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In statistical hypothesis testing, a type I error, or a false positive, is the rejection of the null hypothesis when it is actually true. For example, an innocent person may be convicted. A type II error, or a false negative, is the failure to reject a null hypothesis that is actually false. For example: a guilty person may be not convicted.
The exception proves the rule is a phrase that arises from ignorance, though common to good writers. The original word was preuves, which did not mean proves but tests. [ 4] In this sense, the phrase does not mean that an exception demonstrates a rule to be true or to exist, but that it tests the rule, thereby proving its value.
The rule is often called Chebyshev's theorem, about the range of standard deviations around the mean, in statistics. The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined. For example, it can be used to prove the weak law of large numbers.
Law of averages. The law of averages is the commonly held belief that a particular outcome or event will, over certain periods of time, occur at a frequency that is similar to its probability. [1] [2] Depending on context or application it can be considered a valid common-sense observation or a misunderstanding of probability.
Mathematical fallacy. In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known ...
The law of large numbers provides an expectation of an unknown distribution from a realization of the sequence, but also any feature of the probability distribution.[1] By applying Borel's law of large numbers, one could easily obtain the probability mass function.
Statistical proof is the rational demonstration of degree of certainty for a proposition, hypothesis or theory that is used to convince others subsequent to a statistical test of the supporting evidence and the types of inferences that can be drawn from the test scores. Statistical methods are used to increase the understanding of the facts and ...
The false positive rate depends on the significance level. The specificity of the test is equal to 1 minus the false positive rate. [7] In statistical hypothesis testing, this fraction is given the Greek letter α, and 1 − α is defined as the specificity of the test. Increasing the specificity of the test lowers the probability of type I ...