Statistics worked example

Guedens, W. J. Yperman, J. Mullens, J. et al. Statistical Analysis of Errors A Practical Approach for an Undergraduate Ghemistry Lab, Part 1. The Goncept, /. Chem. Educ. 1993, 70, 776-779 Part 2. Some Worked Examples, /. Chem. Educ. 1993, 70, 838-841. 

But for ordinary data, we would not expect such a sequence to happen. This is the reason most statistics work as general indicators of data performance the special cases that cause them to fail are themselves low-probability occurrences. In this case the problem is not whether or not the data are nonlinear, the problem is that they are nonrandom. This is a perfect example of the data failing to meet a criterion other than the one you are concerned with. Therefore the Durbin-Watson test fails, as would any statistical test fail for such data they are simply not amenable to meaningful statistical calculations. Nevertheless, a blind computation of the Durbin-Watson statistic would give an apparently satisfactory value. But this is a warning that other characteristics of the data can cause it to appear to meet the criteria. 

Table 2.5, together with the subsequent worked examples, illustrates the application of the statistical tests to real laboratory situations. Equation (2.10) is a simplified expression derived on the assumption that the precisions of the two sets of data are not significantly different. Thus the application of the F-test (equation (2.8)) is a prerequisite for its use. The evaluation of t in more general circumstances is of course possible, but from a much more complex expression requiring tedious calculations. Recent and rapid developments in desk top computers are removing the tedium and making use of the general expression more acceptable. The references at the end of the chapter will serve to amplify this point. 

Illustration, by way of worked examples, of the main statistical procedures for the calculation, display and reporting of the results. 

It first introduces the reader to the fundamentals of experimental design. Systems theory, response surface concepts, and basic statistics serve as a basis for the further development of matrix least squares and hypothesis testing. The effects of different experimental designs and different models on the variance-covariance matrix and on the analysis of variance (ANOVA) are extensively discussed. Applications and advanced topics such as confidence bands, rotatability, and confounding complete the text. Numerous worked examples are presented. 

We have deliberately chosen to focus on one particular therapeutic area for the worked examples throughout this book This strategy provides a unified approach in all of the discussions about clinical trials and statistical analyses. Our discussion and worked examples focus on the development of a new investigational drug for the treatment of high blood pressure. The term hypertension is used for this condition, and drugs intended for the treatment of hypertension are called antihypertensive drugs, or simply antihypertensives. 

The two-particle distribution tells us the probability that particles will be found at ri and r2 and will thereby serve as the basis for determining the energy of interaction between different pairs of particles when the best statements we can make about the system are statistical. For example, when describing the energetics of liquids our statements about the structure of that liquid will be couched in the language of distribution functions. In preparation for our later work on the free energy of liquids (see chap. 6), we find it convenient to introduce an auxilliary... 

Working examples of practical algorithms used in the author s laboratory are presented in Chapter 12. The statistical data used to construct them were gathered over a period of several years using the following specimens that were fixed routinely in 10% neutral-buffered formalin primary antibody incubations at 4°C for 16 to 18 hours the Elite avidin-biotin-peroxidase complex method of immunodetection (Vector Laboratories, Burlingame, Calif) and the antibody reagents listed in Table 11.1. 

Hence, in the light of our both accounts of causality, the molecular dynamics model represents causal processes or chains of events. But is the derivation of a molecule s structure by a molecular dynamics simulation a causal explanation Here the answer is no. The molecular dynamics model alone is not used to explain a causal story elucidating the time evolution of the molecule s conformations. It is used to find the equilibrium conformation situation that comes about a theoretically infinite time interval. The calculation of a molecule s trajectory is only the first step in deriving any observable structural property of this molecule. After a molecular dynamics search we have to screen its trajectory for the energetic minima. We apply the Boltzmann distribution principle to infer the most probable conformation of this molecule.17 It is not a causal principle at work here. This principle is derived from thermodynamics, and hence is statistical. For example, to derive the expression for the Boltzmann distribution, one crucial step is to determine the number of possible realizations there are for each specific distribution of items over a number of energy levels. There is no existing explanation for something like the molecular partition function for a system in thermodynamic equilibrium solely by means of causal processes or causal stories based on considerations on closest possible worlds. 

