Hypothesis example

This chapter addresses methods and tools used successfully to identify multiple root causes. Process safety incidents are usually the result of more than one root cause. This chapter provides a structured approach for determining root causes. It details some powerful, widely used tools and techniques available to incident investigation teams including timelines, logic trees, predefined trees, checklists, and fact/hypothesis. Examples are included to demonstrate how they apply to the types of incidents readers are likely to encounter. 

Identify the test statistic and associated probability distribution appropriate to the hypothesis. Examples ... 

The hypothesis example above is very general and the testability could be improved by making it more specific, such as Transthoracic bioimpedance is lower when measured by gel electrodes than measured by textile electrodes using a two-electrode setup if this is the relevant setup we want to test. It is easier to test this hypothesis because it implies only one certain type of measurement, and reduces the chance of an inconclusive result. A... 

Contrary to the impression that one might have from a traditional course in introductory calculus, well-behaved functions that cannot be integrated in closed form are not rare mathematical curiosities. Examples are the Gaussian or standard error function and the related function that gives the distribution of molecular or atomic speeds in spherical polar coordinates. The famous blackbody radiation cuiwe, which inspired Planck s quantum hypothesis, is not integrable in closed form over an arbitiar y inteiwal. 

This genera] scheme could be used to explain hydrogen exchange in the 5-position, providing a new alternative for the reaction (466). This leads us also to ask whether some reactions described as typically electrophilic cannot also be rationalized by a preliminary hydration of the C2=N bond. The nitration reaction of 2-dialkylaminothiazoles could occur, for example, on the enamine-like intermediate (229) (Scheme 141). This scheme would explain why alkyl groups on the exocyclic nitrogen may drastically change the reaction pathway (see Section rV.l.A). Kinetic studies and careful analysis of by-products would enable a check of this hypothesis. 

The larger variance is placed in the numerator. For example, the F test allows judgment regarding the existence of a significant difference in the precision between two sets of data or between two analysts. The hypothesis assumed is that both variances are indeed alike and a measure of the same a. 

As applied in Example 12, the F test was one-tailed. The F test may also be applied as a two-tailed test in which the alternative to the null hypothesis is erj A cr. This doubles the probability that the null hypothesis is invalid and has the effect of changing the confidence level, in the above example, from 95% to 90%. 

The most direct test of the tensile strength hypothesis would be to compare the value of Tq calculated from the closure point of the isotherm by Equation (3.61) with the tensile strength of the bulk liquid determined directly. Unfortunately, experimental measurement of the tensile strength is extremely difficult because of the part played by adventitious factors such as the presence of solid particles and dissolved gases, so that the values in the literature vary widely (between 9 and 270 bar for water at 298 K, for example). 

The difference between retaining a null hypothesis and proving the null hypothesis is important. To appreciate this point, let us return to our example on determining the mass of a penny. After looking at the data in Table 4.12, you might pose the following null and alternative hypotheses... 

Examples of (a) two-tailed, (b) and (c) one-tailed, significance tests. The shaded areas in each curve represent the values for which the null hypothesis is rejected. 

Since significance tests are based on probabilities, their interpretation is naturally subject to error. As we have already seen, significance tests are carried out at a significance level, a, that defines the probability of rejecting a null hypothesis that is true. For example, when a significance test is conducted at a = 0.05, there is a 5% probability that the null hypothesis will be incorrectly rejected. This is known as a type 1 error, and its risk is always equivalent to a. Type 1 errors in two-tailed and one-tailed significance tests are represented by the shaded areas under the probability distribution curves in Figure 4.10. 

If evidence for a determinate error is found, as in Example 4.16, its source should be identified and corrected before analyzing additional samples. Failing to reject the null hypothesis, however, does not imply that the method is accurate, but only indicates that there is insufficient evidence to prove the method inaccurate at the stated confidence level. 

Significance tests, however, also are subject to type 2 errors in which the null hypothesis is falsely retained. Consider, for example, the situation shown in Figure 4.12b, where S is exactly equal to (Sa)dl. In this case the probability of a type 2 error is 50% since half of the signals arising from the sample s population fall below the detection limit. Thus, there is only a 50 50 probability that an analyte at the lUPAC detection limit will be detected. As defined, the lUPAC definition for the detection limit only indicates the smallest signal for which we can say, at a significance level of a, that an analyte is present in the sample. Failing to detect the analyte, however, does not imply that it is not present. 

