What are inferential statistics?

Answer 1

Statistical inference is about testing hypotheses. In order to test a hypothesis, you make a prediction about the observations, contrasting the prediction with what might happen if the hypothesis were not true. The prediction is tested against the observations by calculating a test statistic or inferential statistic. This is a value which is based purely on the observations. If the test statistic is too far from the predicted value then the hypothesis should be rejected in favour of the alternative hypothesis.

What constitutes "too far" depends on the presumed distribution of the variable being tested, as well as the degree of certainty required from the test - the power of the test. The latter is a balance between probability of rejecting the hypothesis when it is true and that of not rejecting it when it is false. These outcomes may be weighted according to the risk or costs that a false decision carries.