• Simple linear regression is a statistical method used to model the relationship between a dependent variable and an independent variable.

• The equation of a simple linear regression model is given by:

  \[y = \beta_0 + \beta_1 x + \epsilon\]

• Where:

  • \(y\) is the dependent variable.
  • \(x\) is the independent variable.
  • \(\beta_0\) is the y-intercept.
  • \(\beta_1\) is the slope of the line.
  • \(\epsilon\) is the error term.

• Using the mtcars dataset, which includes information about different car models.
• We will model the relationship between Miles Per Gallon (mpg) (dependent variable) and Horsepower (hp) (independent variable).

• The mean of a dataset is defined as:

  \[\mu = \frac{1}{N} \sum_{i=1}^{N} x_i\]

• The variance is defined as:

  \[\sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2\]

Where: - \(N\) is the number of observations. - \(x_i\) represents each individual observation in the dataset.

• The coefficient of determination (R²) measures how well the independent variable explains the variability of the dependent variable:

  \[R^2 = 1 - \frac{SS_{res}}{SS_{tot}}\]

Where: - \(SS_{res}\) is the sum of squares of residuals. - \(SS_{tot}\) is the total sum of squares.

```r # Create scatter plot with regression line library(ggplot2)

ggplot(mtcars, aes(x=hp, y=mpg)) + geom_point() + geom_smooth(method=“lm”, se=FALSE, color=“blue”) + labs(title=“Regression Line for MPG vs Horsepower”, x=“Horsepower (hp)”, y=“Miles Per Gallon (mpg)”) + theme_minimal()