One may be faced with the problem of making a definite decision with respect to an uncertain hypothesis which is known only through its observable consequences. A statistical hypothesis test, or more briefly, hypothesis test, is an algorithm to state the alternative (for or against the hypothesis) which minimizes certain risks. Jump to: navigation, search A hypothesis (assumption in ancient Greek) is a proposed explanation for a phenomenon. ...

This article describes the commonly used frequentist treatment of hypothesis testing. From the Bayesian point of view, it is appropriate to treat hypothesis testing as a special case of normative decision theory (specifically a model selection problem) and it is possible to accumulate evidence in favor of (or against) a hypothesis using concepts such as likelihood ratios known as Bayes factors. Statistical regularity has motivated the development of the relative frequency concept of probability. ... Jump to: navigation, search Bayesianism is the philosophical tenet that the mathematical theory of probability applies to the degree of plausibility of a statement. ... Decision theory is an interdisciplinary area of study, related to and of interest to practitioners in mathematics, statistics, economics, philosophy, management and psychology. ... Evidence can mean: Any objectively demonstrable circumstance which tends to indicate or disprove a proposition. ... A likelihood-ratio test is a statistical test relying on a test statistic computed by taking the ratio of the maximum value of the likelihood function under the constraint of the null hypothesis to the maximum with that constraint relaxed. ... In statistics, the use of Bayes factors is a Bayesian alternative to classical hypothesis testing. ...

There are several preparations we make before we observe the data.

1. The hypothesis must be stated in mathematical/statistical terms that make it possible to calculate the probability of possible samples assuming the hypothesis is correct. For example: The mean response to treatment being tested is equal to the mean response to the placebo in the control group. Both responses have the normal distribution with this unknown mean and the same known standard deviation ... (value).
2. A test statistic must be chosen that will summarize the information in the sample that is relevant to the hypothesis. Such a statistic is known as a sufficient statistic. A sufficient statistic for a parameter of a distribution exists if and only if the distribution forms an exponential family. In the example given above, it might be the numerical difference between the two sample means, m₁ − m₂.
3. The distribution of the test statistic is used to calculate the probability sets of possible values (usually an interval or union of intervals). In this example, the difference between sample means would have a normal distribution with a standard deviation equal to the common standard deviation times the factor where n₁ and n₂ are the sample sizes.
4. Among all the sets of possible values, we must choose one that we think represents the most extreme evidence against the hypothesis. That is called the critical region of the test statistic. The probability of the test statistic falling in the critical region when the hypothesis is correct is called the alpha value (or size) of the test.

After the data is available, the test statistic is calculated and we determine whether it is inside the critical region. The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ... In probability and statistics, the standard deviation is the most commonly used measure of statistical dispersion. ... A statistic (singular) is the result of applying a statistical algorithm to a set of data. ... In statistics, one often considers a family of probability distributions for a random variable X (and X is often a vector whose components are scalar-valued random variables, frequently independent) parameterized by a scalar- or vector-valued parameter, which let us call θ. ... In probability and statistics, the exponential family is an important class of probability distributions. ...

If the test statistic is inside the critical region, then our conclusion is one of the following:

1. The hypothesis is incorrect, therefore reject the null hypothesis. (Therefore the critical region is sometimes called the rejection region, while its complement is the acceptance region.)
2. An event of probability less than or equal to alpha has occurred.

The researcher has to choose between these logical alternatives. In the example we would say: the observed response to treatment is statistically significant. In statistics, a result is significant if it is unlikely to have occurred by chance, given that a presumed null hypothesis is true. ...

If the test statistic is outside the critical region, the only conclusion is that

• There is not enough evidence to reject the hypothesis.

This is not the same as evidence in favor of the hypothesis. That we cannot obtain using these arguments, since lack of evidence against a hypothesis is not evidence for it. On this basis, statistical research progresses by eliminating error, not by finding the truth.

See also

• Falsifiability
• Statistical theory
• Applied statistics
• Null hypothesis
• Checking if a coin is fair
• Behrens-Fisher problem

Falsifiability is an important concept in the philosophy of science that amounts to the apparently paradoxical idea that a proposition or theory cannot be scientific if it does not admit the possibility of it being false. ... The theory of statistics includes a number of topics: Statistical models of the sources of data and typical problem formulation: Sampling from a finite population Measuring observational error and refining procedures Studying statistical relations Planning statistical research to measure and control observational error: Design of experiments to determine treatment effects... Applied statistics is the use of statistics and statistical theory in real-life situations. ... In statistics, a null hypotheses is a hypothesis that is presumed true until statistical evidence in the form of a hypothesis test indicates otherwise. ... Sometimes when choosing a coin (particularly for a coin flip), it may be desirable to determine if the coin is fair – that is, if the probability of obtaining a given side (commonly heads or tails) in the toss is 50%. // Posterior probability density function One way of verifying this is... Jump to: navigation, search In statistics, the Behrens-Fisher problem is the problem of interval estimation and hypothesis testing concerning the difference between the means of two normally distributed populations when the variances of the two populations are not assumed to be equal, based on two independent samples. ...

External links

• Hypothesis testing in six sigma
• Bayesian critique of classical hypothesis testing