Simple linear regression studies the relationship between:

• one predictor variable \(x\)
• one response variable \(y\)

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

• Helps understand the relationship between two variables
• Allows prediction of one variable using another
• Gives a simple mathematical model for data
• Example: predicting miles per gallon using car weight

\[ \hat{y} = b_0 + b_1 x \]

\[ b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]

\[ b_0 = \bar{y} - b_1 \bar{x} \]

• Linear relationship
• Independent observations
• Constant variance of errors
• Errors are approximately normal

We use the built-in mtcars dataset.

• wt = car weight
• mpg = miles per gallon

From the fitted model:

• Intercept \(b_0 \approx 37.29\)
• Slope \(b_1 \approx -5.34\)
• \(R^2 \approx 0.753\)

Weight explains about 75.3% of the variation in MPG, and the relationship is negative.

\[ \widehat{mpg} = 37.29 - 5.34(wt) \]

Interpretation:

• For every 1-unit increase in car weight, predicted MPG decreases by about 5.34 on average

We fit a linear model and visualize the relationship.

model <- lm(mpg ~ wt, data = mtcars)

ggplot(mtcars, aes(x = wt, y = mpg)) +
  geom_point() +
  geom_smooth(method = "lm")

• Simple linear regression models the relationship between two quantitative variables
• In this example, weight and MPG have a negative relationship
• Regression is useful for both prediction and interpretation