General Equation of Multiple Linear Regressor
$$ E[Y|X] = f(X)= \beta_0 +\beta_1X_1+...+\beta_pX_p+\epsilon $$
Error in Multiple Linear Regressor
$$ e_i = y_i-\^{y}_i $$
$$ RSS = e^2_1+e^2_2+....+e^2_n=E[(Y-\^{Y})^2 |X] $$
Confidence Intervals:
More Generalized Confidence Intervals
$n$ represents the degree of freedom
$\alpha$ represents the p value.
$p$ represents number of features.
$$ [\;\hat\beta_i\pm t_{n-p-1,\alpha/2}*SE(\hat\beta_i)\;] $$
Hypothesis Testing:
Assume
$$ H_0: \beta_i = 0 \newline H_A:\beta_i\ne0 $$
$$ H_0: \beta_i = \beta \newline H_A:\beta_i\ne \beta $$
We get that
$$ t_{obs} = \frac{\hat\beta_i - \beta}{SE)\hat\beta_i} ;\; given \;\beta_i=\beta\; ;usually \;\beta=0 $$
Reject if:
$$ |t_{obs}| > t_{n-p-1,\alpha/2} $$