HYPOTHESIS TESTING PDF

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What is hypothesis testing? (cont.) The hypothesis we want to test is if H1 is “ likely” true. So, there are two possible outcomes: • Reject H0 and accept H1. The major purpose of hypothesis testing is to choose between two competing null and alternative hypothesis should be stated before any statistical test of. Inferential Statistics and Hypothesis Testing. Four Steps to. Hypothesis Testing. Hypothesis Testing and. Sampling Distributions. Making a.


Hypothesis Testing Pdf

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How Hypothesis Tests. Are Reported in the News. 1. Determine the null hypothesis and the alternative hypothesis. 2. Collect and summarize the data into a. TESTS OF HYPOTHESES. As was mentioned earlier, sometimes we cannot survey or test all persons or objects; therefore, we have to take a sample. From the. Hypothesis Testing. One type of statistical inference, estimation, was discussed in Chapter 5. The other type,hypothesis testing,is discussed in this chapter.

Butts, and G. Pearce eds. Reidel, Dordrecht, , — Ontario; Eds. Hooker and W. Harper , Vol. Reidel, Dordrecht, Holland, , pp. London A , — Reprinted in Zellner Statist Assoc. Google Scholar Hendrickson, A. The third edn. Google Scholar Kalbfleisch, J. Harper eds. Reidel, Dordrecht, Holland, —, Google Scholar Kendall, M. Google Scholar Lehmann, E.

Google Scholar Lindley, D. Google Scholar Minsky, M. Google Scholar Neyman, J. Newman ed. Thus we can say that the suitcase is compatible with the null hypothesis this does not guarantee that there is no radioactive material, just that we don't have enough evidence to suggest there is.

ASC Stats Homepage: Hypothesis Testing

On the other hand, if the null hypothesis predicts 3 counts per minute for which the Poisson distribution predicts only 0. The test does not directly assert the presence of radioactive material.

A successful test asserts that the claim of no radioactive material present is unlikely given the reading and therefore The double negative disproving the null hypothesis of the method is confusing, but using a counter-example to disprove is standard mathematical practice.

The attraction of the method is its practicality. We know from experience the expected range of counts with only ambient radioactivity present, so we can say that a measurement is unusually large. Statistics just formalizes the intuitive by using numbers instead of adjectives. We probably do not know the characteristics of the radioactive suitcases; We just assume that they produce larger readings.

To slightly formalize intuition: This makes no assumptions about the distribution of counts. Many ambient radiation observations are required to obtain good probability estimates for rare events. The test described here is more fully the null-hypothesis statistical significance test. The null hypothesis represents what we would believe by default, before seeing any evidence. Statistical significance is a possible finding of the test, declared when the observed sample is unlikely to have occurred by chance if the null hypothesis were true.

The name of the test describes its formulation and its possible outcome. One characteristic of the test is its crisp decision: A calculated value is compared to a threshold, which is determined from the tolerable risk of error. The following definitions are mainly based on the exposition in the book by Lehmann and Romano: A statistical hypothesis test compares a test statistic z or t for examples to a threshold.

The test statistic the formula found in the table below is based on optimality. For a fixed level of Type I error rate, use of these statistics minimizes Type II error rates equivalent to maximizing power. The following terms describe tests in terms of such optimality:. Statistical hypothesis testing is a key technique of both frequentist inference and Bayesian inference , although 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 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.

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. 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. While hypothesis testing was popularized early in the 20th century, early forms were used in the s.

Modern significance testing is largely the product of Karl Pearson p -value , Pearson's chi-squared test , William Sealy Gosset Student's t-distribution , and Ronald Fisher " null hypothesis ", analysis of variance , " significance test " , while hypothesis testing was developed by Jerzy Neyman and Egon Pearson son of Karl.

Ronald Fisher began his life in statistics as a Bayesian Zabell , but Fisher soon grew disenchanted with the subjectivity involved namely use of the principle of indifference when determining prior probabilities , and sought to provide a more "objective" approach to inductive inference.

Fisher was an agricultural statistician who emphasized rigorous experimental design and methods to extract a result from few samples assuming Gaussian distributions. Neyman who teamed with the younger Pearson emphasized mathematical rigor and methods to obtain more results from many samples and a wider range of distributions. Fisher popularized the "significance test".

