• We want to understand how a numeric outcome changes with a predictor.
• Ex. How does fuel efficiency in miles per gallon (mpg) change as a car gets heavier in weight (wt)?

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

\[ \mathbb{E}[\varepsilon \mid X] = 0, \quad \mathrm{Var}(\varepsilon \mid X)=\sigma^2 \]

\[ SSE(\beta_0,\beta_1)=\sum_{i=1}^{n} (y_i-(\beta_0+\beta_1x_i))^2 \]

## 
## Call:
## lm(formula = mpg ~ wt, data = df)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.5432 -2.3647 -0.1252  1.4096  6.8727 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  37.2851     1.8776  19.858  < 2e-16 ***
## wt           -5.3445     0.5591  -9.559 1.29e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.046 on 30 degrees of freedom
## Multiple R-squared:  0.7528, Adjusted R-squared:  0.7446 
## F-statistic: 91.38 on 1 and 30 DF,  p-value: 1.294e-10

fit <- lm(mpg ~ wt, data = df)

ggplot(df, aes(wt, mpg)) +
  geom_point(size = 2) +
  geom_smooth(method = "lm", se = TRUE)