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Suppose we want to formally test the hypothesis that the distribution F ( x ) of the mechanical device failure data is lognormal with µ = 1.0 and σ = 1.0. In other words, the classical approach to testing H 0 is to fix a significance level α and then require that the test have the property that the probability of a type I error occurring can never be greater than α. The value α, called the level of significance of the test, is usually set in advance, with commonly chosen values being α =. The classical way of accomplishing this is to specify a value α and then require the test to have the property that whenever H 0 is true its probability of being rejected is never greater than α. Hence, with this objective it seems reasonable that H 0 should only be rejected if the resultant data are very unlikely when H 0 is true. Now, as was previously mentioned, the objective of a statistical test of H 0 is not to explicitly determine whether or not H 0 is true but rather to determine if its validity is consistent with the resultant data. The second, called a type II error, results if the test calls for accepting H 0 when it is false. The first of these, called a type I error, is said to result if the test incorrectly calls for rejecting H 0 when it is indeed correct. It is important to note when developing a procedure for testing a given null hypothesis H 0 that, in any test, two different types of errors can result. Thus, this test calls for rejection of the null hypothesis that θ = 1 when the sample average differs from 1 by more than 1.96 divided by the square root of the sample size. There is one table for each probability (tail area), and the values in the table correspond to F values for numerator degrees of freedom ν 1 indicated by column headings, and denominator degrees of freedom ν 2 as row headings. The choice of which variance estimate to place in the numerator is somewhat arbitrary hence the table of probabilities of the F distribution always gives the right tail value.Īppendix Table A.4 gives values of the F distribution for selected degrees of freedom combinations for right tail areas of 0.1, 0.05, 0.025, and 0.01.
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Scientific calculators, spreadsheet applications, and statistical software have more powerful tools for calculating probabilities from this distribution. Fortunately, for most practical problems only a relatively few probability values are needed. 2.Ī different table is needed for each combination of degrees of freedom. The F distribution is defined only for nonnegative values. However, some of the characteristics of the F distribution are of interest: 1. The equation describing the distribution of the F statistic is also quite complex and is of little use to us in this text.
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Also, if both populations have equal variance, that is, σ 1 2 = σ 2 2, the F statistic is simply the ratio S 1 2 ∕ S 2 2. If the variances are estimated in the usual manner, the degrees of freedom are ( n 1 − 1 ) and ( n 2 − 1 ), respectively. The distribution is denoted by F ( ν 1, ν 2 ). The F distribution has two parameters, ν 1 and ν 2.