We will be looking at the relationship between commute times and distance traveled. For this presentation, we have 500 points of data for each of the following cities: Boston, Houston, Minneapolis, and Washington.

\[Time_i = \beta_0 + \beta_1 \cdot Distance_i + \varepsilon_i\]

• \(\beta_0\): intercept (baseline commute time)
• \(\beta_1\): average change in time per unit distance
• \(\varepsilon_i\): random error

\[ H_0: \beta_1 = 0 \]

\[ H_a: \beta_1 \neq 0 \]

• Tests whether distance is linearly related to commute time

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

g <- lm(Time ~ Distance, data = MetroCommutes)

ggplot(data.frame(
  fitted = fitted(g),
  residuals = resid(g)
), aes(x = fitted, y = residuals)) +
  geom_point(alpha = 0.4) +
  geom_hline(yintercept = 0, linetype = "dashed") +
  labs(
    title = "Residuals vs Fitted Values",
    x = "Fitted Commute Time",
    y = "Residuals"
  )

Based on the available data, we can conclude that commute time generally increases as commute distance increases. We used a linear model to display this relationship. Residual analysis demonstrates that the linear model we used was appropriate for this data set.