Intervals of the mean

The sets of technically and statistically acceptable results are represented in the form of bar-charts of which an example is given for Cu in Figure 3.1. The length of a bar corresponds to the 95 % confidence interval of the mean. The certified values... 

The 95 percent confidence intervals of the mean of each coordinate 206Pb/204Pb and 207Pb/204Pb are calculated for m=4, i.e., for 3 degrees of freedom. From standard statistical tables, we obtain t95 3 = 3.18 (two-sided). The 95 percent confidence intervals for the mean are therefore... 

Figure 4.9 The 95 percent confidence ellipse of the mean for the Polynesian data listed in Table 4.3. The horizontal and vertical bars show the 95 percent confidence intervals of the mean calculated independently for each coordinate.

The confidence interval of the mean is sometimes written y tsKyi ). How is this related to Equation 6.5 ... 

Figure 8 Mean serum levels of salicylic acid, gentisic acid, and salicyluric acid of 10 volunteers. Total salicylate represents the sum of the three individual compounds. Error bars indicate 95% confidence interval of the mean. Source From Ref. 99.

In Example 2.3, we have calculated that 14 samples are needed to reach the decision with a 95 percent level of confidence. To be on the safe side, we collected and analyzed 20 samples. The collected samples have the concentrations of lead ranging from 5 to 210mg/kg the mean concentration is 86 mg/kg the standard deviation is 63 mg/kg and the standard error is 14mg/kg. From Appendix 1, Table 2 we determine that the t-value for 19 degrees of freedom (the number of samples less one) and a one-sided confidence interval for a — 0.05 is 1.729. Entering these data into Equation 10, Appendix 1, we calculate the 95 percent confidence interval of the mean 86 24 mg/kg. The upper limit of the confidence interval is 110 mg/kg and it exceeds the action level. Therefore, the null hypothesis Hq p > lOOmg/kg, formulated in Example 2.2 is true, as supported by the sample data. Based on this calculation we make a decision not to use the soil as backfill. 

Confidence limits are a range of values, within which the true mean concentration is expected to lie. Confidence interval is a range of values within confidence limits that is expected to capture the true mean concentration with a chosen probability. For example, a 95 percent confidence interval of the mean is a range of sample concentrations that will capture the true mean concentration 95 percent of the time,... 

In order to compensate for the uncertainty incurred by taking small samples of size n, the / probability distribution shown in Figure 3.2 is used in the calculation of confidence intervals, replacing the normal probability distribution based on z values shown in Figure 3.1. When n > 30, the /-distribution approaches the standard normal probability distribution. For small samples of size n, the confidence interval of the mean is inflated and can be estimated using Equation 3.9... 

Similar expressions for the confidence interval of the mean y, or with any point included in the data, y can be formulated. 

Table S.S The confidence intervals of the mean value and dispersion for the data from Table 5.3.

EXAMPLE 26-1 Ten replicate measurements of lead in a soil sample gave a mean of 0.1462% with a standard deviation of 0.0074%. Calculate the 95% confidence interval of the mean and of the standard deviation. 

Run number Occluded moisture mean value Upper 95% confidence interval of the mean Lower 95% confidence interval of the mean... 

A measure of variability of the estimate can be gained from the standard error, but it can be seen from Equations 12.4 and 12.5 that the magnitude of the standard error is inversely proportional to n that is, the larger the sample size, the smaller will be the standard error. Therefore, without prior knowledge of the sample size, a reported standard error cannot be evaluated. A standard error value of 0.2 indicates a great deal more variability in the estimate if n = 100 than if n = 3. One way around this shortcoming is to report n for every estimate of mean standard error. Another, and better, method is to report confidence intervals of the mean. 

