Population regression model with normally distributed errors \[ y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \qquad \varepsilon_i \sim N(0, \sigma^2), \qquad i = 1 \dots n \]

Mean (population) regression line \[ E[y \mid x] = \beta_0 + \beta_1 x \]

Sum of squared residuals, minimized by least squares \[ S(\beta_0, \beta_1) = \sum_{i=1}^{n} \left( y_i - \beta_0 - \beta_1 x_i \right)^2 \]

Least squares estimates: slope and intercept (respectively) \[ \hat\beta_1 = \frac{\sum_{i=1}^{n} (x_i - \bar x)(y_i - \bar y)}{\sum_{i=1}^{n} (x_i - \bar x)^2} = \frac{S_{xy}}{S_{xx}}, \qquad \hat\beta_0 = \bar y - \hat\beta_1 \bar x \]

Fitted values and residuals \[ \hat y_i = \hat\beta_0 + \hat\beta_1 x_i, \qquad e_i = y_i - \hat y_i \]

fit <- lm(mass ~height, data = sw)
fit_no_jabba <- lm(mass ~ height, data = graph2)
coef(fit)

## (Intercept)      height 
## -11.4868157   0.6240033

coef(fit_no_jabba)

## (Intercept)      height 
## -31.2504692   0.6127301

graph2 <- filter(sw, mass < 1000)
ggplot(graph2, aes(x=height, y=mass)) + 
geom_point(color="darkblue", size = 3) +
geom_smooth(method = "lm", se = TRUE, color = "gold")+
labs(x="Height (cm)", y="Mass (kg)")+
theme_minimal(base_size = 14)