Simple linear regression is a statistical method that analyzes the relationship between two variables, the independent (predictor), and the dependent variable (response variable). The method models both variables on a straight line that serves as the ideal linear relationship between them.
2026-02-08
What is Simple Linear Regression
Simple Linear Regression Model
The relationship between the independent variable and dependent variable:
\[ Y_i = \beta_0 +\beta_1 X_i + \epsilon \] Variables:
- \(Y_i\) = dependent variable
- \(X_i\) = independent variable
- \(\beta_0\) = intercept
- \(\beta_1\) = slope
- \(\epsilon\) = error
Linear Regression Formulas
Regression line estimate:
\[ \hat{Y} = b_0 + b_1 X \] Slope estimate: \[ b_1 = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \] Intercept estimate: \[ b_0 = \bar{y} - b_1 \bar{x} \]
Dataset Information
A data set will be used to demonstrate linear regression through plots and graphs.This data set named Auto MPG is taken from https://archive.ics.uci.edu/dataset/9/auto+mpg. The data set will contain the following automobile information:
- V1 = miles per gallon
- V2 = number of cylinders
- V3 = displacement
- V4 = horsepower
- V5 = weight
- V6 = acceleration
- V7 = model year
- V8 = origin
- V9 = car name
MPG vs Weight ggplot
MPG vs Weight code
ggplot(df, aes(V5, V1)) + geom_point()+
labs(title = "MPG vs Vehicle Weight",
x = "Weight", y = "MPG")
Regression Line ggplot
`geom_smooth()` using formula = 'y ~ x'
