From 5660fd1d265d74a50d53ce4c79fe6186e9115412 Mon Sep 17 00:00:00 2001
From: Raffaele Mancuso <raffaele.mancuso4@unibo.it>
Date: Wed, 21 Aug 2024 23:00:14 +0200
Subject: [PATCH] math proof for question 3

---
 03-linear-regression.Rmd | 12 ++++++++++++
 1 file changed, 12 insertions(+)

diff --git a/03-linear-regression.Rmd b/03-linear-regression.Rmd
index 0874103..2768d14 100644
--- a/03-linear-regression.Rmd
+++ b/03-linear-regression.Rmd
@@ -48,6 +48,18 @@ the $K$ nearest neighbors.
 >         average than high school graduates provided that the GPA is high 
 >         enough.
 
+The model is:
+
+$y = \beta_0 + \beta_1 \text{GPA} + \beta_2 \text{IQ} + \beta_3 \text{Level} + \beta_4 \text{GPA} \text{IQ} + \beta_5 \text{GPA} \text{Level}$
+
+Fixing IQ and GPA, changing Level from 0 to 1 will change the outcome by:
+
+$\Delta y = \beta_3 + \beta_5 \text{GPA}$
+
+$\Delta y > 0 \Rightarrow \beta_3 + \beta_5 \text{GPA} > 0 \Rightarrow \text{GPA} > \dfrac{-\beta3}{\beta_5} = - \dfrac{35}{-10} = 3.5$
+
+From a graphical standpoint:
+
 ```{r, message = FALSE, warning = FALSE}
 library(plotly)
 ```