Confidence Intervals measurements

Unpaired Data Consider two samples, A and B, for which mean values, Xa and Ab, and standard deviations, sa and sb, have been measured. Confidence intervals for Pa and Pb can be written for both samples... 

Four replicate measurements were made at the center of the factorial design, giving responses of 0.334, 0.336, 0.346, and 0.323. Determine if a first-order empirical model is appropriate for this system. Use a 90% confidence interval when accounting for the effect of random error. 

Increa.se the number of mea.surements included in the mea.sure-ment. set by using mea.surements from repeated. sampling. Including repeated measurements at the same operating conditions reduces the impact of the measurement error on the parameter estimates. The result is a tighter confidence interval on the estimates. 

First, the parameter estimate may be representative of the mean operation for that time period or it may be representative of an extreme, depending upon the set of measurements upon which it is based. This arises because of the normal fluc tuations in unit measurements. Second, the statistical uncertainty, typically unknown, in the parameter estimate casts a confidence interv around the parameter estimate. Apparently, large differences in mean parameter values for two different periods may be statistically insignificant. 

Fig. 25. Relationship between the measured interfacial strength and the (negative) Gibbs free energy of mixing, (-AG )o5, for glass beads treated with various silane coupling agents embedded in a PVB matrix. Error bars correspond to 95% mean confidence intervals. Redrawn from ref. [165].

When a small number of observations is made, the value of the standard deviation s, does not by itself give a measure of how close the sample mean x might be to the true mean. It is, however, possible to calculate a confidence interval to estimate the range within which the true mean may be found. The limits of this confidence interval, known as the confidence limits, are given by the expression ... 

Assuming for the moment that a large number of measurements went into a determination of a mean Xmean and a standard deviation s, what is the width of the 95% confidence interval, what are the 95% confidence limits ... 

Figure 1.18. The Student s ( resp. t/Vn for various confidence levels are plotted the curves for p = 0.05 are enhanced. The other curves are for p = 0.5 (bottom), 0.2, 0.1, 0.02, 0.01, 0.002, 0.001, and 0.0001 (top). By plotting a horizontal, the number of measurements necessary to obtain the same confidence intervals for different confidence levels can be estimated. While it takes n - 9 measurements (/ = 8) for a t-value of 7.12 and p = 0.0001, just n = 3 f - 2) will give the same limits on the population for p = 0.02 (line A - C). For the CL on the mean, in order to obtain the same t/ /n for p = 0.02 as for p = 0.0001, it will take n = 4 measurements (line B ) note the difference between points D and ...

For standard deviations, an analogous confidence interval CI(.9jr) can be derived via the F-test. In contrast to Cl(Xmean), ClCij ) is not symmetrical around the most probable value because by definition can only be positive. The concept is as follows an upper limit, on is sought that has the quality of a very precise measurement, that is, its uncertainty must be very small and therefore its number of degrees of freedom / must be very large. The same logic applies to the lower limit. s/ ... 

Figure 2.9. The confidence interval for an individual result CI( 3 ) and that of the regression line s CLj A are compared (schematic, left). The information can be combined as per Eq. (2.25), which yields curves B (and S, not shown). In the right panel curves A and B are depicted relative to the linear regression line. If e > 0 or d > 0, the probability of the point belonging to the population of the calibration measurements is smaller than alpha cf. Section 1.5.5. The distance e is the difference between a measurement y (error bars indicate 95% CL) and the appropriate tolerance limit B this is easy to calculate because the error is calculated using the calibration data set. The distance d is used for the same purpose, but the calculation is more difficult because both a CL(regression line) A and an estimate for the CL( y) have to be provided.

The use of confidence intervals is one way to state the required precision. Confidence limits provide a measure of the variability associated with an estimate, such as the average of a characteristic. Table I is an example of using confidence intervals in planning a sampling study. This table shows the interrelationships of variability (coefficient of variation), the distribution of the characteristic (normal or lognormal models), and the sample frequency (sample sizes from 4 to 365) for a monitoring program. 

Expected Variability of Measurement (Coefficient Distribution of Variation) (Model) 95% Confidence Interval About the Mean Estimate (percent) ... 