Fig. 15.9. Determination of the Multiple Of Normal Activity (MONA). It is assumed that all sera to be tested have parallel dose-response curves (I-lII), the slope (m) of which with simple regression analysis is obtained from the log transformation in Fig. 15.8. The number of times a serum should be diluted over that of the reference serum (MONA) can then be calculated and is independent of the original dilution (i.e., a = b = MONAn). However, to obtain the MONA of the various sera, their EIA-absorbance values are determined at the same dilution (e.g., D ). The confidence interval of the mean of MONA-values, important for seroconversion tests, can similarly be established.

The uncertainty is taken as the half-width of the 95% confidence interval of the mean given in (1). When the reference material is used to assess the performance of a method, the user should refer to the recommendations of the certification report. 

Quote the 95% confidence = STDEV(range) TINV(0.05, rt—1)/SQRT(n) interval of the mean of n data ... 

When establishing confidence intervals from data, analysts are sometimes told to make seven measurements. The magical nature of this number stems from the fact that the standard deviation of seven results is just greater than the 95% confidence interval of the mean. How so Because the 95% confidence interval for n = l is /0.05",6 s/ /1 = (2.45 s)/2.65 (/0.05",6 = 2.45 and /7 = 2.65), and so these nearly cancel leaving s as about the 95% confidence interval of the mean. 

Systematic error in an analytical method must be determined and corrected for. We have seen that systematic error is assessed by making a measurement on a certified reference material (sometimes just referred to as a CRM). The mean of a number of determinations, x, can be used to decide if the systematic error is significant by using the equation for a confidence interval of the mean... 

The confidence interval contains two parts (1) An estimate of the quality of the sample for the estimate, known as the standard error of the mean and (2) the degree of confidence provided by the interval specified, known as the standard or Z-score. The confidence interval of the mean can be calculated by ... 

Examples for reference materials useful for instrument calibration, method validation and development in the field of human materials for which certified or other kinds of concentration values are reported for the 13 trace elements considered in this book (Al, As, Cd, Cr, Cu, Hg, Mn, Ni, Pb, Se, Tl, V and Zn) are given in Table 3. The data are taken from the survey prepared by Cortes Toro et al. (1990) and from other sources (BCR, 1992, Trahey, 1992, Chai Chifang, 1993) which the reader should consult for further details. Most of the columns are self explanatory. Column T contains a code (C = certified, N = noncertified or information value) for the type of reference value specified by the issuing authority. The uncertainty in the concentration value is expressed as a percentage error, but the meaning of this may differ somewhat from one material to another. In most cases it expresses the 95% confidence interval of the mean, but in a few other cases a tolerance interval, or some other definition (sometimes unspecified), may have been used by the producer. 

The interval x 2s contains 95% of the sample data set, and the interval X + 3s contains 99% of the data. In practice, knowing the spread of the data about the mean is valuable, but from a practical standpoint, the interval of the mean is a confidence interval of the mean, not of the data about the mean. Fortunately, the same basic principle holds when we are interested in the standard deviation of the mean, which is s/y/n, and not the standard deviation of the data set, s. Many statisticians refer to the standard deviation of the mean as the standard error of the mean. 

The assessment of bioequivalence is based on 90% confidence intervals for the ratio of the population geometric means (test/reference) for the parameters under consideration. This method is equivalent to two one-sided tests with the null hypothesis of bio-inequivalence at the 5% significance level. Two products are declared bioequivalent if upper and lower limits of the confidence interval of the mean (median) of log-transformed AUC and Cmax each fall within the a priori bioequivalence intervals 0.80-1.25. It is then assumed that both rate (represented by Cmax) and extent (represented by AUC) of absorption are essentially similar. Cmax is less robust than AUC, as it is a single-point estimate. Moreover, Cmax is determined by the elimination as well as the absorption rate (Table 2.1). Because the variability (inter- and intra-animal) of Cmax is commonly greater than that of AUC, some authorities have allowed wider confidence intervals (e.g., 0.70-1.43) for log-transformed Cmax, provided this is specified and justified in the study protocol. 

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