Linear regression is a simple way to see how one variable changes with another. We basically draw a straight line that best fits the data points.

The model looks like this:

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

All this means is that y changes based on x plus some error.

The slope tells us how much y changes when x changes by 1 unit.

\[ \hat{\beta}_1 = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2} \]

Here is the basic code I used to fit the model:

library(ggplot2)   
library(plotly)    

data(mtcars)         # loads the car dataset
model <- lm(mpg ~ wt, data = mtcars)   # fits the regression model, with miles per gallon(mpg) against the weight
model                 # prints the model results

## 
## Call:
## lm(formula = mpg ~ wt, data = mtcars)
## 
## Coefficients:
## (Intercept)           wt  
##      37.285       -5.344

ggplot(mtcars, aes(wt, mpg)) +
  geom_point() +
  labs(x = "Car Weight", y = "Miles Per Gallon")

ggplot(mtcars, aes(wt, mpg)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE) +
  labs(x = "Car Weight",y = "Miles Per Gallon")

plot_ly(x = mtcars$wt, y = mtcars$mpg,
        type = "scatter", mode = "markers") %>%
  layout(xaxis = list(title = "Car Weight"),
         yaxis = list(title = "Miles Per Gallon"))

From this example, we can see that heavier cars tend to have lower mpg. Linear regression helps us understand simple relationships between variables.