Simple linear regression helps us understand the relationship between two variables.

• One variable predicts another
• Shows if there is a pattern between them
• Draws a straight line through data points

Example: Does speed affect required distance to stop?

The equation for a straight line is:

\[Y = mX + b\]

Where:

• \(Y\) = what we want to predict
• \(X\) = what we use to predict
  
• \(b\) = the y-axis intercept of the line
• \(m\) = the slope of the line

##   speed dist
## 1     4    2
## 2     4   10
## 3     7    4
## 4     7   22
## 5     8   16
## 6     9   10

• speed: how fast the car was going (mph)
• dist: how far it took to stop (feet)

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

ggplot(cars, aes(x = speed, y = dist)) +
  geom_point() +
  geom_smooth(method = "lm") +
  labs(title = "Speed vs Distance with Best Fit Line",
       x = "Speed (mph)", 
       y = "Distance (feet)")

The general form of our fitted model is: \[\hat{Y} = \hat{\beta_0} + \hat{\beta_1}X\]

## (Intercept)       speed 
##  -17.579095    3.932409

Our Specific Equation and its Meaning: Distance = -17.6 + 3.9 × Speed

• For every 1 mph faster, cars need approximately 4 more feet to stop
• The line starts at -17.6 feet, which doesn’t make real sense, but that is the math!

This displays if our line is or is not a good fit for the data.

What we Learned:

• Faster cars do take longer to stop
• Stopping distance can be predicted from speed
• Linear regression helps us find patterns within data

A Real World Use-case: Simple Linear Regression data on speed verses stopping distance could help set speed limits or impact brake system designs.