The standard error of the estimate indicates the precision with which the regression model can be expected to predict the dependent Y variable. The standard deviation (or standard error) of each individual coefficient provides a similar measure of precision for the relation between the dependent Y variable and a given X variable. When the standard deviation of a given estimated coefficient is small, a strong relation is suggested between X and Y. When the standard deviation of a coefficient estimate is relatively large, the underlying relation between X and Y is typically weak.
A number of interesting statistical tests can be conducted based on the size of a given estimated coefficient and its standard deviation. These tests are based on alternate versions of the previously described t statistic. Generally speaking, a t test is performed to test whether the estimated coefficient b is significantly different from some hypothesized value. By far, the most commonly tested hypothesis is that b = 0. This stems from the fact that if X and Y are indeed unrelated, then the b slope coefficient for a given X variable will equal zero. If the b = 0 hypothesis can be rejected, then it is possible to infer that b ^ 0 and that a relation between Y and a given X variable does in fact exist. The t statistic with n - k degrees of freedom used to test the b = 0 hypothesis is given by the expression
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