Estimates of the standard deviation of the n measurements will have a confidence interval ... 

The last two calculations of the confidence interval indicate in which range there are 95 out of 100 chances of finding an experimental LEL for this substance. The only legitimate experimental approach to the measurement of LEL is the repetition of measurements and the calculation of the average. The first two sequences show that the fluctuations in LEL in both of these cases cannot be considered to be linked to the uncertainty of measurement. There is a predictable cause, which cancels out all interest in this data. 

One can take as an example the flashpoint method used in the author s laboratory. The measured flashpoints were the subject of repetitions, which gave an accurate confidence interval which was relatively narrow. Therefore these figures can be used with confidence. The following is obtained. 

Stability is then considered as known and defined when Rf Uj is not significantly different from one. However the uncertainty calculated for the ratio RT based on the sum of CVs of two measurements carried out at two temperatures is a CV and not a confidence interval. In fact it does not consider the number of measurements carried out at the two temperatures and the use of this combined CV is not correct. In many cases it is an underestimation, as usually only two or three replicates are made. However, stability should be determined on the basis of a trend analysis, which is of importance also for any shelf life quantification see below. 

In all the above cases we presented confidence intervals for the mean expected response rather than a future observation (future measurement) of the response variable, y0. In this case, besides the uncertainty in the estimated parameters, we must include the uncertainty due to the measurement error (so). 

Results of the covariance analysis for the accuracy of estimates of the relative permeability of water and capillary pressure functions, along with the specified true functions, are shown in Figures 4.1.9 and 4.1.10 (the results for the relative permeability of oil are not included here). The accuracy measures are presented as 95 % confidence intervals. 

In Figure 4 the results from the three different groups are in excellent agreement for butanol concentrations of 90 wt% and greater, although the data from the Russian group scatter somewhat more around our results than do the values interpolated from Westmeier s data.(14.16). At lower amphiphile concentrations the isoperibolic calorimeter measurements are in noticeably better agreement with the data of ref. 16 than with the Russian work (14-16). However, almost all results fall within the 95% confidence interval (dashed lines) for our results. 

The relation between systematic and random deviations as well as the character of outliers is shown in Fig. 4.1. The scattering of the measured values is manifested by the range of random deviations (confidence interval or uncertainty interval, respectively). Measurement errors outside this range are described as outliers. Systematic deviations are characterized by the relation of the true value p and the mean y of the measurements, and, in general, can only be recognized if they are situated beyond the range of random variables on one side. 

Conventionally, a measured result is said to be correct if the true value is situated within the confidence interval of the observed mean (pi, case 1 in Fig. 4.1). If the true value is located outside of the range of random deviations (p2, case 2 in Fig. 4.1) then the result is incorrect. 

In general, this interval includes P 100% of all measured values, i.e., in the case of ft = 20 individual measurements, one value outside the confidence interval corresponds to the statistical expectation for P = 1 — a = 0.95. 

Such a parameter may be, e.g., standard deviation, or a given multiple of it, or a one-sided confidence interval attributed to a fixed level of confidence. In general, uncertainty of measurement comprises many components. These uncertainty components are subdivided into... 

The performance curve presents graphically the relationship between the probability of obtaining positive results PPRy i.e. x > xLSp on the one hand and the content x within a region around the limit of discrimination xDIS on the other. For its construction there must be carried out a larger number of tests (n > 30) with samples of well-known content (as a rule realized by doped blank samples). As a result, curves such as shown in Fig. 4.10 will be obtained, where Fig. 4.10a shows the ideal shape that can only be imagined theoretically if infinitely exact decisions, corresponding to measured values characterized by an infinitely small confidence interval, exist. 

The coefficients a,- are estimated from the results of experiments carried out according to a design matrix such as Table 5.9 which shows a 23 plan matrix. The significance of the several factors are tested by comparing the coefficients with the experimental error, to be exact, by testing whether the confidence intervals Aai include 0 or not. The experimental error can be estimated by repeated measurements of each experiment or - as it is done frequently in a more effective way - by replications at the centre of the plan (so-called zero replications ), see Fig. 5.2. 

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