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# Hypothesis Testing Examples And Solutions Pdf

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*With powerful computers and statistical packages, modelers can now run an enormous number of tests effortlessly. But should they? This article discusses how bank risk modelers should approach statistical testing when faced with tiny data sets.*

Misinterpretation and abuse of statistical tests, confidence intervals, and statistical power have been decried for decades, yet remain rampant. A key problem is that there are no interpretations of these concepts that are at once simple, intuitive, correct, and foolproof. Instead, correct use and interpretation of these statistics requires an attention to detail which seems to tax the patience of working scientists.

Misinterpretation and abuse of statistical tests, confidence intervals, and statistical power have been decried for decades, yet remain rampant. A key problem is that there are no interpretations of these concepts that are at once simple, intuitive, correct, and foolproof.

Instead, correct use and interpretation of these statistics requires an attention to detail which seems to tax the patience of working scientists. This high cognitive demand has led to an epidemic of shortcut definitions and interpretations that are simply wrong, sometimes disastrously so—and yet these misinterpretations dominate much of the scientific literature.

In light of this problem, we provide definitions and a discussion of basic statistics that are more general and critical than typically found in traditional introductory expositions. Our goal is to provide a resource for instructors, researchers, and consumers of statistics whose knowledge of statistical theory and technique may be limited but who wish to avoid and spot misinterpretations.

We emphasize how violation of often unstated analysis protocols such as selecting analyses for presentation based on the P values they produce can lead to small P values even if the declared test hypothesis is correct, and can lead to large P values even if that hypothesis is incorrect. We then provide an explanatory list of 25 misinterpretations of P values, confidence intervals, and power. We conclude with guidelines for improving statistical interpretation and reporting.

One journal now bans all statistical tests and mathematically related procedures such as confidence intervals [ 2 ], which has led to considerable discussion and debate about the merits of such bans [ 3 , 4 ]. Despite such bans, we expect that the statistical methods at issue will be with us for many years to come. We thus think it imperative that basic teaching as well as general understanding of these methods be improved. Toward that end, we attempt to explain the meaning of significance tests, confidence intervals, and statistical power in a more general and critical way than is traditionally done, and then review 25 common misconceptions in light of our explanations.

We also discuss a few more subtle but nonetheless pervasive problems, explaining why it is important to examine and synthesize all results relating to a scientific question, rather than focus on individual findings. We further explain why statistical tests should never constitute the sole input to inferences or decisions about associations or effects.

More detailed discussion of the general issues can be found in many articles, chapters, and books on statistical methods and their interpretation [ 5 — 20 ]. Every method of statistical inference depends on a complex web of assumptions about how data were collected and analyzed, and how the analysis results were selected for presentation. The full set of assumptions is embodied in a statistical model that underpins the method. This model is a mathematical representation of data variability, and thus ideally would capture accurately all sources of such variability.

Many problems arise however because this statistical model often incorporates unrealistic or at best unjustified assumptions. These assumptions are often deceptively simple to write down mathematically, yet in practice are difficult to satisfy and verify, as they may depend on successful completion of a long sequence of actions such as identifying, contacting, obtaining consent from, obtaining cooperation of, and following up subjects, as well as adherence to study protocols for treatment allocation, masking, and data analysis.

There is also a serious problem of defining the scope of a model, in that it should allow not only for a good representation of the observed data but also of hypothetical alternative data that might have been observed. The difficulty of understanding and assessing underlying assumptions is exacerbated by the fact that the statistical model is usually presented in a highly compressed and abstract form—if presented at all.

As a result, many assumptions go unremarked and are often unrecognized by users as well as consumers of statistics. Nonetheless, all statistical methods and interpretations are premised on the model assumptions; that is, on an assumption that the model provides a valid representation of the variation we would expect to see across data sets, faithfully reflecting the circumstances surrounding the study and phenomena occurring within it.

In most applications of statistical testing, one assumption in the model is a hypothesis that a particular effect has a specific size, and has been targeted for statistical analysis. This targeted assumption is called the study hypothesis or test hypothesis , and the statistical methods used to evaluate it are called statistical hypothesis tests.

