Uncertainty propagation confidence intervals

Determine the density at least five times, (a) Report the mean, the standard deviation, and the 95% confidence interval for your results, (b) Eind the accepted value for the density of your metal, and determine the absolute and relative error for your experimentally determined density, (c) Use the propagation of uncertainty to determine the uncertainty for your chosen method. Are the results of this calculation consistent with your experimental results ff not, suggest some possible reasons for this disagreement. 

Uncertainly estimates are made for the total CDF by assigning probability distributions to basic events and propagating the distributions through a simplified model. Uncertainties are assumed to be either log-normal or "maximum entropy" distributions. Chi-squared confidence interval tests are used at 50% and 95% of these distributions. The simplified CDF model includes the dominant cutsets from all five contributing classes of accidents, and is within 97% of the CDF calculated with the full Level 1 model. 

To put equation 44-6 into a usable form under the conditions we wish to consider, we could start from any of several points of view the statistical approach of Hald (see [10], pp. 115-118), for example, which starts from fundamental probabilistic considerations and also derives confidence intervals (albeit for various special cases only) the mathematical approach (e.g., [11], pp. 550-554) or the Propagation of Uncertainties approach of Ingle and Crouch ([12], p. 548). In as much as any of these starting points will arrive at the same result when done properly, the choice of how to attack an equation such as equation 44-6 is a matter of familiarity, simplicity and to some extent, taste. 

The precision uncertainty associated with field sampling is generally much larger than that associated with analytical technique, which is roughly 2% to the 67% confidence interval for the two compounds used as conservative and gas tracers. A technique to determine the precision uncertainty associated with field sampling and incorporated into the mean Kl estimate will therefore be propagated with the first-order, second moment analysis (Abernathy et al., 1985) ... 

In the ordinary weighted least squares method, the most probable values of source contributions are achieved by minimizing the weighted sum of squares of the difference between the measured values of the ambient concentration and those calculated from Equation 1 weighted by the analytical uncertainty of those ambient measurements. This solution provides the added benefit of being able to propagate the measured uncertainty of the ambient concentrations through the calculations to come up with a confidence interval around the calculated source contributions. 

This solution provides two benefits. First, it propagates a confidence interval around the calculated source contributions which reflects the cumulative uncertainty of the input observables. The second benefit provided by this "effective variance" weighting is to give those chemical properties with larger uncertainties, or chemical properties which are not as unique to a source type, less weight in the fitting procedure than those properties having more precise measurements or a truly unique source character. 

The major components of uncertainty are combined according to the rules of propagation of uncertainty, often with the assumption of independence of effects, to give the combined uncertainty. If the measurement uncertainty is to be quoted as a confidence interval, for example, a 95% confidence interval, an appropriate coverage factor is chosen by which to multiply the combined uncertainty and thus yield the expanded uncertainty. The coverage factor should be justified, and any assumptions about degrees of freedom stated. 

Chapter 3 gave rules for propagation of uncertainty in calculations. For example, if we were dividing a mass by a volume to find density, the uncertainty in density is derived from the uncertainties in mass and volume. The most common estimates of uncertainty are the standard deviation and the confidence interval. 

To perform unbiased analyses, a collaboration-wide policy of blindness was established, where cut selections are optimized on a fraction of data or on time-scrambled data set. We present upper confidence limits for null results following the treatment described in (Feldman and Cousins, 1998) and incorporate systematic uncertainties into the calculation of confidence intervals according to (Conrad et al., 2003). The contributions to systematic uncertainties is predominantly due to variations of the optical properties of the ice, the absolute sensitivity of the OM, the neutrino cross section and the muon propagation. The combined systematic uncertainty is typically 30%, although the value varies slightly with the analysis method. 

The next example combines the concepts of uncertainty propagation as well as confidence intervals. The problem is related to measuring the viscosity of a transparent fluid in a falling ball viscometer, in which the time it takes a ball to cross two lines separated by a distance is measured. The ball falls through a tube filled with the fluid and is assumed to reach a steady velocity by the time it reaches the first Une. The velocity is determined by the geometry and density of the ball as well as the density and, most importantly, the viscosity of the liquid ... 

Coincidental or not, also in a PLP study of several methacrylates, the ethyl monomer EMA was found to have the highest activation energy of propagation [15, 28]. (2) Because of the high uncertainty in the exact values of the Mark-Houwink coefficients of EA, the corresponding confidence intervals of equations 4.12 to 4.14 have not been calculated. 

Develop and implement a technique for propagating the estimated uncertainties (confidence intervals) onto the probabilities of occupying the degradation states over time ... 

The Best Estimate Plus Uncertainties (BEPU) analysis is recommended by IAEA in addition or alternatively to the deterministic approach in the safety analysis of nuclear components and systems (IAEA 2009). The aim of the BEPU analysis is to determine a quantile of an output measure of interest (noted R in the following) with a certain level of confidence (usually the 95% quantile obtained with a 95% confidence, denoted Rg gf and to verify that this quantile is below an acceptable limit (acceptance criteria). In order to obtain this quantile, the uncertainty space of input parameters is sampled at random according to their combined probability distribution and a code calculation is performed for each sampled set of parameters. The number of code calculations is determined by the requirement to estimate a tolerance and confidence interval for the quantity of interest. Wilks formula (Wilks 1941) (or Wald formula (Wald 1943) when several criteria must be respected simultaneously) is used to determine the number of calculations to obtain the uncertainty bands and the associated quantile with a given confidence level. In classical BEPU analysis, there is no separation between the aleatory variables and the epistemic variables the epistemic variables, which are often model uncertainties, are generally modeled by uniform probability distributions within intervals provided by expert opinion and propagated in the same way that the aleatory variables by Monte Carlo simulation. 

When all the standard uncertainties of Type A and Type B have been determined in this way, they should be combined to produce the combined standard uncertainty (suggested symbol u), which may be regarded as the estimated standard deviation of the measurement result. This process, often called the law of propagation of uncertainty or root-sum-of-squares, involves taking the square root of the sum of the squares of all the u.. In many practical measurement situations, the probability distribution characterized by the measurement result y and its combined standard uncertainty ufy) is approximately normal (Gaussian). When this is the case, ufy) defines an interval y - ufy) to y + ufy) about the measurement result y within which the value of the measurand Yestimated by y is believed to lie with a level of confidence of approximately 68 percent. That is, it is believed with an approximate level of confidence of 68 percent that y - ufy) < Y < y + ufy), which is commonly written as T = y ufy). 

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