• Linear regression is a statistical method that models the relationship between two variables by fitting a line to the data.
• The line of best fit , given by \[\hat{y} = \beta_0 + \beta_1x + \varepsilon\] is found using the least squares method, which minimizes the squared differences between the observed values (\(y_i\)) and the predicted values (\(\hat{y}_i\)). \[\sum(y_i - \hat{y}_i)^2\]

\[\hat{y} = \beta_0 + \beta_1x + \varepsilon\] Where

• \(\beta_0\) is the y-Intercept
• \(\beta_1x\) is the slope
• \(\varepsilon\) is the random error term

The following code uses ’ geom_smooth(method ‘lm’)’ to automatically fit and draw the least squares regression line.

plot = ggplot(trees, aes(x = Girth, y = Height))+geom_point()+
  geom_smooth(method = 'lm', se = T)

-Below we have the coefficient of determination, \(R^2\), which is low at 0.2697.

-The \(R^2\) value of 0.2697 indicates a weak relationship between Height and Girth, meaning Height alone does not explain much of the variation in Girth.

model <- lm(Girth~Height, data = trees)
summary(model)$r.squared

## [1] 0.2696518