Can someone explain to me how regression is used to make predicitions when applying statistics to real-world decisions?

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It is not so much that regression is used to make predictions. You use regression to find the "best-fit" model -- the model that fits the data best. Regression assures that the model created is closest to the data.

Once you have decided on a type of model (often linear) you use regression to find the best model. Then you can use the model for predictions -- either interpolation (e.g. given population at 2000 and 2010 predict the population at 2005) or extrapolation (given a population model for 1960-2010, predict the population in 2030).

Regression analysis can shed light on how the variables in your model interact. That is, you can use regression analysis to study how changing one variable affects the other variables.

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