## Statistical hypothesis testing

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A statistical hypothesissometimes called confirmatory data analysisis a hypothesis that is testable on the basis of observing a process that is modeled via a set of random variables. Commonly, two statistical data sets are compared, or a data set obtained by sampling is compared against a synthetic data set from an idealized model.

A hypothesis is proposed for the statistical relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis that proposes no obligations of the binary orders academy student between two data sets. The comparison is deemed statistically significant if the relationship between the data sets would be an unlikely realization of the null hypothesis according to a threshold probabilityâ€”the significance level.

Hypothesis 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.

An alternative framework for statistical hypothesis testing is to specify a set of statistical modelsone for each candidate hypothesis, and then use model selection techniques to choose the most appropriate model.

Confirmatory data analysis can be contrasted with exploratory data analysiswhich may not have pre-specified hypotheses. Statistical hypothesis testing is a key technique of both frequentist inference and Bayesian inferencealthough the two types of inference 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. The procedure is 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 obligations of the binary orders academy student 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. Other approaches to decision making, such as Bayesian decision theoryattempt to balance the consequences of incorrect decisions across obligations of the binary orders academy student 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 decisionssome of which have desirable properties. Hypothesis testing, though, 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, i. Such considerations can be used for the purpose of sample size determination prior to the collection of data. In the statistics literature, statistical hypothesis testing plays a fundamental role.

The two processes are equivalent. It allowed a decision to be made without the calculation of a probability. It was adequate for classwork and for operational use, but it was deficient for reporting results. The latter process relied on extensive tables or on computational support not always available.

The explicit calculation of a probability is useful for reporting. The calculations are now trivially performed with appropriate software. The former report is adequate, the latter gives a more obligations of the binary orders academy student explanation of the data obligations of the binary orders academy student the reason why the suitcase is being checked. It is important to note the difference between accepting the null hypothesis and simply failing to reject it.

The "fail to reject" terminology highlights the fact that the null hypothesis is assumed to be obligations of the binary orders academy student from the start of the test; obligations of the binary orders academy student there is a lack of evidence against it, it simply continues to be assumed true.

The phrase "accept the null hypothesis" may suggest it has been proved simply because it has not been disproved, a logical fallacy known as the argument from ignorance. Unless a test with particularly high power is used, the idea of "accepting" the null hypothesis may be dangerous. Nonetheless the terminology is prevalent throughout statistics, where the meaning actually intended is well understood.

The processes described here are perfectly adequate for computation. They seriously neglect the design of experiments considerations. It is particularly critical that appropriate sample sizes be estimated before conducting the experiment. The phrase "test of significance" was coined by statistician Ronald Fisher.

The p -value is the probability that a given result or a more significant result would occur under the null hypothesis. For example, say that a fair coin is tested for fairness the null hypothesis. Obligations of the binary orders academy student a significance level of 0. The p -value does not provide the probability that either hypothesis is correct a common source of confusion.

If the p -value is less than the chosen significance threshold equivalently, if the observed test statistic is in the critical regionthen we say the null hypothesis is rejected at the chosen level of significance. Rejection of the null hypothesis is a conclusion.

This is like a "guilty" verdict in a criminal trial: We might accept the alternative hypothesis and the research hypothesis. If the p -value is not less than the chosen significance threshold equivalently, if the observed test statistic is outside the critical regionthen the evidence is insufficient to support a conclusion. This is similar to a "not guilty" verdict. The researcher typically gives extra consideration to those cases where the p -value is close to the significance level.

Some people find it helpful to think of the hypothesis testing framework as analogous to a mathematical proof by contradiction. In the Lady tasting tea example belowFisher required the Lady to properly categorize all of the cups of tea to justify the conclusion that the result was unlikely to result from chance. His test revealed that if the lady was effectively guessing at random the null hypothesisthere was a 1. Whether rejection of the null hypothesis truly justifies acceptance of the research hypothesis depends on the structure of the hypotheses.

