Statistical Hypothesis Testing

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• Statistical hypothesis testing http://en.wikipedia.org/w/index.php?title=Statistical_hypothesis_testing&printable=yes

A statistical hypothesis test is a method of statistical inference using data from

a scientific study. In statistics, a result is called statistically significant if it has been

predicted as unlikely to have occurred bychance alone, according to a pre-determined

threshold probability, the significance level. The phrase "test of significance" was

coined by statistician Ronald Fisher.[1] These tests are used in determining what

outcomes of a study would lead to a rejection of the null hypothesis for a pre-specified

level of significance; this can help to decide whether results contain enough

information to cast doubt on conventional wisdom, given that conventional wisdom

has been used to establish the null hypothesis. The critical region of a hypothesis test

is the set of all outcomes which cause the null hypothesis to be rejected in favor of

the alternative hypothesis. Statistical hypothesis testing is sometimes

called confirmatory data analysis, in contrast to exploratory data analysis, which

may not have pre-specified hypotheses. In the Neyman-Pearson framework (see

below), the process of distinguishing between the null & alternative hypotheses is

aided by identifying two conceptual types of errors (type 1 & type 2), and by

specifying parametric limits on e.g. how much type 1 error will be permitted.

Contents

1 Variations and sub-classes

2 The testing process

2.1 Interpretation

2.2 Use and importance

2.3 Cautions

3 Example

3.2 Analogy Courtroom trial

3.3 Example 1 Philosopher's beans

3.4 Example 2 Clairvoyant card game

4 Definition of terms

5 Common test statistics

6 Origins and early controversy

• 6.1 Early choices of null hypothesis

7 Null hypothesis statistical significant testing vs hypothesis testing

8 Criticism

8.1 Alternatives to significance testing

9 Philosophy

10 Education

12 References

Variations and sub-classes

Statistical hypothesis testing is a key technique of both Frequentist

inference and Bayesian inference although they have notable differences. Statistical

hypothesis tests define a procedure that controls (fixes) the probability of

incorrectly deciding that a default position (null hypothesis) is incorrect based on how

likely it would be for a set of observations to occur if the null hypothesis were true.

Note that this probability of making an incorrect decision is not the probability that

the null hypothesis is true, nor whether any specific alternative hypothesis is true. This

contrasts with other possible techniques of decision theory in which the null

and alternative hypothesis are treated on a more equal basis. One

naive Bayesian approach to hypothesis testing is to base decisions on the posterior

probability,[2][3] but this fails when comparing point and continuous hypotheses. Other

approaches to decision making, such as Bayesian decision theory, attempt to balance

the consequences of incorrect decisions across all possibilities, rather than

concentrating on a single null hypothesis. A number of other approaches to reaching a

decision based on data are available via decision theory and optimal decisions, some

of which have desirable properties, yet hypothesis testing is a dominant approach to

data analysis in many fields of science. Extensions to the theory of hypothesis testing

include the study of the power of tests, the probability of correctly rejecting the null

hypothesis given that it is false. Such considerations can be used for the purpose

of sample size determination prior to the collection of data.

The testing process

In the statistics literature, statistical hypothesis testing plays a fundamental role.[4] The

usual line of reasoning is as follows:

• 1. There is an initial research hypothesis of which the truth is unknown.

2. The first step is to state the relevant null and alternative hypotheses. This is

important as mis-stating the hypotheses will muddy the rest of the process.

Specifically, the null hypothesis allows to attach an attribute: it should be

chosen in such a way that it allows us to conclude whether the alternative

hypothesis can either be accepted or stays undecided as it was before the test.[5]

3. The second step is to consider the statistical assumptions being made about the

sample in doing the test; for example, assumptions about the statistical

independence or about the form of the distributions of the observations. This is

equally important as invalid assumptions will mean that the results of the test

are invalid.

4. Decide which test is appropriate, and state the relevant test statistic T.

5. Derive the distribution of the test statistic under the null hypothesis from the

assumptions. In standard cases this will be a well-known result. For example

the test statistic might follow a Student's t distribution or

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