Fundamentals of Probability and Statistics for Engineers

316 Fundamentals of Probability and Statistics for Engineersmass function (pmf) p(x; q), where q may be specified or unspecified. We denoteby hypothesis H the hypothesis that the sample represents n values of a randomvariable with pdf f (x; q) or p(x; q). This hypothesis is called a simple hypothesiswhen the underlying distribution is completely specified; that is, the parametervalues are specified together with the functional form of the pdf or the pmf;otherwise, it is a composite hypothesis. To construct a criterion for hypothesestesting, it is necessary that an alternative hypothesis be established againstwhich hypothesis H can be tested. An example of an alternative hypothesis issimply another hypothesized distribution, or, as another example, hypothesisH can be tested against the alternative hypothesis that hypothesis H is not true.In our applications, the latter choice is considered more practical and we shallin general deal with the task of either accepting or rejecting hypothesis H onthe basis of a sample from the population.10.1.1 TYPE-I AND TYPE-II ERRORSAs in parameter estimation, errors or risks are inherent in deciding whether ahypothesis H should be accepted or rejected on the basis of sample information.Tests for hypotheses testing are therefore generally compared in terms of theprobabilities of errors that might be committed. There are basically two typesof errors that are likely to be made – namely, reject H when in fact H is true or,alternatively, accept H when in fact H is false. We formalize the above withDefinition 10.1.D efinition 10. 1. in testing hypothesis H, a Type-I error is committed when His rejected when in fact H is true; a Type-II error is committed when H isaccepted when in fact H is false.In hypotheses testing, an important consideration in constructing statisticaltests is thus to control, insofar as possible, the probabilities of making theseerrors. Let us note that, for a given test, an evaluation of Type-I errors can bemade when hypothesis H is given, that is, when a hypothesized distribution isspecified. In contrast, the specification of an alternative hypothesis dictatesType-II error probabilities. In our problem, the alternative hypothesis is simplythat hypothesis H is not true. The fact that the class of alternatives is so largemakes it difficult to use Type-II errors as a criterion. In what follows, methodsof hypotheses testing are discussed based on Type-I errors only.10.2 CHI-SQUARED GOODNESS-OF-FIT TESTAs mentioned above, the problem to be addressed is one of testing hypothesis Hthat specifies the probability distribution for a population X compared with theTLFeBOOK