Statistical hypothesis testingAdriana Albu，Loredana UngureanuPolitehnica University Timisoara,adrianaa@aut.utt.roPolitehnica University Timisoara,loredanau@aut.utt.roAbstract In this article,we present a Bayesian statistical hypothesis testing inspection, testing theory and the process Mentioned hypothesis testing in the real world and the importance of, and successful test of the Notes.Key words Bayesian hypothesis testing; Bayesian inference;Test of significance IntroductionA statistical hypothesis test is a method of making decisions using data, whether from a controlled experiment or an observational study (not controlled). In statistics, a result is called statistically significant if it is unlikely to have occurred by chance alone, according to a pre-determined threshold probability, the significance level. The phrase "test of significance" was coined by Ronald Fisher: "Critical tests of this kind may be called tests of significance, and when such tests are available we may discover whether a second sample is or is not significantly different from the first."[1]Hypothesis testing is sometimes called confirmatory data analysis, in contrast to exploratory data analysis. In frequency probability,these decisions are almost always made using null-hypothesis tests. These are tests that answer the question Assuming that the null hypothesis is true, what is the probability of observing a value for the test statistic that is at[]least as extreme as the value that was actually observed?) 2 More formally, they represent answers to the question, posed before undertaking an experiment,of what outcomes of the experiment would lead to rejection of the null hypothesis for a pre-specified probability of an incorrect rejection. One use of hypothesis testing is deciding whether experimental results contain enough information to cast doubt on conventional wisdom.Statistical hypothesis testing is a key technique of frequentist statistical inference. The Bayesian approach to hypothesis testing is to base rejection of the hypothesis on the posterior probability.[3][4]Other approaches to reaching a decision based on data are available via decision theory and optimal decisions.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. The critical region is usually denoted by the letter C.One-sample tests are appropriate when a sample is being compared to the population from a hypothesis. The population characteristics are known from theory or are calculated from the population.Two-sample tests are appropriate for comparing two samples, typically experimental and control samples from a scientifically controlled experiment.Paired tests are appropriate for comparing two samples where it is impossible to control important variables. Rather than comparing two sets, members are paired between samples so the difference between the members becomes the sample.Typically the mean of the differences is then compared to zero.Z-tests are appropriate for comparing means under stringent conditions regarding normality and a known standard deviation.T-tests are appropriate for comparing means under relaxed conditions(less is assumed).Tests of proportions are analogous to tests of means (the 50% proportion). Chi-squared tests use the same calculations and the same probability distribution for different applications:Chi-squared tests for variance are used to determine whether a normal population has a specified variance. The null hypothesis is that it does.Chi-squared tests of independence are used for deciding whether two variables are associated or are independent. The variables are categorical rather than numeric. It can be used to decide whether left-handedness is correlated with libertarian politics (or not). The null hypothesis is that the variables are independent.The numbers used in the calculation are the observed and expected frequencies of occurrence (from contingency tables).Chi-squared goodness of fit tests are used to determine the adequacy of curves fit to data. The null hypothesis is that the curve fit is adequate.It is common to determine curve shapes to minimize the mean square error, so it is appropriate that the goodness-of-fit calculation sums the squared errors.F-tests (analysis of variance, ANOVA) are commonly used when deciding whether groupings of data by category are meaningful.If the variance of test scores of the left-handed in a class is much smaller than the variance of the whole class, then it may be useful to study lefties as a group. The null hypothesis is that two variances are the same - so the proposed grouping is not meaningful.The testing processIn the statistical literature, statistical hypothesis testing plays a fundamental role. 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 isimportant as mis-stating the hypotheses will muddy the rest of the process.Specifically, the null hypothesis allows attaching 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.[9]3. The second step is to consider the statistical assumptions being made about thesample in doing the test;for example,assumptions about the statistical independence or about the form of the distributions of the observations. This isequally 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 theassumptions. In standard cases this will be a well-known result. For example the test statistic may follow a Student's t distribution or a normal distribution.6. Select a significance level (α), a probability threshold below which the nullhypothesis will be rejected. Common values are 5% and 1%.7. The distribution of the test statistic under the null hypothesis partitions the possiblevalues of T into those for which the null-hypothesis is rejected, the so called critical region, and those for which it is not. The probability of the critical region is α.8. Compute from the observations the observed value t obs of the test statistic T .9. Decide to either fail to reject the null