Error confidence limits

Define acceptable levels of reliability and decision risks (type 1 and type 2 errors, confidence limits). 

The distribution of the /-statistic (x — /ji)s is symmetrical about zero and is a function of the degrees of freedom. Limits assigned to the distance on either side of /x are called confidence limits. The percentage probability that /x lies within this interval is called the confidence level. The level of significance or error probability (100 — confidence level or 100 — a) is the percent probability that /X will lie outside the confidence interval, and represents the chances of being incorrect in stating that /X lies within the confidence interval. Values of t are in Table 2.27 for any desired degrees of freedom and various confidence levels. 

Establishment of areas where the signal is never detected, always detected, and where results are ambiguous. The upper and lower confidence limits are defined by the probability of a type 1 error (dark shading), and the probability of a type 2 error (light shading). 

The confidence limits of a measurement are the limits between which the measurement error is with a probability P. The probability P is the confidence level and a = 1 - P is the risk level related to the confidence limits. The confidence level is chosen according to the application. A normal value in ventilation would be P = 95%, which means that there is a risk of a = 5 /o for the measurement error to be larger than the confidence limits. In applications such as nuclear power plants, where security is of prime importance, the risk level selected should be much lower. The confidence limits contain the random errors plus the re.sidual of the systematic error after calibration, but not the actual systematic errors, which are assumed to have been eliminated. 

The bias error is a quantity that gives the total systematic error of a measuring instrument under defined conditions. As mentioned earlier, the bias should be minimized by calibration. The repeatability error consists of the confidence limits of a single measurement under certain conditions. The mac-curacy or error of indication is the total error of the instrument, including the... 

If the lower values in the brackets are applied, an additional 0.5 uncertainty (error on 5% risk level) has to be added arithmetically to the flow coefficient confidence limits. The use of flow straighteners is recommended in cases when a nonstandard type of upstream fitting disturbs the flow velocity profile. 

Every measured quantity or component in the main equations, Eqs. (12.30) and (12.31), influence the accuracy of the final flow rate. Usually a brief description of the estimation of the confidence limits is included in each standard. The principles more or less follow those presented earlier in Treatment of Measurement Uncertainties. There are also more comprehensive error estimation procedures available.These usually include, beyond the estimation procedure itself, some basics and worked examples. 

Finally, a 1 1 mixture of acetic and propionic acids containing 2 % of water has been used in order to study the rates of chlorination of polyalkylbenzenes at low temperatures. Second-order rate coefficients were obtained and the values are recorded in Table 58 together with the energies and entropies of activation (which are given with the errors for 95 % confidence limits) from which it was concluded... 

In Section 1.3.2, confidence limits are calculated to define a confidence interval within which the true value p is expected with an error probability of p or less. 

Error bars defined by the confidence limits CL(y,) will shrink or expand, most likely in an asymmetric manner. Since we here presuppose near absence of error from the abscissa values, this point applies only to y-transformations. A numerical example is 17 1 ( 5.9%, symmetric CL), upon logarithmic transformation becomes 1.23045 -0.02633. .. 1.23045 + 0.02482. 

The typical error is here defined as a confidence limit. [See Eqs. (4.16H4.19).]... 

Figure 4.47. Drug assay using HPLC respectively UV Spectroscopy. Correlation of HPLC and UV results obtained on four batches of a cream. The vertical error bars each give the mean + standard deviation of 6 HPLC determinations because the Student s t-factor for five d.f. is nearly equal to /6 (see Section 1.3.2), the bars can also be interpreted as 95% confidence limits. The circles connected by a line indicate the corresponding duplicate UV determinations. The proportionality line passes through the origin and the center of mass for the four coordinates. The drug is slightly overdosed (= 103-104% the traditional UV assay apparently is not as selective as it should be an interference adds about 4% to the result.

One measure of the quality of an estimate of an average Is the confidence limits (or maximum probable error) for the estimate. For averages of Independent samples, the maximum probable error Is... 

The first analysis is one with AS-level precision, the second with TIMS-level precision. The first order 2a error for the resulting 331 ka age is 96 ka, but examination of the distribution of a Monte Carlo simulation (Fig. 2) shows that the actual age distribution is strongly asymmetric, with 95% confidence limits of 158/-79 ka. For either younger ages or more-precise analyses, however, the first-order age errors are more than adequate, as shown by the Monte Carlo results for the same data, but with TIMS-level precision (Fig. 2B). 

Figure 2. Histograms of Monte Carlo simulations for two synthetic analyses (Table 1) of a 330 ka sample. The lower precision analysis (A) has a distinctly asymmetric, non-Gaussian distribution of age errors and a misleading first-order error calculation. The higher precision analysis (B) yields a nearly symmetric, Gaussian age distribution with confidence limits almost identical those of the first-order error expansion.

These results can then be compared to experimental values (at the same temperature) in a number of informative ways. First we can plot the calculated values as a function of temperature and represent results as a line, see Figure 4. The experimental results can be represented as an error band vs temperature plot. Real differences are readily apparent since they must lie outside the 95% confidence limits. Another way to represent the difference is to plot the difference between calculated and experimental values as a function of temperature. In the same graph an estimate and plot of the experimental errors can also be made, see Figure 5. 

Standard Error of Estimate Syx = 0.4328 Slope Confidence Limits Method 1... 

Ref. 31 Xs=0 87 x 10- esu for CHCI3 and Xs=l 06 x 10 esu for hexane. There are no two-photon resonances in the solvents in the frequency range of interest here (31). The resulting values of Y t and y"t> with 90% confidence limits expressed as error bars, are shown in Fig. 9. Each data pair is determined from the... 

Figure 9. Variation of the real (m) and imaginary (o) parts of the two photon hyperpolarizability (yr) with two-photon energy (2aJt) for the yellow solution. The error bars represent 90% confidence limits. All data are taken in a 1111 geometry (all beams polarized ). The solid lines are theoretical fts to the data according to Equation 3. (Reproduced with permission from Ref. 24. Copyright 1979, American Institute of...

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