These models are usually categorized according to the number of supplementary partial differential transport equations which must be solved to supply the modeling parameters. The so-called zero-equation models do not use any differential equation to describe the turbulent quantities. The best known example is the Prandtl (19) mixing length hypothesis ... 

Correlations of nucleation rates with crystallizer variables have been developed for a variety of systems. Although the correlations are empirical, a mechanistic hypothesis regarding nucleation can be helpful in selecting operating variables for inclusion in the model. Two examples are (/) the effect of slurry circulation rate on nucleation has been used to develop a correlation for nucleation rate based on the tip speed of the impeller (16) and (2) the scaleup of nucleation kinetics for sodium chloride crystalliza tion provided an analysis of the role of mixing and mixer characteristics in contact nucleation (17). Pubhshed kinetic correlations have been reviewed through about 1979 (18). In a later section on population balances, simple power-law expressions are used to correlate nucleation rate data and describe the effect of nucleation on crystal size distribution. 

The Minitran system, by 3M Health Care, is a monolithic transdermal system that deUvers nitroglycerin at a continuous rate of 0.03 mg/(cm h) (81). The dmg flux through the skin is higher than the previous two systems thus the Minitran system is a smaller size for equivalent dosing. For example, the 0.1 mg/h dose is achieved with a 3.3 cm system rather than the 5 cm systems of Transderm-Nitro or Nitro-dur. Because the skin is rate-controlling in a monolithic system and the Minitran flux is higher than the similar monolithic Nitro-dur system flux, it appears that 3M Health Care has included an additive to increase the skin flux to 0.03 mg/(cm h). Whereas this information is not apparent in Reference 81, patent information supports the hypothesis (96). 

Under the null hypothesis, it is assumed that the sample came from a population with a proportion po of items having the specified attribute. For example, in tossing a coin the population could be thought of as having an unbounded number of potential tosses. If it is assumed that the coin is fair, this would dictate po = 1/2 for the proportional number of heads in the population. The null hypothesis can take one of three forms ... 

Probability in Bayesian inference is interpreted as the degree of belief in the truth of a statement. The belief must be predicated on whatever knowledge of the system we possess. That is, probability is always conditional, p(X l), where X is a hypothesis, a statement, the result of an experiment, etc., and I is any information we have on the system. Bayesian probability statements are constructed to be consistent with common sense. This can often be expressed in tenns of a fair bet. As an example, I might say that the probability that it will rain tomorrow is 75%. This can be expressed as a bet I will bet 3 that it will rain tomorrow, if you give me 4 if it does and nothing if it does not. (If I bet 3 on 4 such days, I have spent 12 I expect to win back 4 on 3 of those days, or 12). 

A striking example of the importance of narrowing the focus in research, which is what the concept of the parepisteme really implies, is the episode (retailed in Chapter 3, Section 3.1.1) of Eilhard Mitscherlich s research, in 1818, on the crystal forms of potassium phosphate and potassium arsenate, which led him, quite unexpectedly, to the discovery of isomorphism in crystal species and that, in turn, provided heavyweight evidence in favour of the then disputed atomic hypothesis. As so often happens, the general insight comes from the highly specific observation. 

In the arithmetical methods a circular flow cross-section is divided into concentric rings and a central element. The areas of the elements are equal except for the outermost ring, which has only half of that area, A hypothesis is made for the velocity profile for each element. For example the log-linear rule assumes a velocity profile of... 

This procedure constitutes an application of the steady-state approximation [also called the quasi-steady-state approximation, the Bodenstein approximation, or the stationary-state hypothesis]. It is a powerful method for the simplification of complicated rate equations, but because it is an approximation, it is not always valid. Sometimes the inapplicability of the steady-state approximation is easily detected for example, Eq. (3-143) predicts simple first-order behavior, and significant deviation from this behavior is evidence that the approximation cannot be applied. In more complex systems the validity of the steady-state approximation may be difficult to assess. Because it is an approximation in wide use, much critical attention has been directed to the steady-state hypothesis. 

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