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Even though you will not have gone through a formal exercise of stating a pair of hypotheses about the value of a parameter and structuring a formal decision rule for how to reject the null, the group, through its reaction to the unexpected experimental results, has, in fact, performed a basic hypothesis test. By reacting with surprise to the high percentage of zeros observed (identifying a result as highly unlikely, in the tail of the sampling distribution) and suggesting an alternative answer to the question of what the chance of a 0 is, they have gone through the equivalent of a test of the null hypothesis that the probability is one-tenth.You can build on this, explaining that this exercise is typical of one of the major types of activities carried out in statistics, using the results in a sample as the basis for drawing some conclusions about the characteristics of a population of interest, and giving brief explanations of what hypothesis testing and point estimation are. Philosophical Issues in Model DevelopmentFinally, the experiment can be used as the basis for a philosophical discussion of the process of developing a model for a particular real-world situation. In particular, it can be used to illustrate the problems associated with one's perceptions, rather than reality, as the basis for a model and the need to revise models as information is collected and compared with the predictions of the original model. Then, by revealing the actual contents of the box, you can respond to the initial questions that you may have had to duck and illustrate the mln approach to assessing probabilities. SUMMARYA variation on the standard lottery-type box-of-balls experiment has been presented. This experiment has shown itself to be useful, in my experience, in introducing various concepts associated with the assessment of probability, the ideas of point estimation and hypothesis testing, and some of the issues one needs to be aware of in developing a model for an unknown system.For testing the equality of two independent binomial populations the Fisher exact test and the chi-squared test with Yates's continuity correction are often suggested for small and intermediate size samples. The use of these tests is inappropriate in that they are extremely conservative. In this article we demonstrate that, even for small samples, the uncorrected chi-squared test (i.e., the Pearson chi-squared test) and the two-independent-sample r test are robust in that their actual significance levels are usually close to or smaller than the nominal levels. We encourage the use of these latter two tests.KEY WORDS: Fisher exact test; Ill-use of the continuity correction; Robustncss of the t test; Small sample testing.
2 PrefaceThe main purpose of this book is to document the mathematics of propagation of error and regression analysis-two of the most common statistical analysis procedures in the physical sciences. Both procedures transform input data and their uncertainties to output parameters and their uncertainties.There are several important assumptions that make the mathematics rigorous. For example, each input datum is assumed to be a random sample from a probability distribution such as might occur from measurement. The difference between the sample value and the mean of the distribution is taken as the random error and the difference between the true value associated with that datum and the mean is taken as the systematic error. Another assumption involves the transformation between the input and output. Namely, each input error must be small enough that a change to its value (over a range that would include nearly all of its likely values) should cause only proportional changes in the output parameters.Sticking to the mathematics, several important experimental issues are left unaddressed. For example, while real physical variables are often nuanced, we take the simplistic view that the quantity underlying a datum is perfectly well defined and has an exact true value-the sample value one would get if the random and systematic error could be brought to zero. While impractical, this model should be kept in mind when constructing one for a real physical apparatus.The assumptions above are not overly restrictive for experiments in the physical sciences where small measurement errors are common. Together with the principle of maximum likelihood, they lead to a robust set of statistical formulas and procedures. Covariance matrices describe the behavior of the input and output variations, Jacobian matrices describe the linear relationships among those variations, and linear algebra makes the math readable and concise.The target audience for this book is anyone who wants to learn the mathematics of error propagation and regression analysis. I particularly enjoyed learning how applying the principle of maximum likelihood to regression analysis with independent variables governed by the Poisson, binomial, and exponential distributions leads to a chi-square minimization. In each case, the mathematics is quite specific about what to use for the variances appearing in the chi-square calculation so as to achieve the maximum likelihood 4 condition. Figuring out how to propagate an instrument's calibration uncertainty to the uncertainty in any results obtained using that instrument is one of the original problems I wanted to resolve. The mathematics turned out quite simple and straightforward.I hope the reader will find clarity and rigor in the treatment presented and beauty and simplicity in the final results.
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