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

\(\beta_0\) = theoretical y‑intercept
\(\beta_1\) = theoretical slope
\(\varepsilon\) = error term (also called the disturbance or noise). It captures everything the model cannot explain — measurement error, randomness, natural variability, or omitted variables from the regression.

\(\mathbb{E}[Y \mid X = x] = \beta_0 + \beta_1 x\)

\(\mathbb{E}[Y \mid X = x]\) = the expected value of \(Y\) when the predictor \(X\) takes a specific value \(x\).

\(\varepsilon \sim i.i.d. N(0, \sigma^2)\)

\(i.i.d.\) = The errors are independent (one error does not influence another), and identically distributed (all errors come from the same distribution).

\(N(0, \sigma^2)\) = The error follows a normal distribution with mean \(0\) and variance \(\sigma^2\). Also known as the Constant Variance Assumption.

model <- lm(medv ~ rm, data = Boston)
summary(model)

## 
## Call:
## lm(formula = medv ~ rm, data = Boston)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -23.346  -2.547   0.090   2.986  39.433 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -34.671      2.650  -13.08   <2e-16 ***
## rm             9.102      0.419   21.72   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.616 on 504 degrees of freedom
## Multiple R-squared:  0.4835, Adjusted R-squared:  0.4825 
## F-statistic: 471.8 on 1 and 504 DF,  p-value: < 2.2e-16

sorted <- Boston[order(Boston$rm), ]
predicted_values <- predict(model, newdata = sorted)