Nonetheless, it is also possible to test other effect sizes. We may also test hypotheses that the effect does or does not fall within a specific range; for example, we may test the hypothesis that the effect is no greater than a particular amount, in which case the hypothesis is said to be a one - sided or dividing hypothesis [ 7 , 8 ]. Much statistical teaching and practice has developed a strong and unhealthy focus on the idea that the main aim of a study should be to test null hypotheses.

This exclusive focus on null hypotheses contributes to misunderstanding of tests. Adding to the misunderstanding is that many authors including R. A more refined goal of statistical analysis is to provide an evaluation of certainty or uncertainty regarding the size of an effect. The focus of traditional definitions of P values and statistical significance has been on null hypotheses, treating all other assumptions used to compute the P value as if they were known to be correct.

Recognizing that these other assumptions are often questionable if not unwarranted, we will adopt a more general view of the P value as a statistical summary of the compatibility between the observed data and what we would predict or expect to see if we knew the entire statistical model all the assumptions used to compute the P value were correct.

Specifically, the distance between the data and the model prediction is measured using a test statistic such as a t-statistic or a Chi squared statistic. The P value is then the probability that the chosen test statistic would have been at least as large as its observed value if every model assumption were correct, including the test hypothesis. This definition embodies a crucial point lost in traditional definitions: In logical terms, the P value tests all the assumptions about how the data were generated the entire model , not just the targeted hypothesis it is supposed to test such as a null hypothesis.

Furthermore, these assumptions include far more than what are traditionally presented as modeling or probability assumptions—they include assumptions about the conduct of the analysis, for example that intermediate analysis results were not used to determine which analyses would be presented.

It is true that the smaller the P value, the more unusual the data would be if every single assumption were correct; but a very small P value does not tell us which assumption is incorrect. For example, the P value may be very small because the targeted hypothesis is false; but it may instead or in addition be very small because the study protocols were violated, or because it was selected for presentation based on its small size.

Conversely, a large P value indicates only that the data are not unusual under the model, but does not imply that the model or any aspect of it such as the targeted hypothesis is correct; it may instead or in addition be large because again the study protocols were violated, or because it was selected for presentation based on its large size.

The general definition of a P value may help one to understand why statistical tests tell us much less than what many think they do: Not only does a P value not tell us whether the hypothesis targeted for testing is true or not; it says nothing specifically related to that hypothesis unless we can be completely assured that every other assumption used for its computation is correct—an assurance that is lacking in far too many studies.

Nonetheless, the P value can be viewed as a continuous measure of the compatibility between the data and the entire model used to compute it, ranging from 0 for complete incompatibility to 1 for perfect compatibility, and in this sense may be viewed as measuring the fit of the model to the data.

In contrast, the P value is a number computed from the data and thus an analysis result, unknown until it is computed. We can vary the test hypothesis while leaving other assumptions unchanged, to see how the P value differs across competing test hypotheses. Confidence intervals are examples of interval estimates. Hence, the specified confidence level is called the coverage probability. As Neyman stressed repeatedly, this coverage probability is a property of a long sequence of confidence intervals computed from valid models, rather than a property of any single confidence interval.

Many journals now require confidence intervals, but most textbooks and studies discuss P values only for the null hypothesis of no effect. This exclusive focus on null hypotheses in testing not only contributes to misunderstanding of tests and underappreciation of estimation, but also obscures the close relationship between P values and confidence intervals, as well as the weaknesses they share.

Much distortion arises from basic misunderstanding of what P values and their relatives such as confidence intervals do not tell us. Therefore, based on the articles in our reference list, we review prevalent P value misinterpretations as a way of moving toward defensible interpretations and presentations.

We adopt the format of Goodman [ 40 ] in providing a list of misinterpretations that can be used to critically evaluate conclusions offered by research reports and reviews. The P value assumes the test hypothesis is true—it is not a hypothesis probability and may be far from any reasonable probability for the test hypothesis. The P value simply indicates the degree to which the data conform to the pattern predicted by the test hypothesis and all the other assumptions used in the test the underlying statistical model.

The P value for the null hypothesis is the probability that chance alone produced the observed association; for example, if the P value for the null hypothesis is 0. This is a common variation of the first fallacy and it is just as false. To say that chance alone produced the observed association is logically equivalent to asserting that every assumption used to compute the P value is correct, including the null hypothesis.

