Regression analysis is a statistical technique used to understand the relationship between a dependent variable and one or more independent variables. It’s widely utilized in various fields, including economics, finance, social sciences, and engineering. At its core, regression analysis aims to model the relationship between variables and make predictions based on observed data. In this article, we’ll delve into the mathematical underpinnings of regression analysis, exploring its concepts with a detailed example.
Mathematical Foundation
The fundamental equation for a simple linear regression model can be expressed as:
Where:
- y is the dependent variable,
- x is the independent variable,
- β0 is the intercept (the value of y when x=0),
- β1 is the slope (the change in y for a unit change in x),
- ϵ represents the error term, which captures the discrepancy between the observed and predicted values.
The goal of regression analysis is to estimate the coefficients β0 and β1 that minimize the sum of squared differences between the observed and predicted values.
Example Scenario
Consider a scenario where we want to analyze the relationship between hours of study (x) and exam scores (y). We collect data from a sample of students and obtain the following dataset:
Calculation Steps
Step 1: Calculate the means of x and y:
xˉ= (2+3+4+5+6) / 5 = 4
yˉ= (60+70+75+80+85) / 5 = 74
Step 2: Calculate the slope β1:
Substituting the values:
β1 = (2−4)*(60−74)+(3−4)*(70−74)+(4−4)*(75−74)+(5−4)*(80−74)+(6−4)*(85−74) / ((2−4)**2+(3−4)**2+(4−4)**2+(5−4)**2+(6−4)**2)
β1 = (−28−4+1+30+22 ) / (4+1+0+1+4) = 2.1
Step 3: Calculate the intercept β0:
β0=74−(2.1×4)
β0=74−8.4 = 65.6
Step 4: Construct the regression equation:
y = 65.6 + 2.1 x
Interpretation
For every additional hour of study, the expected increase in exam score is 2.1 points. The intercept of 65.6 suggests that a student who studies for zero hours is expected to score 65.6 points.
Conclusion
Regression analysis provides a powerful framework for understanding the relationship between variables and making predictions based on observed data. By applying mathematical principles, we can derive meaningful insights and make informed decisions in various fields of study and practice.
