A statistical model is a formalization of relationships between variables in the form of mathematical equations. A statistical model describes how one or more random variables are related to one or more other variables. The model is statistical as the variables are not deterministically but stochastically related. In mathematical terms, a statistical model is frequently thought of as a pair where is the set of possible observations and the set of possible probability distributions on . It is assumed that there is a distinct element of which generates the observed data. Statistical inference enables us to make statements about which element(s) of this set are likely to be the true one.
Most statistical tests can be described in the form of a statistical model. For example, the Student's t-test for comparing the means of two groups can be formulated as seeing if an estimated parameter in the model is different from 0. Another similarity between tests and models is that there are assumptions involved. Error is assumed to be normally distributed in most models.
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