Thus to claim that the null P value is the probability that chance alone produced the observed association is completely backwards: The P value is a probability computed assuming chance was operating alone.

The absurdity of the common backwards interpretation might be appreciated by pondering how the P value, which is a probability deduced from a set of assumptions the statistical model , can possibly refer to the probability of those assumptions. A small P value simply flags the data as being unusual if all the assumptions used to compute it including the test hypothesis were correct; it may be small because there was a large random error or because some assumption other than the test hypothesis was violated for example, the assumption that this P value was not selected for presentation because it was below 0.

A large P value only suggests that the data are not unusual if all the assumptions used to compute the P value including the test hypothesis were correct. The same data would also not be unusual under many other hypotheses. Furthermore, even if the test hypothesis is wrong, the P value may be large because it was inflated by a large random error or because of some other erroneous assumption for example, the assumption that this P value was not selected for presentation because it was above 0.

A large P value is evidence in favor of the test hypothesis. In fact, any P value less than 1 implies that the test hypothesis is not the hypothesis most compatible with the data, because any other hypothesis with a larger P value would be even more compatible with the data. A P value cannot be said to favor the test hypothesis except in relation to those hypotheses with smaller P values.

Furthermore, a large P value often indicates only that the data are incapable of discriminating among many competing hypotheses as would be seen immediately by examining the range of the confidence interval. A null -hypothesis P value greater than 0. If the null P value is less than 1 some association must be present in the data, and one must look at the point estimate to determine the effect size most compatible with the data under the assumed model.

Statistical significance indicates a scientifically or substantively important relation has been detected. Especially when a study is large, very minor effects or small assumption violations can lead to statistically significant tests of the null hypothesis. Again, a small null P value simply flags the data as being unusual if all the assumptions used to compute it including the null hypothesis were correct; but the way the data are unusual might be of no clinical interest.

One must look at the confidence interval to determine which effect sizes of scientific or other substantive e. Lack of statistical significance indicates that the effect size is small.

A large null P value simply flags the data as not being unusual if all the assumptions used to compute it including the test hypothesis were correct; but the same data will also not be unusual under many other models and hypotheses besides the null.

Again, one must look at the confidence interval to determine whether it includes effect sizes of importance. And again, the P value refers to a data frequency when all the assumptions used to compute it are correct. In addition to the test hypothesis, these assumptions include randomness in sampling, treatment assignment, loss, and missingness, as well as an assumption that the P value was not selected for presentation based on its size or some other aspect of the results.

To see why this description is false, suppose the test hypothesis is in fact true. It does not refer to your single use of the test, which may have been thrown off by assumption violations as well as random errors. This is yet another version of misinterpretation 1. P values are properly reported as inequalities e. This is bad practice because it makes it difficult or impossible for the reader to accurately interpret the statistical result. Only when the P value is very small e.

Statistical significance is a property of the phenomenon being studied, and thus statistical tests detect significance. The effect being tested either exists or does not exist. One should always use two-sided P values. Two-sided P values are designed to test hypotheses that the targeted effect measure equals a specific value e. When, however, the test hypothesis of scientific or practical interest is a one-sided dividing hypothesis, a one-sided P value is appropriate.

For example, consider the practical question of whether a new drug is at least as good as the standard drug for increasing survival time.

This question is one-sided, so testing this hypothesis calls for a one-sided P value. Nonetheless, because two-sided P values are the usual default, it will be important to note when and why a one-sided P value is being used instead.

The disputed claims deserve recognition if one wishes to avoid such controversy. For example, it has been argued that P values overstate evidence against test hypotheses, based on directly comparing P values against certain quantities likelihood ratios and Bayes factors that play a central role as evidence measures in Bayesian analysis [ 37 , 72 , 77 — 83 ].

Nonetheless, many other statisticians do not accept these quantities as gold standards, and instead point out that P values summarize crucial evidence needed to gauge the error rates of decisions based on statistical tests even though they are far from sufficient for making those decisions. See also Murtaugh [ 88 ] and its accompanying discussion. Some of the most severe distortions of the scientific literature produced by statistical testing involve erroneous comparison and synthesis of results from different studies or study subgroups.

