Distribution confidence limits, figure showing

Figure 1.21. Monte Carlo simulation of six groups of eight normally distributed measurements each raw data are depicted as x,- vs. i (top) the mean (gaps) and its upper and lower confidence limits (full lines, middle) the confidence limits CL(s ) of the standard deviation converge toward a = 1 (bottom, Eq. 1.42). The vertical divisions are in units of 1 a. The CL are clipped to +5a resp. 0. .. 5ct for better overview. Case A shows the expected behavior, that is for every increase in n the CL(x,nean) bracket /r = 0 and the CL(i t) bracket a - 1. Cases B, C, and D illustrate the rather frequent occurrence of the CL not bracketing either ii and/or ff, cf. Case B n = 5. In Case C the low initial value (arrow ) makes Xmean low and Sx high from the beginning. In Case D the 7 measurement makes both Cl n = 7 widen relative to the n 6 situation. Case F depicts what happens when the same measurements as in Case E are clipped by the DVM.

Sometimes a measurement involves a single piece of calibrated equipment with a known measurement uncertainty value o, and then confidence limits can be calculated just as with the coin tosses. Usually, however, we do not know o in advance it needs to be determined from the spread in the measurements themselves. For example, suppose we made 1000 measurements of some observable, such as the salt concentration C in a series of bottles labeled 100 mM NaCl. Further, let us assume that the deviations are all due to random errors in the preparation process. The distribution of all of the measurements (a histogram) would then look much like a Gaussian, centered around the ideal value. Figure 4.2 shows a realistic simulated data set. Note that with this many data points, the near-Gaussian nature of the distribution is apparent to the eye. 

Figure 7.10. Student s t-distribution for 2, 10, or 1000 degrees of freedom, showing 95% confidence limit.

For the calculation of simultaneous confidence intervals it is necessary to replace the t value in the above equations with y/2F 2, n — 2) from tables of the F distribution for the same degree of confidence. This is rarely done in practice but Figure 29 shows the superimposition of these limits on the calibration plot. 

Note Node numbers correspond to those on the chronogram (Figure 17.1). Point estimates are from analyses of the all-compatible majority rule consensus tree and posterior probability values are rep ted. The Mode value represents the most likely divergence lime value under the specified model (obtained by local density estimation calculated over the 100 random frees drawn from the posterior distribution of trees and parameters), and the HPD values limits the confidence interval for the estimates. ( node constrained j age estimates show bimodal distribution across the 100 random trees > age distribution with a pronounced right tail across the 100 random frees)... 

From the MC generated cumulative distributions of the c[] parameters as shown in Figures 9.10, 9.11 and 9.12 estimated limits on the parameters at various confidence levels can be readily obtained by neglecting appropriate numbers of lower and upper values from the parameter distributions. The function climits() as used on line 35 of Listing 9.3 can be used to automatically generate such limits. Various confidence level bounds on the parameter values are shown for the three parameters in Figures 9.13 through 9.15. The solid lines show the bounds as estimated by the MC analysis while the dotted lines show the one standard error... 

In many applications, the Weibull analysis is applied to predict the part reliability or unreliability based on limited data with the help of modern computer technology. The limited data will inevitably introduce some statistical uncertainty to the results. Figure 6.11 shows the unreliability as a function of time in a Weibull plot. The six data points reasonably fit on the straight line and verify that the data are a Weibull distribution. The hourglass curves plotted on each side of the Weibull line represent the bounds of 90% confidence intervals for fhis analysis. The width of the intervals depends on the sample size it narrows when more samples are analyzed. 

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