What is Simple Linear Regression?

Simple linear regression models the relationship between:

• Response variable \(Y\)
• Predictor variable \(X\)

It is widely used in engineering, data science, and applied statistics.

The simple linear regression model is:

\[ Y = \beta_0 + \beta_1 X + \varepsilon \]

Where:

• \(\beta_0\) is the intercept
• \(\beta_1\) is the slope
• \(\varepsilon\) is the random error

The estimated regression equation is:

\[ \hat{Y} = \hat{\beta}_0 + \hat{\beta}_1 X \]

The slope estimator is:

\[ \hat{\beta}_1 = \frac{\sum_{i=1}^n (X_i - \bar{X})(Y_i - \bar{Y})} {\sum_{i=1}^n (X_i - \bar{X})^2} \]

We use the built-in dataset mtcars.

• Response variable: mpg (miles per gallon)
• Predictor variable: wt (weight in 1000 lbs)
• Extra variable for 3D plot: hp (horsepower)

library(ggplot2)

ggplot(mtcars, aes(x = wt, y = mpg)) +
  geom_point(size = 3) +
  labs(
    title = "MPG vs Weight",
    x = "Weight (1000 lbs)",
    y = "Miles per Gallon (MPG)"
  ) +
  theme_minimal()

library(ggplot2)
ggplot(mtcars, aes(x = wt, y = mpg)) +
  geom_point(size = 3) +
  geom_smooth(method = "lm", se = TRUE) +
  labs(
    title = "Regression Fit: mpg ~ wt",
    x = "Weight (1000 lbs)",
    y = "Miles per Gallon (MPG)"
  ) +
  theme_minimal()

## `geom_smooth()` using formula = 'y ~ x'

library(plotly)

## 
## Attaching package: 'plotly'

## The following object is masked from 'package:ggplot2':
## 
##     last_plot

## The following object is masked from 'package:stats':
## 
##     filter

## The following object is masked from 'package:graphics':
## 
##     layout

plot_ly(
  data = mtcars,
  x = ~wt,
  y = ~hp,
  z = ~mpg,
  type = "scatter3d",
  mode = "markers",
  marker = list(size = 4))