Simple linear regression models the relationship between dependent and independent variables by using a line of best fit.

To make sure the model is reliable, there are four assumptions that need to be met:

1. There is a linear relationship between the dependent and independent variable.
2. The residuals are independent.
3. The residuals are normally distributed.
4. The residuals of the independent variables have constant variance.

The equation for the line of best fit is:

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

where

• \(\widehat{y}\) is the estimated dependent variable
• \(\beta_0\) is the \(y\)-intercept
• \(\beta_1\) is the slope
• \(x\) is the independent variable

The equations for the \(y\)-intercept and the slope are:

\(\beta_0 = \bar{y} - \beta_1 \bar{x}\)

\(\beta_1 = \frac{\sum (x_i - \bar{x}) (y_i - \bar{y})}{\sum (x_i - \bar{x})^2}\)

where

• \(\bar{y}\) is the mean of the dependent variables
• \(\bar{x}\) is the mean of the independent variables
• \(x_i\) is the values of the independent variable
• \(y_i\) is the values of the dependent variable

Using r dataset trees, a Q-Q Plot can be used to check if the residuals of variable Height are normally distributed.

A simple linear regression plot using dataset trees.

best_fit = lm(Volume ~ Height, data = trees)

plot_ly(data = trees, x = ~Height, y = ~Volume, type = "scatter", 
        mode = "markers", marker = list(color = "black"))%>%
  add_lines(x = ~Height, y = ~fitted(best_fit), 
            line = list(color = "blue"), inherit = FALSE)%>%
  layout(title = "Volume by Height",
         xaxis = list(title = "Height in ft"),
         yaxis = list(title = "Volume in cubic ft"),
         showlegend = FALSE)

A simple linear regression plot with 95% confidence interval bands using dataset trees.

ggplot(data = trees, aes(x = Height, y = Volume))+
      geom_point()+
      geom_smooth(formula = y ~ x ,method = "lm", se = TRUE)+
      labs(title = "Volume by Height With 95% Bands", 
           x = "Height in ft", y = "Volume in cubic ft")

• [Equations on Slide 2 and 3] Libretexts.“5.4.1: Model and Equation for Simple Linear Regression Analysis.” Business LibreTexts, August 25, 2025.