Simple linear regression models the relationship between two variables by fitting a straight line to the observed data.

• One independent variable (x)
  
• One dependent variable (y)
  
• Goal: predict y using x

The equation of the line is:

\(\hat{y} = \beta_0 + \beta_1 x\)

Where:
- \(\beta_0\) is the intercept
- \(\beta_1\) is the slope
- \(\hat{y}\) is the predicted value of y given x

For the linear regression model to be valid, several key assumptions must hold:

• Linearity: The relationship between x and y is linear
  \(y = \beta_0 + \beta_1 x + \varepsilon\)

• Independence: Observations are independent of each other

• Homoscedasticity: The variance of residuals is constant across all levels of x

• Normality of Residuals: The errors \(\varepsilon\) are normally distributed
  \(\varepsilon \sim N(0, \sigma^2)\)

We’ll use the built-in mtcars dataset in R, which contains various car attributes.

For this example, we’ll model:

• mpg (Miles per Gallon) — our dependent variable
• hp (Horsepower) — our independent variable

This will help us explore how a car’s horsepower affects its fuel efficiency.

# Load dataset
data(mtcars)

# Build the linear regression model
model = lm(mpg ~ hp, data = mtcars)

# View model summary
summary(model)

# Add residuals to dataset
mtcars$residuals = residuals(model)

# Create residual plot
ggplot(mtcars, aes(x = hp, y = residuals)) +
  geom_point() +
  geom_hline(yintercept = 0, linetype = "dashed") +
  labs(
    title = "Residuals vs Horsepower",
    x = "Horsepower",
    y = "Residuals"
  )

• Simple linear regression helps us understand and predict relationships between two variables.

• In this case, we explored how horsepower impacts fuel efficiency using the mtcars dataset.

• Regression models are used across many fields, from economics to engineering, for forecasting and decision-making.

• This foundational concept is the building block for more advanced techniques like multiple regression and machine learning.

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