Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat 312: Lecture 23 Categorical Data Moo K. Chung [email protected] April 22, 2003 Concepts 1. Multinomial data. There are k outcomes (categories): C 1 , C 2 , · · · , C k . The probabil- ity of sample X i belonging to category C j is p j , i.e. P ( X i C 1 ) = p 1 , · · · , P ( X i C k ) = p k . 2. The null hypothesis of interest is H 0 : p 1 = c 1 , · · · , p k = c k for given constants c 1 , · · · , c k ( c 1 + · · · + c k = 1). H 1 : p j 6 = c j for some j. 3. Let x 1 , · · · , x n be the observations. Sup- pose that there are N j items in category C j . Note that N 1 + · · · + N k = n . We as- sume N 1 , · · · N j to be distributed as multi- nomial with parameter p 1 , · · · , p k . The ex- pected number of elements in category C j is EN j = np j . 4. Chi- squared goodness-of-fit test. The test procedure involes measuring the relative
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