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Regression - Algebra vs Calculus and Probability


 Regression with ALGEBRA ONLY
Regression with CALCULUS and PROBABILITY 
 y = mx + b    <or>    Yi = b0 + b1*Xi 


 Yi = b0 + b1*Xi + εi


The εi term is the difference, and is where probability and calculus are needed.

 Regression with ALGEBRA ONLY  Regression with CALCULUS and PROBABILITY  
Model summary:

Only includes a Least Squares fit
  • y = 28.139 + 4.2167x

Might also sometimes include R-squared value.
Model summary:

Can use Least Squares, Maximum Likelihood, other


Model assumptions regarding the εi term (model residuals).
  • iid N(0,σ)

 Regression with ALGEBRA ONLY  Regression with CALCULUS and PROBABILITY  

Does NOT include regression confidence bands.


Includes regression confidence bands, which characterize the sampling uncertainty.


 Visually, it appears that the slope is greater than zero.


 We cannot conclude that the slope is not zero.