Statistical hypotheses

In order to detect the presence of gross errors in the proposed measurement and constraint models, the strategy SEGE applies a collective hypothesis statistical... 

Hypothesis (Statistical) a statistical hypothesis is an hypothesis concerning the parameter or form of a probabilistic mechanism which is assumed to generate the observations ... 

The factorial design data were analyzed by use of a linear hypothesis statistical model (20). By this it is assumed that the conversion of carbon monoxide is dependent on all of the factors to the first power only. The following empirical regression equation for the percent of carbon monoxide converted resulted from this model ... 

The null hypothesis (statistical terminology), states that if there are no significant differences in the variances, then the ratio must be close to 1. Reference should therefore be made to the Fisher-Snedecor values of F, established for a variable number of observations (Table 22.3). If the calculated value for F exceeds that found in the table, the means are considered to be significantly different. Since the variability is greater than si, then the second series of measurements is therefore the more precise one. 

To show that the impact of changes in rotations frequency, the mass of feed and balls on the modified coefficients of flattening and asymmetry on mean grain size (m ) can be omitted, it was assumed that irrespective of the change of process parameters, the series of experimental data could be treated as if they had belonged to one population. In order to verify this hypothesis statistical tests were carried out with the aim to confirm or reject the hypothesis. 

In attempting to reach decisions, it is useful to make assumptions or guesses about the populations involved. Such assumptions, which may or may not be true, are called statistical hypotheses and in general are statements about the probability distributions of the populations. A common procedure is to set up a null hypothesis, denoted by which states that there is no significant difference between two sets of data or that a variable exerts no significant effect. Any hypothesis which differs from a null hypothesis is called an alternative hypothesis, denoted by Tfj. 

Next, an equation for a test statistic is written, and the test statistic s critical value is found from an appropriate table. This critical value defines the breakpoint between values of the test statistic for which the null hypothesis will be retained or rejected. The test statistic is calculated from the data, compared with the critical value, and the null hypothesis is either rejected or retained. Finally, the result of the significance test is used to answer the original question. 

The test statistic for evaluating the null hypothesis is called an f-test, and is given as either... 

The variance for the sample of ten tablets is 4.3. A two-tailed significance test is used since the measurement process is considered out of statistical control if the sample s variance is either too good or too poor. The null hypothesis and alternative hypotheses are... 

Documentation of experimental method so that work can be reproduced at a later time Appropriate data handling statistical methods conclusions based on fact, supportable by data Define and execute critical experiments to prove or disprove hypothesis Mechanistic or fundamental interpretation of data preferred Communication of Conclusions to Incorporate Technical Learning in Organization Experimental W rk Done in Support of New or Existing Processes Should be Captured in Process Models... 

Jui y trials represent a form of decision making. In statistics, an analogous procedure for making decisions falls into an area of statistical inference called hypothesis testing. 

If the null hypothesis is assumed to be true, say, in the case of a two-sided test, form 1, then the distribution of the test statistic t is known. Given a random sample, one can predict how far its sample value of t might be expected to deviate from zero (the midvalue of t) by chance alone. If the sample value oft does, in fact, deviate too far from zero, then this is defined to be sufficient evidence to refute the assumption of the null hypothesis. It is consequently rejected, and the converse or alternative hypothesis is accepted. 

The procedure for testing the significance of a sample proportion follows that for a sample mean. In this case, however, owing to the nature of the problem the appropriate test statistic is Z. This follows from the fact that the null hypothesis requires the specification of the goal or reference quantity po, and since the distribution is a binomial proportion, the associated variance is [pdl — po)]n under the null hypothesis. The primary requirement is that the sample size n satisfy normal approximation criteria for a binomial proportion, roughly np > 5 and n(l — p) > 5. 

I. Under the null hypothesis, it is assumed that the respective two samples have come from populations with equal proportions pi = po. Under this hypothesis, the sampling distribution of the corresponding Z statistic is known. On the basis of the observed data, if the resultant sample value of Z represents an unusual outcome, that is, if it falls within the critical region, this would cast doubt on the assumption of equal proportions. Therefore, it will have been demonstrated statistically that the population proportions are in fact not equal. The various hypotheses can be stated ... 

Suppose we have two methods of preparing some product and we wish to see which treatment is best. When there are only two treatments, then the sampling analysis discussed in the section Two-Population Test of Hypothesis for Means can be used to deduce if the means of the two treatments differ significantly. When there are more treatments, the analysis is more detailed. Suppose the experimental results are arranged as shown in the table several measurements for each treatment. The goal is to see if the treatments differ significantly from each other that is, whether their means are different when the samples have the same variance. The hypothesis is that the treatments are all the same, and the null hypothesis is that they are different. The statistical validity of the hypothesis is determined by an analysis of variance. 

The goal of any statistical analysis is inference concerning whether on the basis of available data, some hypothesis about the natural world is true. The hypothesis may consist of the value of some parameter or parameters, such as a physical constant or the exact proportion of an allelic variant in a human population, or the hypothesis may be a qualitative statement, such as This protein adopts an a/p barrel fold or I am currently in Philadelphia. The parameters or hypothesis can be unobservable or as yet unobserved. How the data arise from the parameters is called the model for the system under study and may include estimates of experimental error as well as our best understanding of the physical process of the system. 

In frequentist statistics, probability is instead a long-run relative occurrence of some event out of an infinite number of repeated trials, where the event is a possible outcome of the trial. A hypothesis or parameter that expresses a state of nature cannot have a probability in frequentist statistics, because after an infinite number of experiments there can be no uncertainty in the parameter left. A hypothesis or parameter value is either... 

Because the datay are random, the statistics based on y, S(y), are also random. For all possible data y (usually simulated) that can be predicted from H, calculate p(S(ysim) H), the probability distribution of the statistic S on simulated data y ii given the truth of the hypothesis H. If H is the statement that 6 = 0, then y i might be generated by averaging samples of size N (a characteristic of the actual data) with variance G- = G- (yacmai) (yet another characteristic of the data). 

Note that the bar above y m y m this section denotes the average of y. A bar over a statement or hypothesis A in the previous section was used to denote not-A. Both of these are standard notations in statistics and probability theory, respectively. 

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