He required a null-hypothesis corresponding to a population frequency distribution and a sample. His now familiar calculations determined whether to reject the null-hypothesis or not.

Significance testing did not utilize an alternative hypothesis so there was no concept of a Type II error. The p -value was devised as an informal, but objective, index meant to help a researcher determine based on other knowledge whether to modify future experiments or strengthen one's faith in the null hypothesis. They initially considered two simple hypotheses both with frequency distributions. They calculated two probabilities and typically selected the hypothesis associated with the higher probability the hypothesis more likely to have generated the sample.

Their method always selected a hypothesis. It also allowed the calculation of both types of error probabilities. The defining paper [35] was abstract. Mathematicians have generalized and refined the theory for decades.

The dispute between Fisher and Neyman—Pearson was waged on philosophical grounds, characterized by a philosopher as a dispute over the proper role of models in statistical inference. Events intervened: Neyman accepted a position in the western hemisphere, breaking his partnership with Pearson and separating disputants who had occupied the same building by much of the planetary diameter. World War II provided an intermission in the debate. The dispute between Fisher and Neyman terminated unresolved after 27 years with Fisher's death in Neyman wrote a well-regarded eulogy.

The modern version of hypothesis testing is a hybrid of the two approaches that resulted from confusion by writers of statistical textbooks as predicted by Fisher beginning in the s. Great conceptual differences and many caveats in addition to those mentioned above were ignored. Neyman and Pearson provided the stronger terminology, the more rigorous mathematics and the more consistent philosophy, but the subject taught today in introductory statistics has more similarities with Fisher's method than theirs.

Some Logic and History of Hypothesis Testing

Sometime around , [42] in an apparent effort to provide researchers with a "non-controversial" [44] way to have their cake and eat it too , the authors of statistical text books began anonymously combining these two strategies by using the p -value in place of the test statistic or data to test against the Neyman—Pearson "significance level".

It then became customary for the null hypothesis, which was originally some realistic research hypothesis, to be used almost solely as a strawman "nil" hypothesis one where a treatment has no effect, regardless of the context.

Paul Meehl has argued that the epistemological importance of the choice of null hypothesis has gone largely unacknowledged. When the null hypothesis is predicted by theory, a more precise experiment will be a more severe test of the underlying theory. When the null hypothesis defaults to "no difference" or "no effect", a more precise experiment is a less severe test of the theory that motivated performing the experiment. Pierre Laplace compares the birthrates of boys and girls in multiple European cities.

He states: Thus Laplace's null hypothesis that the birthrates of boys and girls should be equal given "conventional wisdom". Karl Pearson develops the chi squared test to determine "whether a given form of frequency curve will effectively describe the samples drawn from a given population. He uses as an example the numbers of five and sixes in the Weldon dice throw data.

Karl Pearson develops the concept of " contingency " in order to determine whether outcomes are independent of a given categorical factor. Here the null hypothesis is by default that two things are unrelated e.

An example of Neyman—Pearson hypothesis testing can be made by a change to the radioactive suitcase example. If the "suitcase" is actually a shielded container for the transportation of radioactive material, then a test might be used to select among three hypotheses: The test could be required for safety, with actions required in each case.

The Neyman—Pearson lemma of hypothesis testing says that a good criterion for the selection of hypotheses is the ratio of their probabilities a likelihood ratio. A simple method of solution is to select the hypothesis with the highest probability for the Geiger counts observed. The typical result matches intuition: Notice also that usually there are problems for proving a negative. Null hypotheses should be at least falsifiable.

Neyman—Pearson theory can accommodate both prior probabilities and the costs of actions resulting from decisions. The latter allows the consideration of economic issues for example as well as probabilities.

A likelihood ratio remains a good criterion for selecting among hypotheses. The two forms of hypothesis testing are based on different problem formulations.

In the view of Tukey [51] the former produces a conclusion on the basis of only strong evidence while the latter produces a decision on the basis of available evidence. While the two tests seem quite different both mathematically and philosophically, later developments lead to the opposite claim. Consider many tiny radioactive sources. The hypotheses become 0,1,2, There is little distinction between none or some radiation Fisher and 0 grains of radioactive sand versus all of the alternatives Neyman—Pearson.

