Number of samples required

The basic assumption, in analyzing data statistically, is that the samples are representative of the populations from which they are withdrawn. However, there is an uncertainty associated with any measured property and this can be estimated using the confidence interval (Cl). This can be expressed in general terms as  

At the 95% confidence level M= 1.96, i.e. there is a five in one hundred chance that the true mean lies outside these limits. The 90% confidence interval is smaller M= 1.645) and is less likely to have the true value of within its limits. The 99% confidence interval is larger (M= 2.576) but is more likely to contain the true value ofx within its limits. 

For pharmaceutical applications, a value r = 2 is used to denote working quality and a value / = 3 (99.9% confidence level) is used for total quality [35] The statistical reliability of analytical data can be improved by increasing the homogeneity of the sample (reducing cr), increasing the size of the sample, or increasing taken (increasing ). 

In most instances the population standard deviation is not known and must be estimated from the sample standard deviation (s). Substitution of s for cr in equation (1.1) with M = 1.96 does not result in a 95% confidence interval unless the sample number is infinitely large (in practice 30). When s is used, multipliers, whose values depend on sample number, are chosen from the /-distribution and the denominator in 

Assuming a normal distribution of variance, the number of samples required, to assume at the 95% confidence level that the median is known to 4, is given by 

---

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... 

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. 

In the case of sampling according to the scheme given in Fig. 2.4a, the critical number of samples required at least for reliable analytical results can be derived from Eq. (2.4) (Danzer [1995b]) ... 

Despite the difficulties of on-line automation, the need to develop such systems is considerable. The increase in the number of different compounds that must be determined and the number of samples required for a meaningful survey or laboratory study make it essential to improve the quality and throughput of samples. There are a number of stages in fully automating trace organic analysis. Autosampler LC or GC-data systems as GC-MS or GC-ion trap detector (ITD) are well established and require no further elaboration here [191, 203, 495]. 

Table II Calculation by Visman Equation of Number of Samples Required for Determination of Dieldrin with a Sampling Uncertainty of 50% Relative Standard Deviation in Test Data of Table I, Using Pairs of Large Cores Taken Perpendicular to Spray-Track Direction.

Complementary to IC, the capillary electrophoresis (CE) technique is useful for both anions and cations. The method is significantly faster than IC for screening and is relatively easy to automate which is advantageous when large numbers of samples require analysis. Although CE is currendy similar in terms of sensitivity to IC it is a relatively new technique and significant improvements in both selectivity and sensitivity continue to be made [26]. 

Table 12.4 Comparison of four different experimental design types, with respect to the number of samples required and the number of actual design variable levels that are used for the case of 2 specified levels of 3, 4 and 5 design variables...

For example, if the coefficient of variation is 2 and the level is twice the other then seventy-three samples are required to achieve a false negative rate of 5 percent. To achieve a false negative rate of 1 percent, one hundred and twelve samples would be required. As the acceptable difference between levels increase, the required number of sample required decreases. When one level is one hundred times the other and the coefficient of variation is 1, only three samples are required to achieve a false negative rate of 1 percent. 

The advantages of these methods are based on their simplicit - tlie number of samples required to construct the models is relatively small, statistics are available that can be used to validate the models, and it is easier to describe these methods to the users of the models. The classical methods are also multivariate in nature and, therefore, have good diagnostics tools that can be used to detect violations of the assumptions both during the calibration and prediction phases. 

Target fill < 5 ml the minimum number of samples required, per lot, is not less than ten. An approximately equal increment of samples is to be collected from each autoclave load of the lot. 

In short the approach based upon the concept of a limiting distribution offers a viable alternative to that based upon tolerance sets. The stated objective of reducing the number of samples required for making correct decisions has been achieved. Additional refinements in the selection of parameters for the limiting distribution should further enhance its applicability in evaluating acute exposures. 

The analyst must have reasonable answers to these questions in order to participate intelligently in decisions involving the number of samples required, the accuracy necessary, and the variability to be expected so he can provide analytical measurements adequately tailored to the importance of the decision required ... 

Table 8.4 Comparison of four different types of experimental design with regard to the number of samples required...

Design type Number of samples required (2 levels) Number of... 

Establishing the frequency (i.e., is toxicity consistent or transient between sample), degree (i.e., magnitude) and persistency (i.e., how toxicity changes over time) of toxicity will be important, since these factors can provide insight into the type of substance responsible for toxicity, and can also influence subsequent TRE activities. The actual number of samples required to assess these factors will be site-specific and depend predominantly on effluent variability. 

The cost of sampling and analysis can be minimized by determining the minimum number of samples required for representative description of the area under investigation. 

The number of samples required for representative assessment can be determined as follows. The relative confidence interval for a given mean value x from the sample size of n measured values is given by ... 

In Tab. 10-1 the calculated number of samples n required is demonstrated for probabilities of an error of the first kind of a = 0.1 and a = 0.25. With the exception of cadmium and lead the number of samples required is less than or equal to 10 for a probability of an error of the first kind of 25%. When the intake is well below the provisional tolerable weekly maximum, as in the case investigated [HAHN et al., 1992], the sample size for representative assessment can be reduced considerably. 

The selection of the threshold H can be based on considerations of average run length (ARL), the average number of samples required to detect a disturbance of specified magnitude. For example, suppose that the objective is to be able to detect if the sample mean x has shifted from the target by a small amount 8. The slack parameter K is usually specified as K = 0.58. For the ideal situation (e.g., normally distributed, uncorrelated disturbances), ARL values have been tabulated for different values of 8, K, and H. Table 8-7 summarizes ARL... 

Liess and Schulz12 have given a formula for predicting the number of samples required for an assumed uncertainty ... 

A simple approach to estimating the number of samples is to repeat the sample preparation and analysis to calculate an overall standard deviation, Using Student s t distribution, the number of samples required to achieve a given confidence level is calculated as... 

Relative standard deviation of repeat HPLC analysis of a drug metabolite standard was between 2 and 5%. Preliminary measurements of several serum samples via solid-phase extraction cleanup followed by HPLC analyses showed that the analyte concentration was between 5 and 15 mg/L and the standard deviation was 2.5 mg/L. The extraction step clearly increased the random error of the overall process. Calculate the number of samples required so that the sample mean would be within +1.2 mg/L of the population mean at the 95% confidence level. 

---