Probability statistical confidence

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. 

The mathematical methods used for interpolation and extrapolation of the data obtained from accelerated tests, as described in Chapters 8 and 9, include both the mechanistic and the empirical. Arrhenius formula, based on chemical rate kinetics and relating the rate of degradation to temperature, is used very widely. Where there are sufficient data, statistical methods can be applied and probabilities and confidence limits calculated. For many applications a high level of precision is unnecessary. The practitioners of accelerated weathering are only too keen to tell you of its quirks and inaccuracies, but this obscures... 

As the /statistic deviates further from one, there is a greater probability that the two standard deviations are different. To quantitate this probability, one needs to consult an f-table that provides the critical values off as a function of the degrees of freedom for the two distributions (typically A - 1 for each of the data series), and the desired statistical confidence (90%, 95%, 99%, etc.). 

The probability that this overall effect is statistically significant The statistical confidence limits on the overall effect... 

Strictly speaking, a lifetime prediction should be done with a specified failure probability and confidence level, see for instance Ronold and Echtermeyer [30], or Sutherland and Veers [31], but for the sake of simplicity in comparing the mean values of different models to the experimentally determined mean lifetime, these statistics and the associated myriad of possibilities for the choice of distribution types and statistical methods have been omitted. Moreover, for the comparison of models, comparing the mean of the data is sufficient. 

A linear fit of I versus cos 0 or I versus sin 0 will reveal whether orientation is consistent for the collected spectra. If no systematic deviation from linearity is observed, there are probably no gross experimental artifacts. The sample charging artifact described in Section 4.2.2, however, can sometimes result in a linear fit because the illuminated spot size increases trigonometrically with incident angle. The standard deviations of the fit parameters provide some statistical uncertainty of the orientation measurement. With a sufficient number of points, a confidence interval can be determined. Small differences in orientation can then be judged on a statistical confidence basis. The parameters A and a can also be fit directly using nonlinear least-squares methods. 

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

A numerical illustration will show the lack of power of the technique. Suppose we have a pair of human cousins and that each has 60 detectable bands, which is also n (the average number of bands), using the combination of the two Jeffreys probes. A good estimate of x is 20%. From this, one can easily calculate that the expected bandsharing between cousins is 29.44%. The most probable number of bands shared for two such cousins is thus 18. If they were unrelated the most likely number of bands shared would be 12. If we have a = b = 60, and c = 18, we can use (1) above to calculate the likelihood ratio. The number resulting is 8.75. Thus, the most probable data set if the individuals are cousins is only 8.75 times more probable than it would be if they were unrelated. Therefore it will normally be impossible to establish with statistical confidence that cousins are related. 

S = standard deviation a = statistical confidence level t = statistical t probability distribution... 

When epidemiologists compare two human populations, one defined as being at risk and the other defined as the control, they begin by hypothesizing that there is no difference in disease frequency between the two populations. They then collect data to decide whether their hypothesis is correct or incorrect. The hypothesis of no difference between two populations is called the null hypothesis. The null hypothesis is accepted if it is decided that there is no difference between the two populations, and it is rejected if it is decided that there is a difference. There is a finite probability of committing an error and rejecting the null hypothesis when it should be accepted and of accepting the nnll hypothesis when it should be rejected. The decision to accept or reject the null hypothesis is associated with a specified level of statistical confidence in the data. For example, if the null hypothesis is rejected at the 0.95 confidence level, there is a 95% chance that the decision is correct (i.e., that there really is a difference between the study and control populations) and a 5% chance that the decision to reject the null hypothesis is erroneous (i.e., that there really is no difference between the two populations). 

This section discusses the calculation of the uncertainty in the damage quantification stage, by estimating the uncertainty in the value of damage parameter. The classical statistics-based approach calculates statistical confidence intervals on the value of damage parameter, while the Bayesian statistics-based approach directly calculates the probability distribution of the value of the damage parameter. 

But decision making in the real world isn t that simple. Statistical decisions are not absolute. No matter which choice we make, there is a probability of being wrong. The converse probability, that we are right, is called the confidence level. If the probability for error is expressed as a percentage, 100 — (% probability for error) = % confidence level. 

Statistically, a similar Indication of precision could be achieved by utilising the 95% probability level if the results fell on a "Gaussian" curve, viz., the confidence would lie within two standard deviations of the mean. R 2 x SD = 56.3 24.8... 

In the introduction to this section, two differences between "classical" and Bayes statistics were mentioned. One of these was the Bayes treatment of failure rate and demand probttbility as random variables. This subsection provides a simple illustration of a Bayes treatment for calculating the confidence interval for demand probability. The direct approach taken here uses the binomial distribution (equation 2.4-7) for the probability density function (pdf). If p is the probability of failure on demand, then the confidence nr that p is less than p is given by equation 2.6-30. 

One way to choose the value of p is as follows. Assume that the distribution of squared residuals is normal, as is often done in crystallography. Then tables are available [17] which give the probability p that a particular experiment will give a X2 less than p. The value of p can be chosen according to the desired confidence level, p. Of course, other ways to choose are possible. Indeed, other choices for the agreement of statistic are possible. 

The statistical fundamentals of the definition of CV and LD are illustrated by Fig. 7.8 showing a quasi-three-dimensional representation of the relationship between measured values and analytical values which is characterized by a calibration straight line y = a + bx and their two-sided confidence limits and, in addition (in z-direction) the probability density function the measured values. 

The model itself can be tested against the sum of squared residuals c2=4.01. If, as a first approximation, we admit that intensities are normally distributed (which may not be too incorrect since all the values seem to be distant from zero by many standard deviations), c2 is distributed as a chi-squared variable with 5 — 3 = 2 degrees of freedom. Consulting statistical tables, we find that there is a probability of 0.05 that a chi-squared variable with two degrees of freedom exceeds 5.99, a value much larger than the observed c2. We therefore accept to the 95 percent confidence level the hypothesis that the linear signal addition described by the mass balance equations is correct, o... 

The first precise or calculable aspect of experimental design encountered is determining sufficient test and control group sizes to allow one to have an adequate level of confidence in the results of a study (that is, in the ability of the study design with the statistical tests used to detect a true difference, or effect, when it is present). The statistical test contributes a level of power to such a detection. Remember that the power of a statistical test is the probability that a test results in rejection of a hypothesis, H0 say, when some other hypothesis, H, say, is valid. This is termed the power of the test with respect to the (alternative) hypothesis H. ... 

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