The major Neyman—Pearson paper of [35] also considered composite hypotheses ones whose distribution includes an unknown parameter.

An example proved the optimality of the Student's t -test, "there can be no better test for the hypothesis under consideration" p Neyman—Pearson theory was proving the optimality of Fisherian methods from its inception. Fisher's significance testing has proven a popular flexible statistical tool in application with little mathematical growth potential.

Neyman—Pearson hypothesis testing is claimed as a pillar of mathematical statistics, [52] creating a new paradigm for the field. It also stimulated new applications in statistical process control , detection theory , decision theory and game theory.

Both formulations have been successful, but the successes have been of a different character. The dispute over formulations is unresolved. Science primarily uses Fisher's slightly modified formulation as taught in introductory statistics.

Statisticians study Neyman—Pearson theory in graduate school. Mathematicians are proud of uniting the formulations. Philosophers consider them separately. Learned opinions deem the formulations variously competitive Fisher vs Neyman , incompatible [33] or complementary. The terminology is inconsistent.

Hypothesis testing can mean any mixture of two formulations that both changed with time. Any discussion of significance testing vs hypothesis testing is doubly vulnerable to confusion.

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Fisher thought that hypothesis testing was a useful strategy for performing industrial quality control, however, he strongly disagreed that hypothesis testing could be useful for scientists.

The two methods remain philosophically distinct. The preferred answer is context dependent. Criticism of statistical hypothesis testing fills volumes [54] [55] [56] [57] [58] [59] citing — primary references [ citation needed ]. Much of the criticism can be summarized by the following issues:.

Critics and supporters are largely in factual agreement regarding the characteristics of null hypothesis significance testing NHST: While it can provide critical information, it is inadequate as the sole tool for statistical analysis. Successfully rejecting the null hypothesis may offer no support for the research hypothesis. The continuing controversy concerns the selection of the best statistical practices for the near-term future given the often poor existing practices.

Critics would prefer to ban NHST completely, forcing a complete departure from those practices, while supporters suggest a less absolute change. Controversy over significance testing, and its effects on publication bias in particular, has produced several results.

The American Psychological Association has strengthened its statistical reporting requirements after review, [69] medical journal publishers have recognized the obligation to publish some results that are not statistically significant to combat publication bias [70] and a journal Journal of Articles in Support of the Null Hypothesis has been created to publish such results exclusively.

Major organizations have not abandoned use of significance tests although some have discussed doing so. The numerous criticisms of significance testing do not lead to a single alternative.

A unifying position of critics is that statistics should not lead to a conclusion or a decision but to a probability or to an estimated value with an interval estimate rather than to an accept-reject decision regarding a particular hypothesis. It is unlikely that the controversy surrounding significance testing will be resolved in the near future. Its supposed flaws and unpopularity do not eliminate the need for an objective and transparent means of reaching conclusions regarding studies that produce statistical results.

Critics have not unified around an alternative. Other forms of reporting confidence or uncertainty could probably grow in popularity.

One strong critic of significance testing suggested a list of reporting alternatives: On one "alternative" there is no disagreement: Fisher himself said, [26] "In relation to the test of significance, we may say that a phenomenon is experimentally demonstrable when we know how to conduct an experiment which will rarely fail to give us a statistically significant result.

It doesn't exist. The easiest way to decrease statistical uncertainty is by obtaining more data, whether by increased sample size or by repeated tests. Nickerson claimed to have never seen the publication of a literally replicated experiment in psychology. Bayesian inference is one proposed alternative to significance testing. Nickerson cited 10 sources suggesting it, including Rozeboom Psychologist John K.

Kruschke has suggested Bayesian estimation as an alternative for the t -test. Neither the prior probabilities nor the probability distribution of the test statistic under the alternative hypothesis are often available in the social sciences.

Advocates of a Bayesian approach sometimes claim that the goal of a researcher is most often to objectively assess the probability that a hypothesis is true based on the data they have collected. The probability a hypothesis is true can only be derived from use of Bayes' Theorem , which was unsatisfactory to both the Fisher and Neyman—Pearson camps due to the explicit use of subjectivity in the form of the prior probability. Hypothesis testing and philosophy intersect. Inferential statistics , which includes hypothesis testing, is applied probability.

