $Y = β0 + β1*X1 + β2*X2 + ... + βn*Xn + ε$, where $n$ is the number of independent variables.

#### 3.1.2 Assumptions of the Linear Regression Model

In linear regression models, it is usually assumed that the data satisfies several assumptions:
1. **Linear Relationship**: There is a linear relationship between the independent variables and the dependent variable;
2. **Independence and Identical Distribution of Random Error Terms**: The error terms meet the assumption of being independently and identically distributed;
3. **Homoscedasticity (Constant Variance)**: The error terms have a constant variance;
4. **Normality of Residuals**: The model residuals follow a normal distribution.

### 3.2 Normality Assumption in Linear Regression

#### 3.2.1 Meaning of the Normality Assumption

In linear regression, the normality assumption requires that the model residuals follow a normal distribution. If the residuals do not conform to a normal distribution, it may lead to bias in parameter estimation, thereby affecting the predictive accuracy of the model.

#### 3.2.2 Impact of the Normality Assumption on Linear Regression

- **Validity of Parameter Estimation**: When the residuals of th