95% confidence limit

Confidence limits are partial integrations over a probability density function. There are two special cases failure with time and failure with demand. 

Suppose N identical components with an exponential distribution (constant A) are on test the test is terminated at T, with M failures. What is the confidence that A is the true failure rate. 

The derivation will not be provided. Suffice it to say that the failures in a time interval may be modeled using the binomial distribution. As these intervals are reduced in size, this goes over to the Poisson distribution and the MTTF is chi-square distributed according to equation 2.9-31, where = 2 A N T and the degrees of freedom,/= 2(M+i). 

Confidence is calculated as the partial integral over the chi-squared distribution, i.e., the partial integral over equation 2.5-31 which is equation 2.5-32. where is the cumulative 

It is important to note that the chi-squared estimator provides upper bounds on A for the case of zero failures. For example, a certain type of nuclear plant may have 115 plant-years of experience using 61 control rods. If there has never been a failure of a control rod, what is A for 50% (median) and 90% confidence  

To be more certain that our quoted limits encompassed the true value, perhaps we should quote two or three times the standard uncertainty. Whatever limits we choose, we still need to quantify the likelihood of the true value being 

Coverage factor Area within confidence limits (%) 

The confidence hmit quoted in this manner may be referred to as the expanded uncertainty. This particular result has confidence limits that are symmetrical about the mean because we have assumed that the distribution of the measurements is Normal. If the distribution were skewed in any way, or perhaps if we were aware that the measurement was possibly, for some reason, biased high (or low), then the lower and upper confidence limits would not be identical. 

It is a common practice to quote confidence limits as a percentage of the value rather than as standard deviation. For example. Table 5.2 demonstrates a calculation of weighted mean (which will be explained in due course). If we take the first weighted mean result, we might quote it as  

The advantage is that, expressed in this manner, the uncertainty of the result is immediately obvious whereas 

If a result is quoted as having an uncertainty of 1 standard deviation, an equivalent statement would be the 68.3% confidence limits are given by Xmean 1 Sjc, the reason being that the area under a normal distribution curve between z = -1.0 to z = 1.0 is 0.683. Now, confidence limits on the 68% level are not very useful for decision making because in one-third of all 

Assuming for the moment that a large number of measurements went into a determination of a mean Xmean and a standard deviation s, what is the width of the 95% confidence interval, what are the 95% confidence limits  

A table of cumulative probabilities (CP) lists an area of 0.975002 for z -1.96, that is 0.025 (2.5%) of the total area under the curve is found between +1.96 standard deviations and +°°. Because of the symmetry of the normal distribution function, the same applies for negative z-values. Together p = 2 0.025 = 0.05 of the area, read probability of observation, is outside the 95% confidence limits (outside the 95% confidence interval of -1.96 Sx. .. + 1.96 Sx). The answer to the preceding questions is thus 

With z = 1.96 = 2 for the 95% confidence level, this is the explanation for the often-heart term two sigma about some mean. 

Unless otherwise stated, the expressions of CL() and Cl() are forthwith assumed to relate to the 95% confidence level. 

It is impossible to determine p and a from a limited set of measurements. We can use statistics to express the probability that the true value p lies within a certain range of the measured average mean x. That probability is called a CL and is usually expressed as a percentage (e.g., the CL is 95%). The term confidence limit refers to the extremes of the confidence interval (the range) about X within which p is expected to fall at a given CL. 

When i is a good approximation for o, we can state a CL or confidence limit for our results based on the Gaussian distribution. The CL is a statanent of how close the sample mean lies to the population mean. For a single measurement, we let s = a. The CL is then the certainty that p = x zo. For example, if z = 1, we are 68.3% confident thatx lies within 0 of the true value if we set z = 2, we are 95.5% confident that x lies within 2a of the trae value. For N measurements, the CL for p = x z m  

In most cases, s is not a good estimate of a because we have not made enough replicate analyses. In this case, the CL is calculated using a statistical probability parameter. Student s t. The parameter t is defined as t = (x - p)/x and the CL for p = x ts/yfN. An abbreviated set of t values is given in Table 1.9 complete tables can be found in mathematics handbooks or statistics books. 

As an example of how to use Table 1.9 and CLs, assume that we have made five replicate determinations of the pesticide dichlorodiphenyltrichloroethane (DDT) in a water sample using GC. The five results are given in the spreadsheet as follows, along with the mean and standard deviation. How do we report our results so that we are 95% confident that we have reported the true value  

The number of determinations is five, so there are four degrees of freedom. The t value for the 95% CL with A - 1 = 4 is 2.78, according to Table 1.9. Therefore, 

The manufacturer of any product would like to know what the probability is that any one item will deviate from the nominal value by a certain amount. Or, setting some acceptable value of x, call it the manufacturer would like to know what is the probability that x will be bigger than x. Questions of this type come under the subject of quality control.  

In terms of the standard normal distribution, Eq. 2.76 takes the form 

Pit k) = probability that x will exceed x by k standard deviations 

Consider k =. The probability that x will exceed where x = x + a is 15.9 percent. If x is some property of a manufactured product, it is said that the confidence limit is, in this case, 1 — 0.159 = 0.841 or 84.1 percent, i.e., 84.1 percent of the specimens will have x x (Fig. 2.11). If k = 2, the probability 

In actual construction or fabrication of an item, the Gaussian distribution is determined by checking the variable x for a large number of specimens. An average value of x is calculated, 

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Fig. X-7. Advancing and receding contact angles of octane on mica coated with a fluo-ropolymer FC 722 (3M) versus the duration of the solid-liquid contact. The solid lines represent the initial advancing and infinite time advancing and receding contact lines and the dashed lines are 95% confidence limits. (From Ref. 75.)...

