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Examples and Problems in Mathematical Statistics by Shelemyahu Zacks

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CHAPTER 4

Testing Statistical Hypotheses

PART I: THEORY

4.1 THE GENERAL FRAMEWORK

Statistical hypotheses are statements about the unknown characteristics of the distributions of observed random variables. The first step in testing statistical hypotheses is to formulate a statistical model that can represent the empirical phenomenon being studied and identify the subfamily of distributions corresponding to the hypothesis under consideration. The statistical model specifies the family of distributions relevant to the problem. Classical tests of significance, of the type that will be presented in the following sections, test whether the deviations of observed sample statistics from the values of the corresponding parameters, as specified by the hypotheses, cannot be ascribed just to randomness. Significant deviations lead to weakening of the hypotheses or to their rejection. This testing of the significance of deviations is generally done by constructing a test statistic based on the sample values, deriving the sampling distribution of the test statistic according to the model and the values of the parameters specified by the hypothesis, and rejecting the hypothesis if the observed value of the test statistic lies in an improbable region under the hypothesis. For example, if deviations from the hypothesis lead to large values of a nonnegative test statistic T(X), we compute the probability that future samples of the type drawn will yield values of T(X) at least as large as the presently ...

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