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One of the most powerful tools in statistics is the linear model. It is based on the idea that you can explain an outcome by looking at some input variables and an error term. It is clear that a tall person is likely to be heavy. That being said, the relationship is not perfect as other unrelated genetic factors also influence weight. In the linear model, the outcome would be a person's weight, the person's height would be the explanatory variable, and the other factors would be the error term. In this example we get an equation like $$\left(weight\right) = \beta_0 + \beta_1\left(height\right) + \epsilon$$. The value of $$\beta_1$$ tells us exactly how strongly a person's height affects his or her weight. This class will focus on linear regression. If we were omniscient beings, it would be possible to look at all possible heights and weights in order to directly calculate the value of $$\beta_1$$. Unfortunately, this is not usually possible and we are forced to build an estimator $$\hat{\beta_1}$$ of $$\beta_1$$ using ordinary least squares (OLS) or other methods. In this class we will prove theorems to justify the use of statistical methods in the real world. One particularly famous result that we will prove is the Gauss-Markov theorem which shows that the $$\hat{\beta}$$ estimators given by OLS are in fact the best linear unbiased estimators under certain conditions.