Statistics sample size

The quantity of sample required comprises two parts the volume and the statistical sample size. The sample volume is selected to permit completion of all required analytical procedures. The sample size is the necessary number of samples taken from a stream to characterize the lot. Sound statistical practices are not always feasible either physically or economically in industry because of cost or accessibiUty. In most sampling procedures, samples are taken at different levels and locations to form a composite sample. If some prior estimate of the population mean, and population standard deviation. O, are known or may be estimated, then the difference between that mean and the mean, x, in a sample of n items is given by the following ... 

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 ... 

Relationship between chemical sample size and statistical sample size... 

Equation (2.2) is similar to a statistical formula relating sampling variation to statistical sample size. When statisticians refer to sample size, they mean the number of units in the physical sample. When chemists refer to sample size, they mean the mass, weight, or volume of the physical sample. This is a basic difference in the use of the terminology as well as in what is being sampled. We discuss here the relationship between the two and how they affect the variation of our estimates. 

CHEMICAL SAMPLE SIZE AND STATISTICAL SAMPLE SIZE 13... 

This relationship is very important for two reasons. First, the statistical sample size n (number of units) and the chemical sample size Ms (mass) are directly proportional. They increase or decrease together when the mass Mu of each unit is fixed. If we increase the mass Mg of the sample, we are also increasing n. That is, we are sampling more units. We saw from (2.3) that increasing the statistical sample size n (the number of units in the sample)... 

People will differ in perception based on technical considerations. For example, some people have difficulty applying probability and numbers. People will differ in how they interpret statistics, sample sizes in studies and numbers that represent risk or benefits. Some people will simply overrate the danger of a chemical, for example, because it has had adverse publicity. There are many examples. Some case studies in other chapters provide illustrations. 

Doppler effect function, Eq. (19) total neutron angular momentum Boltzmann constant multiplication constant neutron orbital angular momentum likelihood function, Eq. (81) statistical sample size in Section IX isotopic number density statistical distribution function, Eq. (75) defined by Eq. (91)... 

Limitations concerning the sample of the participants need to be considered and this limits the generalizability of the results. At the same time, the statistical sample size of the small and micro enterprises (1-9 workers) is under-represented. 

In general, we assume that the statistics of counting can be adequately described by the Poisson distribution. When we calculated the various decision limits, we effectively assumed, for simplicity, the Normal distribution for the counts. We know, however, that Poisson statistics are only applicable when the probability of detection of the decay of any particular radioactive atom within the count period is small and when the statistical sample size is large. There are a number of circumstances when these conditions may not be met and we should consider whether the statistical treatment above is still valid. 

Figure 2.10 Variation of the density of sheets with the longitudinal distance. Statistics sample size 64, polymer volume fraction c() = 0.2, e = — 1, 10 independent samples, 10 time steps. (From Ref [37].)...

Vitha, M. F. Carr, P. W. A Laboratory Exercise in Statistical Analysis of Data, /. Chem. Educ. 1997, 74, 998-1000. Students determine the average weight of vitamin E pills using several different methods (one at a time, in sets of ten pills, and in sets of 100 pills). The data collected by the class are pooled together, plotted as histograms, and compared with results predicted by a normal distribution. The histograms and standard deviations for the pooled data also show the effect of sample size on the standard error of the mean. 

The procedure for testing the significance of a sample proportion follows that for a sample mean. In this case, however, owing to the nature of the problem the appropriate test statistic is Z. This follows from the fact that the null hypothesis requires the specification of the goal or reference quantity po, and since the distribution is a binomial proportion, the associated variance is [pdl — po)]n under the null hypothesis. The primary requirement is that the sample size n satisfy normal approximation criteria for a binomial proportion, roughly np > 5 and n(l — p) > 5. 

The role of quality in reliability would seem obvious, and yet at times has been rather elusive. While it seems intuitively correct, it is difficult to measure. Since much of the equipment discussed in this book is built as a custom engineered product, the classic statistical methods do not readily apply. Even for the smaller, more standardized rotary units discussed in Chapter 4, the production runs are not high, keeping the sample size too small for a classical statistical analysis. Run adjustments are difficult if the run is complete before the data can be analyzed. However, modified methods have been developed that do provide useful statistical information. These data can be used to determine a machine tool s capability, which must be known for proper machine selection to match the required precision of a part. The information can also be used to test for continuous improvement in the work process. 

Equipment failure rate data points carry varying degrees of uncertainty expressed by two measures, confidence and tolerance. Confidence, the statistical measurement of uncertainty, expresses how well the experimentally measured parameter represents the actual parameter. Confidence in the data increases as the sample size is increased. 

In a data set it may be desirable to ask the question Is any one value significantly different from the others in the sample A t statistic (for n — 1 degrees of freedom where the sample size is n) can be calculated that takes into account the difference of the magnitude of that one value (xj and the mean of the sample (x ) ... 

The number of subjects planned to be enrolled, if more than one site the numbers of enrolled subjects projected for each trial site should be specified. Reason for choice of sample size include calculations of the statistical power of the trial, the level of significance to be used and the clinical justification. 

Reconfiguration of Data. Drug safety data from different sources are often pooled or combined in databases. Reasons for combining data vary. In the case of premarketing studies, data from different sites are routinely combined because one site may not be able to recruit enough patients for a study. Data from different studies are often combined to increase sample size and therefore statistical power for detecting an uncommon adverse event. 

Sturm R, Uniitzer J, Katon W (1999). Effectiveness research and implications for study des n sample size and statistical power. Gen Hosp Psychiatry 21, 274—83. 

The value of spruce-oil chemistry in sorting out problems of hybridization and introgression—major factors in Picea taxonomy—was succinctly summarized by von Rudloff who defined three situations (1) Terpene variation is limited such that it is not possible to use these characters in studies of introgression this is the case in eastern North America where the ranges of black spruce and red spruce overlap. (2) Sufficient variation in terpene profiles exists for the compounds to be useful markers in systematic studies as seen in white spruce. Brewer s spruce, and Sitka spruce. (3) Tree-to-tree variation in terpene content is so variable that use in che-mosystematic studies is precluded, or at least requires very large sample sizes for statistical reliability, as seen with Engelmann s spruce. 

Often there is a desire to compare responses to multiple treatments rather than simply evaluate active against a placebo control. For instance, it may be useful to evaluate several doses or to assess a product against another marketed product. Increasing the number of treatments will increase the sample size required overall, but will also increase the number of subjects required per treatment arm because the number of statistical comparisons is larger. If all between-group comparisons are to be made, the number of statistical tests increases dramatically as the number of treatment arms increases. With two groups, only one comparison is possible. With three groups, the number... 

Statistical methods are often employed to determine the study sample size and optimize power. Outlining the methods for calculating sample size and power for clinical trials is beyond the scope of this chapter. Interested readers are referred to texts by Chow and Liu (1998), Hulley and Cummings (1988), and Shuster (1990) for specific information on sample size and power estimation methods. 

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