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Unformatted text preview: Ch.7: Estimation 1 Introduction The objective of data collection is to learn about the distribution, or aspects of the distribution, of some characteristic of the units of a population of interest. In Chapter 6, we saw how to estimate certain key descriptive parameters of a population distribution, such as the mean, the variance, percentiles, and probabilities (or proportions), from the corresponding sample quantities. In this chapter we will learn another approach to the estimation of such population parameters. This approach is based on the assumption that the population distribution belongs in a certain family of distribution models, and on methods for fitting a particular family of distribution models to data. Different fitting methods will be presented and discussed. The estimators obtained from this other approach will occasionally differ from the esti- mators we saw in Chapter 6. For example, under the assumption that the population distribution is normal, estimators of population percentiles and proportions depend only on the sample mean and sample variance, and thus differ from the sample percentiles and proportions; the assumption of a uniform distribution yields an estimator of the popu- lation mean value which is different from the sample mean; the assumption of Poisson distribution yields an estimator of the population variance which is different from the sample variance. Another learning objective of this chapter is to develop criteria for selecting the best among different estimators of the same quantity, or parameter. For example, should the alternative estimators, which were mentioned in the preceding paragraph, be preferred over those which were discussed in Chapter 6? The same criteria can also help us de- cide whether a stratified sample is preferable to simple random sample for estimating the population mean or a population proportion. Finally, in this chapter we will learn how to report the uncertainty of estimators through their standard error and how that leads to confidence intervals for estimators which have (or have approximately) a normal distribution. The above estimation concepts will be developed here in the context of a single sample, but will be applied in later chapters to samples from several populations. 1 2 Overview, Notation and Terminology Many families of distribution models, including all we have discussed, depend on a small number of parameters; for example, a Poisson distribution model is identified by the single parameter , and normal models are identified by two parameters, and 2 . Such families of distribution models are called parametric . An approach to extrapolating sample information to the population is to assume that the population distribution is a member of (or belongs in) a specific parametric family of distribution models, and then fit the assumed family to the data, i.e. identify the member of the parametric family that best fits the data....
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- Fall '00