On a modification of the hoeffding-blum-kiefer-rosenblatt independence criterion
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The test statistic of Hoeffding-Blum-Kiefer-Rosenblatt for testing the hypothesis of independenceis modified so as to get a procedure which is invariant with respect to strictly monotone transformations of the components. It is shown by simulation that the limiting null distribution providesanapproximation which is more accurateKeywords:
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Taking advantage of the possibility of fuzzy test statistic falling in the rejection region, a statistical hypothesis testing approach for fuzzy data is proposed in this study. In contrast to classical statistical testing, which yields a binary decision to reject or to accept a null hypothesis, the proposed approach is to determine the possibility of accepting a null hypothesis (or alternative hypothesis). When data are crisp, the proposed approach reduces to the classical hypothesis testing approach.
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Abstract Null hypothesis significance testing (NHST) is an inferential statistical method for deciding whether a well‐specified hypothesis, identified as the null hypothesis , is to be regarded as true for a population from which a given set of data has been obtained by random sampling. In the usual procedure the data from a particular dependent (i.e., measured) variable are first summarized by a single number called a test statistic. Usually, some assumptions must then be made in order to find the relative likelihoods of all possible values of that test statistic when the null hypothesis is true, and thus to find the null hypothesis distribution (NHD). The next step is to calculate the probability of obtaining one's actual test statistic from the NHD, or one that is even further from the mean of the NHD. From what is called the “frequentist” point of view, that probability, called a p value , tells us the proportion of times that the NHD would yield a test statistic at least as inconsistent with the null hypothesis as the one you obtained, over many exact replications of your study. In the accept‐support (AS) form of NHST, researchers actually want to obtain a p value that is close to its maximum of 1.0, because the null hypothesis being tested is consistent with the theory that motivated the study.
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Hypothesis testing is a statistical method used to evaluate the validity of a potential outcome within a defined significance level, comparing it with an alternative hypothesis. It involves establishing null and alternative hypotheses, making assumptions, calculating a test statistic, and selecting a significance level. The decision to accept or reject the null hypothesis is based on the observed test statistic. Terminologies include null and alternative hypotheses, critical region, critical value, errors, p-value, power of a test, and more. Hypothesis testing is crucial in statistical inference and offers insights into relationships between variables. Practical applications range from courtroom trials to gender ratio analysis and behavioral effects studies. However, it's important to note the potential limitations and biases in the application of hypothesis testing.
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Factors Determining the Power of a Statistical Test for the Difference between Means and Proportions
One of the most essential issues in research problems design is statistical power of a test is. The problem motivating this topic is to identify the factors and relationships among the components of power analysis for a study. In this paper, we presented testing procedures of hypothesis for means and proportions in different sample situations. Hypothesis testing requires several stages, including specifying the null and alternative or research hypothesis, selecting and computing an appropriate test statistic, setting up a decision rule to reach a conclusion. Some related concepts such as sample size and confidence intervals were demonstrated, and illustrations on theoretical data would be carried. Results and conclusions on the basis of the discussions reflected the relationship among power analysis components and factors that influence the statistical power of a test would be shown.
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Summary To perform a test of significance of a null hypothesis, a test statistic is chosen which is expected to be small if the hypothesis is false. Then the significance level of the test for an observed sample is the probability that the test statistic, under the assumptions of the hypothesis, is as small, or smaller than, its observed value. A “good” test statistic is taken to be one which is stochastically small when the null hypothesis is false. Optimal test statistics are defined using this criterion and the relationship of these methods to the Neyman‐Pearson theory of hypothesis testing is considered.
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Learning about statistics is a lot like learning about science: the learning is more meaningful if you can actively explore. This second installment of Explorations in Statistics delves into test statistics and P values, two concepts fundamental to the test of a scientific null hypothesis. The essence of a test statistic is that it compares what we observe in the experiment to what we expect to see if the null hypothesis is true. The P value associated with the magnitude of that test statistic answers this question: if the null hypothesis is true, what proportion of possible values of the test statistic are at least as extreme as the one I got? Although statisticians continue to stress the limitations of hypothesis tests, there are two realities we must acknowledge: hypothesis tests are ingrained within science, and the simple test of a null hypothesis can be useful. As a result, it behooves us to explore the notions of hypothesis tests, test statistics, and P values.
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