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chap18_2010 - Chapter 18 Chi-Square Tests χ 2 Distribution...

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Unformatted text preview: Chapter 18 Chi-Square Tests χ 2 Distribution • Let x 1 , x 2 , .. x n be a random sample from a normal distribution with μ and σ 2 , and let s 2 be the sample variance, then the random variable (n-1)s 2 / σ 2 has χ 2 distribution with n-1 degrees of freedom. • Probability Density Function, with k degrees of freedom, • Mean • Variance • Mode = k-2 (when k ≥ 3) ) 2 ( 2 1 ) ( 2 / 1 ) 2 / ( 2 Γ =-- x e x k x f x k k k x E = ) ( k x V 2 ) ( = χ 2 Distribution fr.academic.ru/pictures/frwiki/67/Chi-square_.. χ 2 Distribution • Goodness-of-fit Tests • Tests of Independence • Tests of Homogeneity Multinominal Experiments A Multinomial experiment is a statistical experiment that has the following properties: • It consists of n repeated trials (repetitions). • Each trial can result in one of k possible outcomes. • The trials are independent. • The probabilities of the various events remain constant for each trial. Goodness-of-fit Test • Observed Frequencies (O i ): Frequencies obtained from the actual performance of an experiment. • Goodness-of-fit Test: Test of null hypothesis that the observed frequencies follow certain pattern or theoretical distributions, expressed by the Expected Frequencies (E i ). Goodness-of-fit Test for Multinominal Experiments • Degree of freedom = k -1, where k is the number of categories • Chi-square goodness-of-fit test is always a right-tailed test • Sample size should be large enough so that the expected frequency for each category is at least 5....
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This note was uploaded on 12/03/2011 for the course EIN 5226 taught by Professor Staff during the Fall '11 term at FIU.

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chap18_2010 - Chapter 18 Chi-Square Tests χ 2 Distribution...

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