One-sided tolerance limit

Table 4.5 Selected values for f-distribution with v degrees of freedom for a range of one-sided tolerance limit...

Now the reduction for the 99.99 % probability of survival has grown to 10 % from 241.7 MPa down to 216 MPa. As last, the designer wants to know how far or close these 140 samples are from the population size to use the finite population correction (FPC) given by Eq. 4.9. To that purpose he calculates the overall volume of all the work pieces that must be built with that steel to assess N. The overall volume is equal to 2.2 x 10 mm. Since the volume of the traction specimen used is equal to 7,854 mm the number N of specimens of the entire population is N = 2.2 x 10 /7.854 = 280,112. He concludes that 140 specimens represent only a mere 0.05 % of the entire population size and he cannot apply any correction. This simple example evidences the difference that may arise when using a sample of limited size. Statistics results depend on the sample size. We may try, then, to use the Lieberman one-sided tolerance limit. The mean value of the logarithms of experimental data is og au) = 2.717 and 5 = 0.02175 from Table 4.5 we infer that the value of k for a sample of size 140 and a probability of survival of 99.95 % is k = 3.356 therefore ... 

The sampling variance of the material determined at a certain mass and the number of repetitive analyses can be used for the calculation of a sampling constant, K, a homogeneity factor, Hg or a statistical tolerance interval (m A) which will cover at least a 95 % probability at a probability level of r - a = 0.95 to obtain the expected result in the certified range (Pauwels et al. 1994). The value of A is computed as A = k 2R-s, a multiple of Rj, where is the standard deviation of the homogeneity determination,. The value of fe 2 depends on the number of measurements, n, the proportion, P, of the total population to be covered (95 %) and the probability level i - a (0.95). These factors for two-sided tolerance limits for normal distribution fe 2 can be found in various statistical textbooks (Owen 1962). The overall standard deviation S = (s/s/n) as determined from a series of replicate samples of approximately equal masses is composed of the analytical error, R , and an error due to sample inhomogeneity, Rj. As the variances are additive, one can write (Equation 4.2) ... 

To calculate tolerance limits, two values must first be specified C the proportion of the population to be covered (the "coverage") and P the confidence coefficient. For given values of C and P, one or two sided tolerance limits for the population can be calculated. 

During the past year, chloroperoxidase (CPO) was found to catalyze the smooth asymmetric epoxidation of functionalized cii-alkenes, such as the unsaturated ester 32. The reaction appears to be limited to 2-alkenes (i.e., methyl group on one side of the alkene), although some branching on the longer alkyl chain is tolerated. Allylic alcohols are oxidized to the corresponding unsaturated aldehydes but without epoxide formation <99TL1641>. 

Systemic Effects. Although acetazolamide is effective as an ocular hypotensive agent, a significant number of side effects limit its clinical usefulness (Box 10-5). Maximal doses of CAIs produce intolerable effects in 30% to 80% of patients. The incidence of side effects varies with the dose and the fitrmulation however, when all side effects are considered, the incidence probably approaches 100% in patients taking either acetazolamide tablets or the 500-mg sustained-release capsules. One study demonstrated that only 26% of patients could tolerate acetazolamide tablets beyond 6 weeks, whereas 58% of patients could tolerate prolonged use of the sustained-release formulation. 

The second and preferred method is to apply appropriate statistical analysis to the dataset, based on linear regression. Both EU and USFDA authorities assume log-linear decline of residue concentrations and apply least-squares regression to derive the fitted depletion line. Then the one-sided upper tolerance limit (95% in EU and 99% in USA) with a 95% confidence level is computed. The WhT is the time when this upper one-sided 95% tolerance limit for the residue is below the MRL with 95% confidence. In other words, this definition of the WhT says that at least 95% of the population in EU (or 99% in USA) is covered in an average of 95% of cases. It should be stressed that the nominal statistical risk that is fixed by regulatory authorities should be viewed as a statistical protection of farmers who actually observe the WhT and not a supplementary safety factor to protect the consumer even if consumers indirectly benefit from this rather conservative statistical approach. 

Lay-up and spray-up. The largest of parts (mine sweeper hulls) can be made by lay-up and spray-up. However, the ability to create fine detail is limited and close tolerances are not possible. Machining is necessary to create holes and trim the parts. The construction is laminated with polyester with glass reinforcement, and it is the reinforcement application method which defines the name of the process. Open molds are used therefore, one side of each part is rough and unfinished. Mold costs are low however, they only last for a small number of parts. Thus many molds would be required for high volumes, although the pattern needs to be made only once. Since these are very slow processes, piece... 

Table I presents a summary of the results of the KENO-IV Monte Cario nuclear criticality calculations for the facilities and accidents considered. It. should be noted that the KENO-IV results given in Table I must be adjusted upward by adding the method uncertainty (U) (Reb. 5, 6). U is obtained using the formula U = [(K S ) + (KjSj) , where K is the one-sided 95-95 tolerance limit factor of the benchmaiik criticality calculations, S is the standard deviation of the benchmark calculations, Kj ( I -7) is the onesided 95-95 tolerance limit factor for the inidividual KENO calculations performed in this study, and S2 is the standard demtion of the individual KENO.calculation. U is thus composed of two parts. The first (K S ) is due to the spread in the individual values of keff calculated for the benchmark criticals. The seepnd (KiS is due to the statistics of the specific KENO calculation, wtuchi results from the fact that less than an infinite sample of neutron historic is calcula. There is a 95% probability with 95% confidence that the true k-effective will be less than or equal to the adjusted value given in Table I.

The mean conditional probability that an I C cable in the fire compartment is damaged during the fire scenario was estimated to be 1.76 E-02 based on failure criterion 1 and 4.10 E-03 based on criterion 2. Figure 5 shows the epistemic uncertainty of the cable damage probability for each criterion. It can be seen that the epistemic probability distributions due to both criteria differ significantly. Since just a sample of 60 values is available from the simulations of MCDET and FDS, the one-sided upper (95%, 95%) tolerance limit was calculated to quantify the epistemic uncertainty of the cable damage probability (see section 3.2). With failure criterion 1, the (95%, 95%) tolerance limit is 0.5. The (95%, 95%) tolerance limit based on criterion 2 is 1.53 E-02. 

Lieberman, G.J. Tables for one-sided statistical tolerance limits. Indust. Qual. Contr. 14(10), 7-9 (1958)... 

These two definitions reflect two sides of the same situation. In this book, the term critical effect(s) will be used for the hazard/effect considered as being the essential one(s) for the purpose of the risk characterization, e.g., for the establishment of a health-based guidance value, permissible exposure level, or Reference Dose. It should be noted that the critical effect could be a local as well as a systemic effect. It should also be recognized that the critical effect for the establishment of a tolerable exposure level is not necessarily the most severe effect of the chemical substance. For example, although a substance may cause a serious effect such as liver necrosis, the critical effect for the establishment of, e.g., an occupational exposure limit could be a less serious effect such as respiratory tract irritation, because the irritation occurs at a lower exposure level. 

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