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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 

Deterministic

 Yi = b0 + b1*Xi + εi

Stochastic

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

Includes:


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.

 Conclusions

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

 Conclusions

 We cannot conclude that the slope is not zero.

 

 





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