Table 2.5, together with the subsequent worked examples, illustrates the application of the statistical tests to real laboratory situations. Equation... 

The sampling of a number of items is discussed in the chapters on food, water and counting. It is obvious from the fact that sterility testing is destructive, that one cannot test the whole batch, and therefore sampling on a statistically significant basis must be carried out. The statistics are discussed briefly with a worked example, and also in the BP. ... 

With automated size analyzers and ready computer access, various analogous descriptive parameters can be much more easily and accurately calculated by moment statistics. Most sedimentology texts provide introductions to moment measures, but see the discussions in Friedman et al, (1992) and Friedman Johnson (1982) for particularly lucid descriptions of the computation technique, Lindholm (1987), Lewis (1984), and Griffiths (1967) provide several worked examples using sieve-derived size data. Swan et al. (1979) report the results of an interesting comparison study of moment and graphic statistics. 

This section summarizes the classical, equilibrium, statistical mechanics of many-particle systems, where the particles are described by their positions, q, and momenta, p. The section begins with a review of the definition of entropy and a derivation of the Boltzmann distribution and discusses the effects of fluctuations about the most probable state of a system. Some worked examples are presented to illustrate the thermodynamics of the nearly ideal gas and the Gaussian probability distribution for fluctuations. 

The meaning of an acceptance of H, is that each and every data set mean Xj is judged to be statistically indistinguishable from the grand mean Xq then the data are pooled, with Vg as the variance of Xq j with rfo/g = (wj. — 1) degrees of freedom. In the worked example... 

Mullins, E. 2003. Statistics for the Quality Control Chemistry Laboratory, Royal Society of Chemistry, Cambridge. (Despite the title this text contains a good deal of material on basic statistical concepts, and many worked examples.)... 

Since the caicuiation involved in using this statistic to test for normaiity is relatively complicated, it wiii not be described here. (A reference to a worked example is given at the end of the chapter.) The principle of the chi-squared test is more easily understood by means of the following example. 

The availability of this tremendous computing power naturally makes it all the more important that the scientist applies statistical methods rationally and correctly. To limit the length of the book, and to emphasize its practical bias, we have made no attempt to describe in detail the theoretical background of the statistical tests described. But we have tried to make it clear to the practising analyst which tests are appropriate to the types of problem likely to be encountered in the laboratory. There are worked examples in the text, and exercises for the reader at the end of each chapter. Many of these are based on the data provided by research papers published in The Analyst. We are deeply grateful to Mr. Phil Weston, the Editor, for... 

A competing concept of documenting medically (dermatologically) relevant descriptors, i.e., exposures and their duration and intensity, would be to classify not the individual occupation of the patient, but the type of industry in which he or she works. Examples of such classifications and some of their applications, e.g., employers liability insurance or governmental labor statistics, are listed below. A classification system provided by the United Nations (UN) for this purpose is the international standard industrial classification (ISIC) (United Nations 1990), as shown below. 

The work of Bliss (1951) on the use of statistical methods in vitamin research is a comprehensive publication well worth the detailed attention of food technologists interested in this type of analysis. Many designs are discussed in detail, accompanied by computation directions and worked examples of the methods. 

It seems obvious that the nine MCA and nine MCB scales in this chapter are interdependent to some extent. A systematic exploration of the similarities and differences between two given scales requires 72 correlations (36 for basicity and 36 for affinity) to be performed. If two given scales are found to be not, or poorly, correlated, further statistical work is necessary to understand the chemistry underlying the differences, for example by splitting up the sample of bases into families. 

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