Rejecting the hypothesis that a large paw print originated from a bear does not immediately prove the existence of Bigfoot. Hypothesis testing emphasizes the rejection, which is based on a probability, rather than the acceptance, which requires extra steps of logic. Statistics are helpful in analyzing most collections of data. This is equally true of hypothesis testing which can justify conclusions even when no scientific theory exists. In the Lady tasting tea example, it was "obvious" that no difference existed between milk poured into tea and tea poured into milk.

The data contradicted the "obvious". Real world applications of hypothesis testing obligations of the binary orders academy student Statistical hypothesis testing plays an important role in the whole of statistics and in statistical inference.

For example, Lehmann in a review of the fundamental paper by Neyman and Pearson says: Significance testing is used as a substitute for the traditional comparison of predicted value and experimental result at the core of the scientific method. When theory is only capable of predicting the sign of a relationship, a directional one-sided hypothesis test can be configured so that only a statistically significant result supports theory. This form of theory appraisal is the most heavily criticized application of hypothesis testing.

The successful hypothesis obligations of the binary orders academy student is associated with a probability and a type-I error rate. The conclusion might be wrong. The conclusion of the test is only as solid as the sample upon which it is based. The design of the experiment is critical. A number of unexpected effects have been observed including:. A statistical analysis of misleading data produces misleading conclusions.

The issue of data quality can be more subtle. In forecasting for example, there is no agreement on a measure of forecast accuracy. In the absence of a obligations of the binary orders academy student measurement, no decision based on measurements will be without controversy.

The book How to Lie with Statistics [16] [17] is the most popular book on statistics ever published. Many claims are made on the basis of samples too small to convince. If a report does not mention sample size, be doubtful. Obligations of the binary orders academy student testing acts as a filter of statistical conclusions; only those results meeting a probability threshold are publishable. Economics also acts as a publication filter; only those results favorable to the author and funding source may be submitted for publication.

The impact obligations of the binary orders academy student filtering on publication is termed publication bias. A related problem is that of multiple testing sometimes linked to data miningin which a variety of tests for a variety of possible effects are applied to a single data set and only those yielding a significant result are reported. These are often dealt with by using multiplicity correction procedures that control the family wise error rate FWER or the false discovery rate FDR.

Those making critical decisions based on the results of a hypothesis test are prudent to look at the details rather than the conclusion alone. In the physical sciences most results are fully accepted only when independently confirmed. The general advice concerning statistics is, "Figures never lie, but liars figure" anonymous.

In a famous example of hypothesis testing, known as the Lady tasting tea[19] Dr. Muriel Bristola female colleague of Fisher claimed to be able to tell whether the tea or the milk was added first to a cup. Fisher proposed to give her eight cups, four of each variety, in random order. One could then ask what the probability was for her getting the number she got correct, but just by chance.

The null hypothesis was that the Lady had no such ability. The test statistic was a simple count of the number of successes in selecting the 4 cups. Fisher asserted that no alternative hypothesis was ever required. The lady correctly identified every cup, [20] which would be considered a statistically significant result. A statistical test procedure is comparable to a criminal trial ; a defendant is considered not guilty as long as his or her guilt is not proven.

The prosecutor tries to prove the guilt of the defendant. Only when there is enough evidence for the prosecution is the defendant convicted.

It is the alternative hypothesis that one hopes to support. The hypothesis of innocence is only rejected when an error is very unlikely, because one doesn't want to convict an innocent defendant.

Such an error is called error of the first kind i. As a consequence of this asymmetric behaviour, an error of the second kind acquitting a person who committed the crimeis more common. A criminal trial can be regarded as obligations of the binary orders academy student or both of two decision processes: In one view, the defendant is judged; in the other view the performance of the prosecution which bears the burden of proof is judged.

A hypothesis test can be regarded as either a judgment of a hypothesis or as a judgment of evidence.