Confidence sampling

When a single sorbent is not sufficiently efficient in capturing a wide range of VOCs, combinations of sorbents are employed to increase the range of compounds that can be confidently sampled. Consequently, multibed sorbents made up of Anasorb GCBl, Carbotrap and CarbopackB have been employed in some validated methods [30]. Similarly, multibed sorbents consisting of CarbopackC, CarbopackB and Carbosieve Sill [27] and CarbopackB and Carbosieve Sill [10] have been used to trap a wide diversity of indoor VOCs. 

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. 

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 ability to demonstrate that two samples have different amounts of analyte is an essential part of many analyses. A method s sensitivity is a measure of its ability to establish that such differences are significant. Sensitivity is often confused with a method s detection limit. The detection limit is the smallest amount of analyte that can be determined with confidence. The detection limit, therefore, is a statistical parameter and is discussed in Chapter 4. 

Confidence intervals also can be reported using the mean for a sample of size n, drawn from a population of known O. The standard deviation for the mean value. Ox, which also is known as the standard error of the mean, is... 

In Section 4D.2 we introduced two probability distributions commonly encountered when studying populations. The construction of confidence intervals for a normally distributed population was the subject of Section 4D.3. We have yet to address, however, how we can identify the probability distribution for a given population. In Examples 4.11-4.14 we assumed that the amount of aspirin in analgesic tablets is normally distributed. We are justified in asking how this can be determined without analyzing every member of the population. When we cannot study the whole population, or when we cannot predict the mathematical form of a population s probability distribution, we must deduce the distribution from a limited sampling of its members. 

Earlier we introduced the confidence interval as a way to report the most probable value for a population s mean, p, when the population s standard deviation, O, is known. Since is an unbiased estimator of O, it should be possible to construct confidence intervals for samples by replacing O in equations 4.10 and 4.11 with s. Two complications arise, however. The first is that we cannot define for a single member of a population. Consequently, equation 4.10 cannot be extended to situations in which is used as an estimator of O. In other words, when O is unknown, we cannot construct a confidence interval for p, by sampling only a single member of the population. 

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. 

There is a temptation when analyzing data to plug numbers into an equation, carry out the calculation, and report the result. This is never a good idea, and you should develop the habit of constantly reviewing and evaluating your data. For example, if analyzing five samples gives an analyte s mean concentration as 0.67 ppm with a standard deviation of 0.64 ppm, then the 95% confidence interval is... 

In the previous section we noted that the result of an analysis is best expressed as a confidence interval. For example, a 95% confidence interval for the mean of five results gives the range in which we expect to find the mean for 95% of all samples of equal size, drawn from the same population. Alternatively, and in the absence of determinate errors, the 95% confidence interval indicates the range of values in which we expect to find the population s true mean. 

The probabilistic nature of a confidence interval provides an opportunity to ask and answer questions comparing a sample s mean or variance to either the accepted values for its population or similar values obtained for other samples. For example, confidence intervals can be used to answer questions such as Does a newly developed method for the analysis of cholesterol in blood give results that are significantly different from those obtained when using a standard method or Is there a significant variation in the chemical composition of rainwater collected at different sites downwind from a coalburning utility plant In this section we introduce a general approach to the statistical analysis of data. Specific statistical methods of analysis are covered in Section 4F. 

Relationship between confidence intervals and results of a significance test, (a) The shaded area under the normal distribution curves shows the apparent confidence intervals for the sample based on fexp. The solid bars in (b) and (c) show the actual confidence intervals that can be explained by indeterminate error using the critical value of (a,v). In part (b) the null hypothesis is rejected and the alternative hypothesis is accepted. In part (c) the null hypothesis is retained. 

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. 

Unpaired Data Consider two samples, A and B, for which mean values, Xa and Ab, and standard deviations, sa and sb, have been measured. Confidence intervals for Pa and Pb can be written for both samples... 

Finally, we have seen that the detection limit is a statistical statement about the smallest amount of analyte that can be detected with confidence. A detection limit is not exact because its value depends on how willing we are to falsely report the analyte s presence or absence in a sample. When reporting a detection limit, you should clearly indicate how you arrived at its value. 

Three replicate determinations are made of the signal for a sample containing an unknown concentration of analyte, yielding values of 29.32, 29.16, and 29.51. Using the regression line from Examples 5.10 and 5.11, determine the analyte s concentration, Ca, and its 95% confidence interval. 

Construct an appropriate standard additions calibration curve, and use a linear regression analysis to determine the concentration of analyte in the original sample and its 95% confidence interval. 

In the previous section we considered the amount of sample needed to minimize the sampling variance. Another important consideration is the number of samples required to achieve a desired maximum sampling error. If samples drawn from the target population are normally distributed, then the following equation describes the confidence interval for the sampling error... 

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 ... 

Determine the %CO for each sample, and report the mean value and the 95% confidence interval. 

How Many Samples. A first step in deciding how many samples to collect is to divide what constitutes an overexposure by how much or how often an exposure can go over the exposure criteria limit before it is considered important. Given this quantification of importance it is then possible to calculate, using an assumed variabihty, how many samples are required to demonstrate just the significance of an important difference if one exists (5). This is the minimum number of samples required for each hypothesis test, but more samples are usually collected. In the usual tolerance limit type of testing where the criteria is not more than some fraction of predicted exceedances at some confidence level, increasing the number of samples does not increase confidence as much as in tests of means. Thus it works out that the incremental benefit above about seven samples is small. 

Measurement Method Selection. A measurement method should meet sampling strategy requirements to the degree that the data can be used for decision making. This does not mean that it must be the optimum method with respect to all requirements. The range of methods available is limited and it is often necessary to select a method deficient in one or more attributes but which can yield data from which conclusions can be drawn with the desired degree of confidence. Some of the attributes to be considered in selecting a method foUow. 

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. 

Success Testing. Acceptance life tests ate sometimes planned with no failures allowed. This gives the smallest sample size necessary to demonstrate a rehabiUty at a given confidence level The rehabiUty is demonstrated relative to the test employed and the testing period. 

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