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. 

Confidence limits for an estimate of the variance may be calculated as follows. Eor each group of samples a standard deviation is calculated. These estimates of cr possess a distribution called the ) distribution ... 

The upper and lower confidence limits for the standard deviation are obtained by dividing (A — 1)U by two entries taken from Table 2.28. The estimate of variance at the 90% confidence limits is for use in the entries Xoo5 X095 (for 5% and 95%) with N degrees of freedom. 

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. 

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). 

Da.ta. Ana.lysls. First, the raw data must be converted to concentrations over an appropriate time span. When sample periods do not correspond to the averaging time of the exposure limit, some assumptions must be made about unsampled periods. It may be necessary to test the impact of various assumptions on the final decision. Next, some test statistics (confidence limit, etc) (Fig. 3) are calculated and compared to a test criteria to make an inference about a hypotheses. 

Fig. 3. Confidence limits for exposure levels. A, noncompliance B, possible overexposure C, compliance. STD is the standard value, LCL and UCL represent lower and upper confidence levels, between which it is 95% certain that the tme exposure Hes, and and correspond to two separate...

Here again the quantity is the (1 — /3) percentile of a chi square distribution with V degrees of freedom. If only a 100(1 — a)% lower confidence limit is desired, it can be calculated from... 

The confidence limits for the rehabihty function can be found from equation 30. 

Since the minimum life is critical in this apphcation, a confidence limit estimate would be more appropriate, which can be calculated with the help of equation 39. For a 90% confidence limit, the required value of F is... 

Confidence Limit Estimate. An exact 100(1 — a)% lower confidence limit on the rehabUity is given by... 

Examp/e 8. There are 40 components placed on an accelerated 80-hlife test. A 75% lower confidence limit on the rehabUity is desired. To use equation 51, a value of F must be looked up. In this case, n = 40 and y = 37, and the required value is... 

The LD q is calculated from data obtained by using small groups of animals and usually for only a few dose levels. Therefore, there is an uncertainty factor associated with the calculation. This can be defined by determining the 95% confidence limits for the particular levels of mortaUty of interest (Fig. 7). The 95% confidence limits give the dose range for which there is only a 5% chance that the LD q will be outside. 

Fig. 7. Typical 95% confidence limits foi dose—moitality legiession data.

In defining acute level toxicity foi the purposes of comparing different materials, the LD q itself is not sufficient but the LD q and the 95% confidence limits should be quoted as a minimum. For example, and as demonstrated in Figure 8, two materials (A and B) with different LD q values, but overlapping 95% confidence limits, ate to be considered not statistically significantly different with respect to mortahty at the 50% level this it based on the fact that there is a statistical probabiUty that the LD q of one material could He in the 95% confidence limits of the other, and vice versa. Conversely, when there is no overlap in 95% confidence limits, as shown with material C, it may be concluded that the LD q values ate statistically significantly different. 

Material C, however, has 95% confidence limits at the LD q level which do not overlap those of A or B it is statistically significantly mote lethaHy toxic... 

Fig. 9. The two materials, A and B, have overlapping 95% confidence limits at the LD q level. Because the slopes of the dose—mortahty regression lines for both materials are similar, there is no statistically significant difference in mortahty at the LD q and LD q levels. Both materials may be assumed to be lethahy equitoxic over a wide range of doses, under the specific conditions of the test.

Acute toxicity studies are often dominated by consideration of lethaUty, including calculation of the median lethal dose. By routes other than inhalation, this is expressed as the LD q with 95% confidence limits. For inhalation experiments, it is convenient to calculate the atmospheric concentration of test material producing a 50% mortaUty over a specified period of time, usually 4 h ie, the 4-h LC q. It is desirable to know the nature, time to onset, dose—related severity, and reversibiUty of sublethal toxic effects. 

If there is a lack of specific, appropriate data for a process facility, there can be considerable uncertainty in a frequency estimate like the one above. When study objectives require absolute risk estimates, it is customary for engineers to want to express their lack of confidence in an estimate by reporting a range estimate (e.g., 90% confidence limits of 8 X 10 per year to 1 X 10 per year) rather than a single-point estimate (e.g., 2 X 10per year). For this reason alone it may be necessary for you to require that an uncertainty analysis be performed. 

I Eelsenstein. Confidence limits on phylogenies An approach using the bootstrap. Evolution 39 783-791, 1985. 

Confidence limits are also drawn on Figure 2.15(a) to give boundaries of Cpi for a given q determined from the analysis, which are within 95%. The relationship between q and Cp is described by a power law after linear regression giving ... 

Figure 2.15 Empirical relationships between (a) and Cp and (b) q and Cp (with 95% confidence limits)...

Further statistical procedures can be applied to determine the confidence limits of the results. Generally, only the values for the mean and standard deviation would be reported. The reader is referred to any good statistical text to expand on the brief analysis presented here. 

The lower bound confidence limit is the probability that a parameter, x, is less than some value x . The upper bound confidence limit is the probability that a parameter, x, is greater than some value x . Figure 2.5-1 shows that confidence may be obtained from the discrete curve by simply adding the probabilities below or above x, for the lower or upper bound confidence respectively. If the curve is continuous it must be integrated above or below x. These results are normalized, by dividing the partial integral or partial sum by the full integral of the curve or complete sum. 

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