How Many Samples to Collect

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 

In Example 7.6 we found that an analysis for the inorganic ash content of a breakfast cereal required a sample of 1.5 g to establish a relative standard deviation for sampling of 2.0%. How many samples are needed to obtain a relative sampling error of no more than 0.80% at the 95% conhdence level  

Because the value of t depends on s, and the value of s is not yet known, we begin by letting n = °° and use the associated value of f. Erom Appendix IB, the value for f is 1.96. Substituting known values into equation 7.7 

Letting s = 24, the value of f from Appendix IB is 2.075. Recalculating s gives 

Since two successive calculations give the same value for n, an iterative solution has been found. Thus, 27 samples are needed to achieve the desired sampling error. 

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Equation 7.8 does not have a unique solution because different combinations of and give the same overall error. The choice of how many samples to collect and how many times each sample should be analyzed is determined by other concerns, such as the cost of collecting and analyzing samples, and the amount of available sample. 

More attention to selecting and obtaining a representative sample. The design of a statistically based sampling plan and its implementation are discussed earlier, and in more detail than in other textbooks. Topics that are covered include how to obtain a representative sample, how much sample to collect, how many samples to collect, how to minimize the overall variance for an analytical method, tools for collecting samples, and sample preservation. 

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. 

Even If the cost of estimation error cannot be quantified as this model requires, effective allocation of resources may be possible when detailed knowledge of sources of variation Is available. In this case, a replication strategy can be based on variance component and cost Information. For example, consider the problem of deciding how many samples to collect and how many analyses to perform on each sample Let... 

Another approach for experimental design for inverse modeling is termed a natural design. This is where many samples are collected over a period of time until one has confidence that the variation has been adequately represented. In some cases, these natural designs are the only choice because neither R nor c can be controlled. Successful use of this approach requires some knowledge of the sj stem in order to make an intelligent assessment of how many samples are required and how long to sample. One rule of thumb is to have at least three times as many samples as the expected rank of the system. (Rank is a concept that is discussed in Chapters 4 and 5.)... 

How many samples do we need to collect in order to establish that the mean concentration of lead is below the action level of /00 mgjkg with a confidence level of 95 percent ... 

How many samples do we need to collect for a given analysis How large must the sample be to insure that it is representative These types of questions can be answered by statistics. We also need to have a basic knowledge of statistics to understand the limitations in the other steps in method development, so we will now briefly introduce the statistical concepts and calculations used by analytical chemists. 

Can It be extracted and analyzed How should the urine be collected 24-hr, urines or a timed grab sample And finally, how many sampling periods (days) should the experiment entail In order to make statistically significant results likely ... 

Suppose you are to analyze a solid where the particles containing analyte represent only 1 X 10 % of the population. How many particles must be collected to give a relative sampling variance of 1% ... 

The emption of Mount St. Helens on May 18, 1980, provided geologists with a unique opportunity to study the action of volcanos. Gas samples from the plume were collected and analyzed for toxic heavy metals. To collect mercury (Hg), gas samples were passed over a piece of gold metal, which binds Hg atoms veiy tightly. The mass of the metal increased as it absorbed Hg from the plume. From a plume-gas sample containing 200 g of ash, 3.60 ft g of Hg was deposited on the gold. How many moles of mercury were present in the gas sample How many atoms is this ... 

How many products will be collected (statistical sample size), i.e., will the tail of the distribution need to be defined, or will the mean sufficiently address the issue of concern ... 

How, Then, Does OSHA Issue So Many Citations To support a citation, a compliance officer must demonstrate with a high degree of confidence that at least one employee was exposed above the standard on the day of the inspection. Assuming that day-to-day variability in exposures is approximately equal to the worker-to-worker variability on the day of the inspection, then 1t would be possible, in principle, for a compliance officer to collect n statistically independent samples. Under these conditions, the probability that the workplace will be found in compliance is equal to the probability that all n samples are less than the AL. Let Pn(OK) represent the probability of compliance after n samples are collected and P(0K) represent the probability that a single sample is less than AL. Then one has ... 

What factors can be used to predetermine the quality and utility of a method An analyst must consider the following questions Do I need a proximate analytical method that will determine all the protein, or carbohydrate, or lipid, or nucleic acid in a biological material Or do I need to determine one specific chemical compound among the thousands of compounds found in a food Do I need to determine one or more physical properties of a food How do I obtain a representative sample What size sample should I collect How do I store my samples until analysis What is the precision (reproducibility) and accuracy of the method or what other compounds and conditions could interfere with the analysis How do I determine whether the results are correct, as well as the precision and accuracy of a method How do I know that my standard curves are correct What blanks, controls and internal standards must be used How do I convert instrumental values (such as absorbance) to molar concentrations How many times should I repeat the analysis And how do I report my results with appropriate standard deviation and to the correct number of significant digits Is a rate of change method (i.e., velocity as in enzymatic assays) or a static method (independent of time) needed ... 

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