Confidence interval cover

A confidence interval is said to cover a value at the given confidence value a, if the given value lies within the given confidence bounds. Consider the confidence interval shown in Fig. 2.8. Points b and c are inside the confidence interval, and hence it would be said that the confidence interval covers the given values. On the other hand, points a and d are outside the confidence interval, which implies that the points are not covered by the confidence interval. A point inside the confidence interval can be considered to be equal to the value expressed by the confidence interval. For example, if the true value is 5 and the estimated value is 6 3 (95% confidence interval), then it can be concluded that the estimated value covers the true value, and hence it is likely that the values are the same. On the other hand, if we had 10 2 (95% confidence interval) and the same trae value, then we can conclude that the true value and the estimated value are different. 

The probabilistic nature of a confidence interval provides an opportunity to ask and answer questions comparing a sample s mean or variance to either the accepted values for its population or similar values obtained for other samples. For example, confidence intervals can be used to answer questions such as Does a newly developed method for the analysis of cholesterol in blood give results that are significantly different from those obtained when using a standard method or Is there a significant variation in the chemical composition of rainwater collected at different sites downwind from a coalburning utility plant In this section we introduce a general approach to the statistical analysis of data. Specific statistical methods of analysis are covered in Section 4F. 

The frequency interpretation of the interval estimates on the unknown amounts is given by the following ( 27 ) With at least 1- a confidence, based on the sampling characteristics of the observations on the standards, at least P proportion of the interval estimates made from a particular calibration will contain the true amounts. The Bonferroni inequality insures the 1-a confidence since the confidence interval about the regression line and the upper bound on cr are each performed using a 1- a/2 confidence coefficient. Hence, the frequency interpretation states that at least (1-a) proportion of the standard calibrations are such that at least P proportion of the intervals produced by the method cover the true unknown amounts. For the remaining a proportion of standard calibrations the proportion of intervals which cover the true unknown values may be less than P. 

It is not possible at this stage to say precisely what we mean by small and large in this context, we need the concept of the confidence interval to be able to say more in this regard and we will cover this topic in the next chapter. For the moment just look upon the standard error as an informal measure of precision high values mean low precision and vice versa. Further if the standard error is small, it is likely that our estimate x is close to the true mean, p,. If the standard error is large, however, there is no guarantee that we will be close to the true mean. 

As discussed in the previous section the standard error simply provides indirect information about reliability, it is not something we can use in any specific way, as yet, to tell us where the truth lies. We also have no way of saying what is large and what is small in standard error terms. We will, however, in the next chapter cover the concept of the confidence interval and we will see how this provides a methodology for making use of the standard error to enable us to make statements about where we think the true (population) value lies. 

The ICH recommends that repeatability be assessed using a minimum of nine determinations covering the specified range for the procedure (e.g., three concentrations/three replicates as in the accuracy experiment) or using a minimum of six determinations at 100% of the test concentration. Reporting of the standard deviation, relative standard deviation (coefficient of variation), and confidence interval is required. The assay values are independent analyses of samples that have been carried through the complete analytical procedure from sample preparation to final test result. Table 1 provides an example set of repeatability data. 

The early chapters (1-5) are fairly basic. They cover data description (mean, median, mode, standard deviation and quartile values) and introduce the problem of describing uncertainty due to sampling error (SEM and 95 per cent confidence interval for the mean). In theory, much of this should be familiar from secondary education, but in the author s experience, the reality is that many new students cannot (for example) calculate the median for a small data set. These chapters are therefore relevant to level 1 students, for either teaching or revision purposes. 

Estimates of this type are called tolerance intervals. Such intervals are in a form of x Ks(x) similar to the previously described confidence intervals of X ts(x)/-Jn. The value of K (which is a function of n, a, and y) is selected in such a manner that it can be said with a probability of 100-a (corresponding to an error of a percent) that the interval will cover at least a proportion y of the population, ... 

From the electropherogram in Fig. 2, no interference from the formulation excipients could be observed at the migration times of ENX and IS. The limit of the detection (LOD) was 3.85 x 10 M, while the limit of quantification (LOQ) was 1.16 X 10 M. The results indicate good precision. Method accuracy was determined by analyzing a placebo (mixture of excipients) spiked with ENX at three concentration levels (n = 6) covering the same range as that used for linearity. Mean recoveries with 95% confidence intervals are given in Table 3. 

We also looked at confidence interval construction based on standard errors of parameter estimates. If the asymptotic 90% confidence intervals are truly accurate, then they should cover the true parameter 90% of the time. We examined the confidence interval coverages in simulation scenario I, shown in Figure 16.2. 

The certification of a property value in a material leads to a certified value, which is typically the mean of several determinations or the result of a metrologically valid preparation procedure, e.g. weighing. The confidence interval or uncertainty limits of this mean value are also determined. The two basic analytical parameters, mean and uncertainty, are included in a certificate of analysis. The presentation and the additional information, which should also be given in the certificate, are listed in the ISO Guide 31 [32] and cover in particular ... 

Choose sufficient calibration concentrations to cover the range and give a suitable confidence interval on measurement... 

An overview of statistical methods covers mean values, standard deviation, variance, confidence intervals, Student s t distribution, error propagation, parameter estimation, objective functions, and maximum likelihood. 

Next we covered analysis of data. We used probability and random variables to model the irreproducibie part of the experiment. For models that are linear in the parameters, we can perform parameter estimation and construct exact confidence intervals analytically. For models that are nonlinear in the parameters, we compute para ter estimates and construct approximate confidence intervals using nonlinear optimization methods. 

Note 5. The relative standard deviation [RSD] of based on observed counts is for Poisson data, or about 6% for -E(N)=60. Equivalent precision for based on replication would require about 2fi or 120 degrees of freedom. The same is true for confidence intervals for n, hence, based on counts, vs based on replication. For more detail, Including the use of x bo derive both types (counts, replication) of Cl s see Ref. and the monograph by Cox and Lewis (100). Adequacy of the large count (normal) approximation, and the exact treatment for extreme low-level data (n 10 or less) are covered in Ref. and the references therein. 

Confidence intervals may be alternatively formulated in terms of a factor selected so that the calculated interval covers the mean fi a certain percent (proportion) of the time. 

Statistical confidence is the probability that a particular confidence interval (as calculated from sample data) covers the true value of the statistical parameter. 

First, a partial least squares (PLS) model was built based on the FTIR spectra of 37 lab-prepared calibration samples, covering amine concentration of 1.5 to 2.5 mol/liter, CO2 concentrations of 0.1 to 1.1 mol/liter, and SO2 concentrations of 0.0 to 0.2 mol/liter. The predictive performance of this model was subsequently assessed by applying it to the FTIR spectra of 15 separate lab-prepared validation samples within the same concentration ranges, yielding a predictive accuracy of 0.05 mol/liter for the three components (95% confidence interval). 

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