Example: Predicting Salary based on Experience

Problem Statement

A company wants to predict the salary of a new employee based on their years of experience. We have data from 5 current employees. We will use Simple Linear Regression to find the best-fitting line: $y = mx + b$.

1. The Data

Employee	Years of Experience ($x$)	Salary in $1000s ($y$)
A	1	30
B	2	35
C	3	50
D	4	60
E	5	75

2. Calculating the Model

We need to find the slope ($m$) and y-intercept ($b$) that minimizes the error.

The formula for slope $m$ is: $$ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} $$

Step-by-Step Calculation:

$n = 5$ (number of data points)
$\\sum x = 1 + 2 + 3 + 4 + 5 = 15$
$\\sum y = 30 + 35 + 50 + 60 + 75 = 250$
$\\sum xy = (1\cdot30) + (2\cdot35) + (3\cdot50) + (4\cdot60) + (5\cdot75) = 30 + 70 + 150 + 240 + 375 = 865$
$\\sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 4 + 9 + 16 + 25 = 55$

Plugging these into the formula for $m$: $$ m = \frac{5(865) - (15)(250)}{5(55) - (15)^2} $$ $$ m = \frac{4325 - 3750}{275 - 225} = \frac{575}{50} = 11.5 $$

Now finding the intercept $b$ using $b = \bar{y} - m\bar{x}$: $$ \bar{y} = 250 / 5 = 50 $$ $$ \bar{x} = 15 / 5 = 3 $$ $$ b = 50 - 11.5(3) = 50 - 34.5 = 15.5 $$

Final Equation

Salary = 11.5(Experience) + 15.5

This means for every extra year of experience, the salary increases by $11,500, starting from a base of $15,500.

3. Making a Prediction

If we hire someone with 6 years of experience: $$ y = 11.5(6) + 15.5 $$ $$ y = 69 + 15.5 = 84.5 $$

We would predict a salary of $84,500.