Reliability factor, confidence intervals

So the reliability factor for interval estimates will come from the Z distribution. Then a two-sided (1 - a)% confidence interval for the difference in sample proportions, p - p is ... 

Once the reliability of a replicate set of measurements has been established the mean of the set may be computed as a measure of the true mean. Unless an infinite number of measurements is made this true mean will always remain unknown. However, the t-factor may be used to calculate a confidence interval about the experimental mean, within which there is a known (90%) confidence of finding the true mean. The limits of this confidence interval are given by ... 

Vertzoni et al. (30) recently clarified the applicability of the similarity factor, the difference factor, and the Rescigno index in the comparison of cumulative data sets. Although all these indices should be used with caution (because inclusion of too many data points in the plateau region will lead to the outcome that the profiles are more similar and because the cutoff time per percentage dissolved is empirically chosen and not based on theory), all can be useful for comparing two cumulative data sets. When the measurement error is low, i.e., the data have low variability, mean profiles can be used and any one of these indices could be used. Selection depends on the nature of the difference one wishes to estimate and the existence of a reference data set. When data are more variable, index evaluation must be done on a confidence interval basis and selection of the appropriate index, depends on the number of the replications per data set in addition to the type of difference one wishes to estimate. When a large number of replications per data set are available (e.g., 12), construction of nonparametric or bootstrap confidence intervals of the similarity factor appears to be the most reliable of the three methods, provided that the plateau level is 100. With a restricted number of replications per data set (e.g., three), any of the three indices can be used, provided either non-parametric or bootstrap confidence intervals are determined (30). 

We have already identified the factors that govern the reliability of a sample - variability within the data and the sample size (Chapter 4). It is these same factors that influence the width of a 95 per cent confidence interval... 

It is virtually impossible to detect all of the voids by taking photographs therefore, a few samples have been prepared and presented here. With a confidence interval of 95%, 10 photos for each section have been taken. For a good balance of accuracy and reliability, 150X magnification factor was foimd to be a good zoom factor. To measure the damage induced in the specimen, the area of voids should be calculated and divided by whole area of the photo. 

For a sample size of 100 (99 df) the reliability factors for two-sided 90%, 95%, and 99% confidence intervals are 1.66, 1.98, and 2.63. The implication of these three values is that, all other things being equal (that is, x, s, and ri), requiring greater confidence in the interval estimate results in wider interval estimates. The more confidence that is required, the less reliable is the single sample estimate, and therefore greater numerical uncertainty is expressed in the interval estimate. This very important point is illustrated in the following example. 

With these numbers calculated, all that is left to compute the three confidence intervals are the reliability factors associated with each. For the 90% confidence interval, the value of the reliability factor will be the value of t that cuts off the upper 5% of the area (half the size of a) under the t distribution with 99 df. This value is 1.66 and can be verified from a table of values or from statistical software. Note that the t value of -1.66 is the value of t that cuts off the lower 5% of the area (half of the size of a) under the t distribution with 99 df. The reliability factors listed previously for the two-sided 95% and 99% confidence intervals can also be used to compute the following interval estimates ... 

To summarize, the computational aspects of confidence intervals involve a point estimate of the population parameter, some error attributed to sampling, and the amount of confidence (or reliability) required for interpretation. We have illustrated the general framework of the computation of confidence intervals using the case of the population mean. It is important to emphasize that interval estimates for other parameters of interest will require different reliability factors because these depend on the sampling distribution of the estimator itself and different calculations of standard errors. The calculated confidence interval has a statistical interpretation based on a probability statement. 

In Chapter 6 we described the basic components of hypothesis testing and interval estimation (that is, confidence intervals). One of the basic components of interval estimation is the standard error of the estimator, which quantifies how much the sample estimate would vary from sample to sample if (totally implausibly) we were to conduct the same clinical study over and over again. The larger the sample size in the trial, the smaller the standard error. Another component of an interval estimate is the reliability factor, which acts as a multiplier for the standard error. The more confidence that we require, the larger the reliability factor (multiplier). The reliability factor is determined by the shape of the sampling distribution of the statistic of interest and is the value that defines an area under the curve of (1 - a). In the case of a two-sided interval the reliability factor defines lower and upper tail areas of size a/2. 

The statistical analysis approach is to calculate 95% confidence intervals for the proportion of participants in each group (placebo and combined active) reporting a headache. This analysis approach is reasonable because the sample size is sufficiently large (that is, the values, pn, in each group are at least five). Satisfying this assumption enables us to use the Z distribution for the reliability factor. 

The third component of the interval estimate is the reliability factor. As we are calculating a two-sided 95% confidence interval, we select the value of Z from Table 8.3 corresponding to a of 0.05, that is, 1.96. 

As we have chosen a confidence level of 95%, the corresponding value of Z (the reliability factor) is 1.96. Finally, the 95% confidence interval is calculated as follows ... 

The reliability of the result is determined in ISO 1973 by calculating the 95% confidence interval of the mean and expressing this as a percentage of the mean value. If this is greater than 2%, the number of bundles tested is increased. The confidence interval is calculated using Student s t factor taken from statistical tables, but of course it is much easier if one simply uses the tables and equation in ISO 2602. 

Using the sensitivity factors from Figure 4 and the characteristics from Table 2, the standard deviation of the reliability index can be calculated according to equations (9) and (14). To illustrate the results, the dotted lines in Figure 3 show the interval [pg - Og Pg+ Og]. This interval contains approximately 70% probability. As the sensitivity factors remain approximately constant in time, also the width of this confidence interval remains approximately constant. The coefficient of variation Og/pg varies from 0.110 at t = 0 to 0.108 at t = 50. 

The advantage of our method is clearly illustrated in the decrease of the confidence interval. This illustration provides a better integration of the Eurocode 2 in the reliability decision of structures by adding the corrosion factor the failure can be then defined as a level of risk related to changes in the performance of the structure for a given interval of time. 

Uncertainties linked to the parameters - for various reasons, the available information on dependability is uncertain a small sample leading to a wide confidence interval, extrapolation of data from one installation to another, etc. Certain other parameters (delayed appearance of physical factors, time available after losing a system before undesirable effects ensue, etc.) connected with design or operation are also with uncertainties. Dependability is defined as the ability of an entity to perform one or several required functions under given conditions. This concept can encompass reliability, availability, maintainability, safety, durability, etc - or combinations of these abilities. Generally speaking, dependability is considered to be the science of failures and faults. 

To end up this Section it is convenient to spend a few more lines to firmly dispel any impression that the problem is still unresolved, and that the cavities in use are not reliable. Fortunately, the situation is different our early proposal of using an universal factor A = 1.2 for all atoms in aqueous solutions gives errors on the average below 1 kcal/mol. Further refinements, as those we have mentioned, reduce the error to lower bounds. Passing to other solvents (which are less sensitive than water to the specific A (A) values), A ranges in the interval 1.15-1.5 (with few exceptions). The recommended values reported in the literature can be used with confidence. 

The first target is interpreted to mean that for, say SOO years accumulated operation of a PLC, it is tolerable for a design error to result in one failure. The period of 500 years is somewhat arbitrary, but is chosen such that this type of failure could be claimed not to dominate system unreliability. It is therefore claimed that 5000 years accumulated experience should provide a sufficient basis to claim an tqrpropriale level of reliability. No specific justification for this is provided except that the period for which experience is required is some 10 times the target MTBF. This factor is judged to be appropriate to give some confidence in the claim (if the failures occurred at random intervals, such a period would lead to betto than 85% confidence) and to make some allowance for the lack of maturity of software reliability modelling. 

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