A simple linear regression model is a mathematical equation between two linear variables:

• An independent (predictor) variable - x

• A dependent (response) variable - y

Examples

• Relationship between height and weight

• Relationship between salary and expenditure

• Relationship between car payment and car insurance

The simple linear regression model\(^(1)\):

\[ \hat{y} = b_0 + b_1 x \]

• x = independent variable
  

• y = dependent variable

• \(b_0\) = intercept
  

• \(b_1\) = slope

The equation is \[ \hat{y} = b_0 + b_1 x \]

• \(b_0 = \bar{y} - b_1 x\) — intercept
• \(b_1 = r \left( \frac{s_y}{s_x} \right)\) — slope
  
• r = correlation coefficient

Alternate computational equation \[ b_1 = \frac{ \sum xy - \frac{(\sum x)(\sum y)}{n} }{ \sum x^2 - \frac{(\sum x)^2}{n} } = \frac{S_{xy}}{S_{xx}} \]

library(ggplot2)

ggplot(faithful, aes(x = eruptions, y = waiting)) +
  geom_point(color = "red") +
  geom_smooth(method = "lm", se = FALSE, color = "cyan") +
  labs(
    title = "faithful: Waiting Time vs. Eruption Duration",
    x = "Eruption Duration (min)",
    y = "Waiting Time (min)"
  )

1. Simple Linear Regression (LibreTexts)