Confidence levels

From the probability distributions for each of the variables on the right hand side, the values of K, p, o can be calculated. Assuming that the variables are independent, they can now be combined using the above rules to calculate K, p, o for ultimate recovery. Assuming the distribution for UR is Log-Normal, the value of UR for any confidence level can be calculated. This whole process can be performed on paper, or quickly written on a spreadsheet. The results are often within 10% of those generated by Monte Carlo simulation. 

The basic condition of the Standard application - the availability of stable coupled probabilistic or the multiple probabilistic relations between then controlled quality indexes and magnetic characteristics of steel. All the probabilistic estimates, used in the Standard, are applied at confidence level not less than 0,95. General requirements to the means of control and procedure of its performance are also stipulated. Engineers of standard development endeavoured take into consideration the existed practice of technical control performance and test at the enterprises that is why the preparation of object control for the performance of nondestructive test can be done during the process of ordinary acceptance test. It is suggested that every enterprise is operated in correspondence with direct and non-destructive tests, obtained exactly at it, for detailed process chart and definite product type, however the tests have long since been performed after development of the Standard displayed that process gives way to unification. 

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. 

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. 

There will be incidences when the foregoing assumptions for a two-tailed test will not be true. Perhaps some physical situation prevents p from ever being less than the hypothesized value it can only be equal or greater. No results would ever fall below the low end of the confidence interval only the upper end of the distribution is operative. Now random samples will exceed the upper bound only 2.5% of the time, not the 5% specified in two-tail testing. Thus, where the possible values are restricted, what was supposed to be a hypothesis test at the 95% confidence level is actually being performed at a 97.5% confidence level. Stated in another way, 95% of the population data lie within the interval below p + 1.65cr and 5% lie above. Of course, the opposite situation might also occur and only the lower end of the distribution is operative. 

As applied in Example 12, the F test was one-tailed. The F test may also be applied as a two-tailed test in which the alternative to the null hypothesis is erj A cr. This doubles the probability that the null hypothesis is invalid and has the effect of changing the confidence level, in the above example, from 95% to 90%. 

The confidence limits for the slope are given by fc where the r-value is taken at the desired confidence level and (A — 2) degrees of freedom. Similarly, the confidence limits for the intercept are given by a ts. The closeness of x to X is answered in terms of a confidence interval for that extends from an upper confidence (UCL) to a lower confidence (LCL) level. Let us choose 95% for the confidence interval. Then, remembering that this is a two-tailed test (UCL and LCL), we obtain from a table of Student s t distribution the critical value of L (U975) the appropriate number of degrees of freedom. 

The second complication is that the values of z shown in Table 4.11 are derived for a normal distribution curve that is a function of O, not s. Although is an unbiased estimator of O, the value of for any randomly selected sample may differ significantly from O. To account for the uncertainty in estimating O, the term z in equation 4.11 is replaced with the variable f, where f is defined such that f > z at all confidence levels. Thus, equation 4.11 becomes... 

Values for t at the 95% confidence level are shown in Table 4.14. Note that t becomes smaller as the number of the samples (or degrees of freedom) increase, approaching z as approaches infinity. Additional values of t for other confidence levels can be found in Appendix IB. 

If the significance test is conducted at the 95% confidence level (a = 0.05), then the null hypothesis will be retained if a 95% confidence interval around X contains p,. If the alternative hypothesis is... 

The probability of a type 1 error is inversely related to the probability of a type 2 error. Minimizing a type 1 error by decreasing a, for example, increases the likelihood of a type 2 error. The value of a chosen for a particular significance test, therefore, represents a compromise between these two types of error. Most of the examples in this text use a 95% confidence level, or a = 0.05, since this is the most frequently used confidence level for the majority of analytical work. It is not unusual, however, for more stringent (e.g. a = 0.01) or for more lenient (e.g. a = 0.10) confidence levels to be used. 

The critical value for f(0.05,4), as found in Appendix IB, is 2.78. Since fexp is greater than f(0.05, 4), we must reject the null hypothesis and accept the alternative hypothesis. At the 95% confidence level the difference between X and p, is significant and cannot be explained by indeterminate sources of error. There is evidence, therefore, that the results are affected by a determinate source of error. 

If evidence for a determinate error is found, as in Example 4.16, its source should be identified and corrected before analyzing additional samples. Failing to reject the null hypothesis, however, does not imply that the method is accurate, but only indicates that there is insufficient evidence to prove the method inaccurate at the stated confidence level. 

The sampling constant for the radioisotope " Na in a sample ( homogenized human liver has been reported as approximate 35 g. (a) What is the expected relative standard deviation fo sampling if f.O-g samples are analyzed (b) How many f.O-g samples need to be analyzed to obtain a maximum sampling error of 5% at the 95% confidence level ... 

Because the T-ratio is larger than T(0.05, 9, 9), which is 3.179, we conclude that the systematic errors of the analysts are significant at the 95% confidence levels. The estimated precision for a single analyst is... 

Determine if there is any evidence for a systematic error in the method at the 95% confidence level. 

This value for fexp is smaller than the critical value of 2.26 for f(0.05, 9). Thus, there is no evidence for a systematic error in the method at the 95% confidence level. 

The t-values in this table are for a two-tailed test. For a one-tailed test, the a values for each column are half of the stated value, column for a one-tailed test is for the 95% confidence level, a = 0.05. For example, the first... 

The precision of measurement does not appear to be very high. Confidence levels in the precision may be made by use of "Student f" Tables. 

From standard Student tables, the value for t l/fT= 1.049 at the 95% confidence level. Thus, mean value = 56.3 13.0 (95% confidence level) and one could be confident that 95% of measured values would fall in the range 69.3 - 43.3. This is a large range and is not very precise. 

HETP values obtained in this way have been compared to measured values in data banks (69) and statistical analysis reveals that the agreement is better when equations 79 and 80 are used to predict and than with the other models tested. Even so, a design at 95% confidence level would require a safety factor of 1.7 to account for scatter. 

Confidence level >99% vs control, unless otherwise noted. 

If a heat exchanger is sized usiag the mean values of the design parameters, then the probabiUty, or the confidence level, of the exchanger to meet its design thermal duty is only 50%. Therefore, in order to increase the confidence level of the design, a proper uncertainty analysis must be performed for all principal design parameters. 

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