How Does Linear Regression Work?

In linear regression, our goal is to learn a function $f$ that can predict an outcome $y$ from some input $x$. Let's use a real example:

Imagine we have data collected from different students:

Student	Hours of study	Score
1	2	75%
2	3.5	82%
3	1.5	68%
4	4	90%

In mathematical notation, we write this as:

• Input (study hours): $x$
• Output (test score): $y$
• Our prediction function: $f(x) = y$

Why This Matters

Once we have our best-fit line, we can:

1. Predict scores for new study times.
2. Understand how much extra studying helps, through $\beta_1$.
3. Estimate a "baseline" score through $\beta_0$.

The Linear Model

The "linear" in linear regression means we're looking for a straight line that best fits our data. This line can be written as:

$f(x) = \beta_0 + \beta_1x$

Where:

• $\beta_0$ is the y-intercept (the predicted score for 0 hours of study)
• $\beta_1$ is the slope (how much the score increases for each hour of study)

For example, we might find that: $f(x) = 60 + 7x$

This would mean:

• A student who doesn't study at all, that is, $x=0$, might get a 60%.
• For each hour of studying, we expect the score to increase by 7 points.

The challenge is that real data is messy! Some students might study for 3 hours and get 85%, while others study the same amount and get 78%. Our task is to find the line that best represents the general trend - the "best fit" line through all these points.