Both probability and its application are intertwined with philosophy. Philosopher David Hume wrote, "All knowledge degenerates into probability. The most common application of hypothesis testing is in the scientific interpretation of experimental data, which is naturally studied by the philosophy of science.

Fisher and Neyman opposed the subjectivity of probability. Their views contributed to the objective definitions. The core of their historical disagreement was philosophical. Many of the philosophical criticisms of hypothesis testing are discussed by statisticians in other contexts, particularly correlation does not imply causation and the design of experiments. Hypothesis testing is of continuing interest to philosophers. Statistics is increasingly being taught in schools with hypothesis testing being one of the elements taught.

Some writers have stated that statistical analysis of this kind allows for thinking clearly about problems involving mass data, as well as the effective reporting of trends and inferences from said data, but caution that writers for a broad public should have a solid understanding of the field in order to use the terms and concepts correctly.

Such fields as literature and divinity now include findings based on statistical analysis see the Bible Analyzer. An introductory statistics class teaches hypothesis testing as a cookbook process.

Hypothesis testing is also taught at the postgraduate level. Statisticians learn how to create good statistical test procedures like z , Student's t , F and chi-squared. Statistical hypothesis testing is considered a mature area within statistics, [74] but a limited amount of development continues. An academic study states that the cookbook method of teaching introductory statistics leaves no time for history, philosophy or controversy. Hypothesis testing has been taught as received unified method.

Surveys showed that graduates of the class were filled with philosophical misconceptions on all aspects of statistical inference that persisted among instructors. From Wikipedia, the free encyclopedia.

For the computer science notion of a "critical section", sometimes called a "critical region", see critical section. Main article: Human sex ratio. Lady tasting tea. Test statistic. See also: Estimation statistics. Statistics education. Statistics portal. It is the role of the analyst to collate the hypotheses and determine with the key stakeholders the hypotheses that should be tested.

The testing of each hypothesis should also result in information that can inform a response. The hypotheses that are determined therefore need to be clear and specific. Practice suggests that three to five hypotheses is the preferred number to test, simply because there is unlikely to be analytical capacity to test any more in the timeframe required.

Statistical hypothesis testing

The primary objective is to come to some conclusions that provide evidence that does or does not support each hypothesis. The type of analysis to conduct is determined by the hypothesis that is tested. For example, if a hypothesis states that a recent increase in residential burglaries is related to an increase in homes being left insecure due to residents leaving windows open because of a recent prolonged period of hot weather , the analysis will need to identify if there has been a recent increase in overnight temperature that coincided with the increase in burglaries, and if the volume and proportion of insecure burglaries has increased in line with the burglary increase.

This process will also identify where there are intelligence gaps and any data may need to be collected in order to test a hypothesis e.

Key stakeholders rather than just analysts should be involved in interpreting the analysis. This is best conducted by holding a meeting to discuss the findings. It is at this stage that the key stakeholders should use the analysis to help decide how the problem can be addressed. Experience suggests that the better the problem is understood, the easier it is to determine specific tactics and strategies that will counter the issues the analysis has identified.

The overview did not show any clear seasonal pattern to the burglaries, but certain areas of Oldham had experienced bigger increases than others.

In summary, there was little evidence to support the first two hypotheses. Analysis for hypothesis 3 determined that repeat victimisation levels had fallen, but near repeats had increased to explain 1 in 5 of the additional burglaries. Figure 1 shows the results of the analysis conducted to test hypothesis 4. Comparing the time of day that burglaries were committed in Winter months to those in Summer months revealed that the difference in offending between midday and 9pm explained the entire burglary increase.

This finding led the CSP to refine their Autumn crime awareness programme so that it was more specific to those communities at risk of burglary.Karl Pearson develops the concept of " contingency " in order to determine whether outcomes are independent of a given categorical factor. Philosophers consider them separately. With this increasing demand for analysis and the subsequent standardisation of intelligence products has come the production of templates that aim to determine a consistent structure and content to these materials.

Holt, Reinhart and Winston of Canada, Toronto, , pp. He states: It then became customary for the null hypothesis, which was originally some realistic research hypothesis, to be used almost solely as a strawman "nil" hypothesis one where a treatment has no effect, regardless of the context. Spectral density estimation Fourier analysis Wavelet Whittle likelihood.

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