Published on November 8, by Rebecca Bevans. Revised on February 15, Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is most often used by scientists to test specific predictions, called hypotheses, that arise from theories. Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps. Table of contents State your null and alternate hypothesis Collect data Perform a statistical test Decide whether the null hypothesis is supported or refuted Present your findings. After developing your initial research hypothesis the prediction that you want to investigate , it is important to restate it as a null H o and alternate H a hypothesis so that you can test it mathematically.

Give any two examples of collecting data from day-to-day life. Solution: A. Increase in population of … If marks obtained by students in a class test is given as per below: 55 36 95 73 60 42 25 78 75 A die is rolled, find the probability that an even number is obtained. Let us first write the … Two coins are tossed, find the probability that two heads are obtained.

The logic of hypothesis testing, as compared to jury trials page 3. This simple Here are some examples of the very widely used t test. The t test through of a significant change? SOLUTION: Let's examine the steps to a standard solution.

Test of a single population mean. H a tells you the test is left-tailed. The picture of the p -value is as follows:. Assume the p -value is 0.

Published on November 8, by Rebecca Bevans. Revised on February 15, Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics.

*Test of a single population mean.*

These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax. Some of the following statements refer to the null hypothesis, some to the alternate hypothesis. Over the past few decades, public health officials have examined the link between weight concerns and teen girls' smoking. Researchers surveyed a group of randomly selected teen girls living in Massachusetts between 12 and 15 years old.

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У них было много общего: настойчивость, увлеченность своим делом, ум. Иногда ей казалось, что Стратмор без нее пропадет; ее любовь к криптографии помогала коммандеру отвлечься от завихрений политики, напоминая о молодости, отданной взламыванию шифров. Но и она тоже многим была обязана Стратмору: он стал ее защитником в мире рвущихся к власти мужчин, помогал ей делать карьеру, оберегал ее и, как сам часто шутил, делал ее сны явью. Хотя и ненамеренно, именно Стратмор привел Дэвида Беккера в АНБ в тот памятный день, позвонив ему по телефону. Мысли Сьюзан перенеслись в прошлое, и глаза ее непроизвольно упали на листок бумаги возле клавиатуры с напечатанным на нем шутливым стишком, полученным по факсу: МНЕ ЯВНО НЕ ХВАТАЕТ ЛОСКА, ЗАТО МОЯ ЛЮБОВЬ БЕЗ ВОСКА. Дэвид прислал его после какой-то мелкой размолвки. Несколько месяцев она добивалась, чтобы он объяснил, что это значит, но Дэвид молчал.

Послышался голос с сильным немецким акцентом: - Ja. Беккер молчал. - Ja. Дверь слегка приоткрылась, и на него уставилось круглое немецкое лицо. Дэвид приветливо улыбнулся. Он не знал, как зовут этого человека. - Deutscher, ja.

Больше нечему. - Вирус. - Да, какой-то повторяющийся цикл. Что-то попало в процессор, создав заколдованный круг, и практически парализовало систему.

Но все доказательства к этому моменту будут уничтожены, и Стратмор сможет сказать, что не знает, о чем речь. Бесконечная работа компьютера. Невзламываемый шифр. Но это полный абсурд. Неужели Хейл никогда не слышал о принципе Бергофского.

Шаги быстро приближались. Беккер еще сильнее вцепился во внутреннюю часть проема и оттолкнулся ногами. Тело налилось свинцовой тяжестью, словно кто-то изо всех сил тянул его. Беккер, стараясь преодолеть эту тяжесть, приподнялся на локтях. Теперь он был на виду, его голова торчала из оконного проема как на гильотине.

*Скорее всего Хейл держит там копию ключа.*

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Calfucir L. 01.06.2021 at 20:53testing hypothesis, test statistic, P-value. Text Book: Basic Concepts and what we believe is true if our sample data cause us Solution. 1-Data: variable is age, n=10, =27,σ2=20,α= 2-Assumptions: the population is approximately.

Steven H. 03.06.2021 at 19:32Statistical Test – uses the data obtained from a sample to make a decision about whether the null hypothesis should be rejected. Test Value (test statistic) – the.