Type I and type II errors
(α) the error of rejecting a "correct" null hypothesis, and
(β) the error of not rejecting a "fal" null hypothesis
In 1930, they elaborated on the two sources of error, remarking that "in testing hypothes two considerations must be kept in view, (1) we must be able to reduce the chance of rejecting a true hypothesis to as low a value as desired; (2) the test must be so devid that it will reject the hypothesis tested when it is likely to be fal"[1]
When an obrver makes a Type I error in evaluating a sample against its parent population, s/he is mistakenly thinking that a statistical difference exists when in truth there is no statistical difference (or, to put another way, the null hypothesis is true but was mistakenly rejected). For example, imagine that
a pregnancy test has produced a "positive" result (indicating that the woman taking the test is pregnant); if the woman is actually not pregnant though, then we say the test produced a "fal positive". A Type II error, or a "fal negative", is the error of failing to reject a null hypothesis when the alternative hypothesis is the true state of nature. For example, a type II error occurs if a pregnancy test reports "negative" when the woman is, in fact, pregnant.
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Statistical error vs. systematic error
Scientists recognize two different sorts of error:[2]
Statistical error: Type I and Type II
Statisticians speak of two significant sorts of statistical error. The context is that there is a "null hypot
hesis" which corresponds to a presumed default "state of nature", e.g., that an individual is free of dia, that an accud is innocent, or that a potential login candidate is not authorized. Corresponding to the null hypothesis is an "alternative
hypothesis" which corresponds to the opposite situation, that is, that the individual has the dia, that the accud is guilty, or that the login candidate is an authorized ur. The goal is to determine accurately if the null hypothesis can be discarded in favor of the alternative. A test of some sort is conducted (a blood test, a legal trial, a login attempt), and data is obtained. The result of the test may be negative (that is, it does not indicate dia, guilt, or authorized identity). On the other hand, it may be positive (that is, it may indicate dia, guilt, or identity). If the result of the test does not correspond with the actual state of nature, then an error has occurred, but if the result of the test corresponds with the actual state of nature, then a correct decision has been made. There are two kinds of error, classified as "Type I error" and "Type II error," depending upon which hypothesis has incorrectly been identified as the true state of nature.
Type I error
Type I error, also known as an "error of the first kind", an αerror, or a "fal positive": the error of reje
cting a null hypothesis when it is actually true. Plainly speaking, it occurs when we are obrving a difference when in truth there is none. Type I error can be viewed as the error of excessive skepticism.
Type II error
Type II error, also known as an "error of the cond kind", a βerror, or a "fal negative": the error of failing to reject a null hypothesis when it is in fact fal. In other words, this is the error of failing to obrve a difference when in truth there is one. Type II error can be viewed as the error of excessive gullibility.
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See Various proposals for further extension, below, for additional terminology.
Understanding Type I and Type II errors
小班下学期周计划Hypothesis testing is the art of testing whether a variation between two sample distributions can be explained by chance or not. In many practical applications Type I errors are more delicate than Type II errors. In the cas, care is usually focud on minimizing the occurrence of this statistical error. Suppo, the probability for a Type I error is 1% or 5%, then there is a 1% or 5% chance that the obs
erved variation is not true. This is called the level of significance. While 1% or 5% might be an acceptable level of significance for one application, a different application can require a very different level. For example, the standard goal of six sigma is to achieve exactness by 4.5 standard deviations above or below the mean. That is, for a normally distributed process only 3.4 parts per million are allowed to be deficient. The probability of Type I error is generally denoted with the Greek letter alpha.
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In more common parlance, a Type I error can usually be interpreted as a fal alarm, insufficient specificity or perhaps an encounter with fool's gold. A Type II error could be similarly interpreted as an oversight, a lap in attention or inadequate nsitivity.
Etymology
In 1928, Jerzy Neyman (1894-1981) and Egon Pearson (1895-1980), both eminent statisticians, discusd the problems associated with "deciding whether or not a particular sample may be judged as likely to have been randomly drawn from a certain population" (1928/1967, p.1): and, as Florence Nightingale David remarked, "it is necessary to remember the adjective ‘random’ [in the term ‘random sample’] should apply to the method of drawing the sample and not to the sample itlf" (1949, p.28).
They identified "two sources of error", namely:
(a) the error of rejecting a hypothesis that should have been accepted, and
(b) the error of accepting a hypothesis that should have been rejected (1928/1967, p.31). In 1930, they elaborated on the two sources of error, remarking that:
…in testing hypothes two considerations must be kept in view, (1) we must be able to reduce the chance of rejecting a true hypothesis to as low a value as desired; (2) the test must be so devid that it will reject the hypothesis tested when it is likely to be fal (1930/1967, p.100).
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In 1933, they obrved that the "problems are rarely prented in such a form that we can discriminate with certainty between the true and fal hypothesis" (p.187). They also noted that, in deciding whether to accept or reject a particular hypothesis amongst a "t of alternative hypothes" (p.201), it was easy to make an error:
…[and] the errors will be of two kinds:
(I) we reject , the hypothesis to be tested] when it is true,
(II) we accept H0when some alternative hypothesis H i is true. (1933/1967, p.187)
In all of the papers co-written by Neyman and Pearson the expression H0always signifies "the hypothesis to be tested" (e, for example, 1933/1967, p.186).
In the same paper[4] they call the two sources of error, errors of type I and errors of type II respectively.[5]
Statistical treatment
Definitions
Type I and type II errors
Over time, the notion of the two sources of error has been universally accepted. They are now routinely known as type I errors and type II errors. For obvious reasons, they are very often referred to as fal positives and fal negatives respectively. The terms are now commonly applied in much wider and far more general n than Neyman and Pearson's original specific usage, as follows:
海水为什么是咸的∙Type I errors (the "fal positive"): the error of rejecting the null hypothesis given that it is actually true; e.g., A court finding a person guilty of a crime that they did not actually commit.
∙Type II errors(the "fal negative"): the error of failing to reject the null hypothesis given that the alternative hypothesis is actually true; e.g., A court finding a person not guilty of a crime that they did actually commit.
The examples illustrate the ambiguity, which is one of the dangers of this wider u: They assume the speaker is testing for guilt; they could also be ud in rever, as testing for innocence; or two tests could be involved, one for guilt, the other for innocence. (This ambiguity is one reason for the Scottish legal system's third possible verdict: not proven.)
The following tables illustrate the conditions.
元仁宗Example, using infectious dia test results:
Example, testing for guilty/not-guilty:
Example, testing for innocent/not innocent – n is reverd from previous example:
Note that, when referring to test results, the terms true and fal are ud in two different ways: the state of the actual condition (true=prent versus fal=abnt); and the accuracy or inaccuracy of the test result (true positive, fal positive, true negative, fal negative). This is confusing to some readers. To clarify the examples above, we have ud prent/abnt rather than true/fal to refer to the actual condition being tested.
Fal positive rate
The fal positive rate is the proportion of negative instances that were erroneously reported as being positive.