hypothesis or reject it in favor of thealternative. The decision rule is to reject the null hypothesis H 0 if the observed value t obs is in the critical region, and to accept or "fail to reject" the hypothesis otherwise.Use and ImportanceStatistics 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. Real world applications of hypothesis testing include : Testing whether more men than women suffer from nightmaresEstablishing authorship of documentsEvaluating the effect of the full moon on behaviorDetermining the range at which a bat can detect an insect by echoDeciding whether hospital carpeting results in more infectionsSelecting the best means to stop smokingChecking whether bumper stickers reflect car owner behaviorTesting the claims of handwriting analystsStatistical hypothesis testing plays an important role in the whole of statistics and in statistical inference. For example, Lehmann (1992) in a review of the fundamental paper by Neyman and Pearson (1933) says: "Nevertheless, despite their shortcomings, the new paradigm formulated in the 1933 paper, and the many developments carried out within its framework continue to play a central role in both the theory and practice of statistics and can be expected to do so in the foreseeable future".Significance testing has been the favored statistical tool in some experimental social sciences (over 90% of articles in the Journal of Applied Psychology during the early1990s). Other fields have favored the estimation of parameters. Editors often consider significance as a criterion for the publication of scientific conclusions based on experiments with statistical results.CautionsThe successful hypothesis test is associated with a probability and a type-I error rate. The conclusion might be wrong.[7] [8]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:The Clever Hans effect. A horse appeared to be capable of doing simple arithmetic. The Hawthorne effect. Industrial workers were more productive in better illumination, and most productive in worse.The Placebo effect. Pills with no medically active ingredients were remarkably effective.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 consensus measurement, no decision based on measurements will be without controversy.The book How to Lie with Statistics is the most popular book on statistics everpublished. It does not much consider hypothesis testing, but its cautions are applicable,including: Many claims are made on the basis of samples too small to convince. If a report does not mention sample size, be doubtful.Hypothesis 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 of filtering on publication is termed publication bias. A related problem is that of multiple testing (sometimes linked to data mining), in 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.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).ControversySince significance tests were first popularized many objections have been voiced by prominent and respected statisticians. The volume of criticism and rebuttal has filled books with language seldom used in the scholarly debate of a dry subject. Much of the criticism was published more than 40 years ago. The fires of controversy have burned hottest in the field of experimental psychology. Nickerson surveyed the issues in the year 2000. He included 300 references and reported 20 criticisms and almost as many recommendations, alternatives and supplements. The following section greatly condenses Nickerson's discussion, omitting many issues.Results of the controversyThe controversy has produced several results. The American PsychologicalAssociation has strengthened its statistical reporting requirements after review, medicaljournal publishers have recognized the obligation to publish some results that are notstatistically significant to combat publication bias and a journal (Journal of Articles inSupport of the Null Hypothesis ) has been created to publish such results exclusively. Textbooks have added some cautions and increased coverage of the tools necessary to[28] [10] .estimate the size of the sample required to produce significant results. Major organizations have not abandoned use of significance tests although they have discussed doing so. References[1] R. A. Fisher (1925). Statistical Methods for Research Workers, Edinburgh: Oliver and Boyd, 1925, p.43.[2] Cramer, Duncan; Dennis Howitt (2004). The Sage Dictionary of Statistics.p. 76. ISBN0-7619-4138-X.[3] Schervish,M(1996)Theory of Statistics, p. 218.Springer ISBN 0-387-94546-6[4] Kaye, David H.; Freedman, David A. (2011). "Reference Guide on Statistics". Reference manual on scientific evidence (3rd ed.). Eagan, MN Washington, D.C: West National Academies Press. p. 259. ISBN978-0-309-21421-6.[5] C. S. Peirce (August 1878). "Illustrations of the Logic of Science VI: Deduction, Induction, and Hypothesis".Popular Science Monthly13.[6] Fisher, Sir Ronald A. (1956) [1935]. "Mathematics of a Lady Tasting Tea". In James Roy Newman. The World of Mathematics, volume 3 [Design of Experiments]. Courier Dover Publications. ISBN978-0-486-41151-4.[7] Box, Joan Fisher (1978). R.A. Fisher, The Life of a Scientist. New York: Wiley. p. 134. ISBN0-471-09300-9[8] Lehmann, E.L.; Romano,Joseph P. (2005). Testing Statistical Hypotheses (3E ed.). New York: Springer. ISBN0-387-98864-5.[9] Adèr,J.H. (2008). Chapter 12: Modelling. In H.J. Adèr & G.J. Mellenbergh (Eds.) (with contributions by D.J. Hand), Advising on Research Methods: A consultant's companion (pp. 183–209). Huizen,The Netherlands:Johannes van Kessel Publishing[10] Triola, Mario (2001). Elementary statistics (8 ed.). Boston: Addison-Wesley. p. 388. ISBN 0-201-61477-4.摘要济南大学泉城学院毕业论文外文资料翻译American Journal of Mathematics, 2007,126(5): 2387-2425统计假设检验Adriana Albu，Loredana UngureanuPolitehnica University Timisoara,adrianaa@aut.utt.roPolitehnica University Timisoara,loredanau@aut.utt.ro在这篇文章中，我们给出统计假设检验的贝叶斯检验，介绍了检验理论和其过程。