Stat400lec36 - Test statistic 2 2 2 1 10 2 20 1 10 20 k k k k y np y np y np q np np np-= Critical region P-value= Example Ping Ma Lecture 36 Fall

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Statistics 400 Section 8.4 Chi-square Goodness of Fit Tests Central Limit Theorem for Binomial Distribution Y 1 ~ b ( n, p 1 ) and Y 2 =n-Y 1 p 2 =1-p 1 Z= 1 1 1 1 (1 ) Y np np p - - N(0,1) “sufficiently large”: np 1 5 and n(1-p 1 ) 5 2 2 2 1 1 1 1 1 1 1 1 1 1 1 ( ) ( ) ( ) (1 ) (1 ) Y np Y np Y np Q np p np n p - - - = = + - - ~ Fact: 2 2 1 1 2 2 1 1 2 ( ) ( ) Y np Y np Q np np - - = + Q 1 measure the closeness of the observed numbers to the corresponding expected numbers (goodness of fit) The above result can be easily generalized to Multinomial distribution. The joint pmf of 1 2 , ,..., k Y Y Y is 1 2 1 2 1 2 1 2 ! ( , ,..., ) ... ! !... ! k y y y k k k n f y y y p p p y y y = 2 2 2 1 1 2 2 1 1 2 ( ) ( ) ( ) ... k k k k Y np Y np Y np Q np np np - - - - = + + + ~ Ping Ma Lecture 36 Fall 2005 - 1 -

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1 2 k Y1 Y2 …. Yk Test the hypothesis H0: p i =p i0 i=1,2,…,k Ha: at least one of p i ≠p i0
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Unformatted text preview: Test statistic 2 2 2 1 10 2 20 1 10 20 ( ) ( ) ( ) ... k k k k y np y np y np q np np np----= + + + Critical region P-value= Example Ping Ma Lecture 36 Fall 2005- 2 -A bag of candy colored brown, orange, green and yellow. Test the null hypothesis at a significance level of 0.05 that the machine filling these bags treats the four colors of candy equally, that is, test H0: p B =p O =p G =p Y =1/4. The observed values are listed as following, Brown Orange Green Yellow 42 64 53 65 Ping Ma Lecture 36 Fall 2005- 3 -...
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This note was uploaded on 07/24/2008 for the course STAT 400 taught by Professor Tba during the Fall '05 term at University of Illinois at Urbana–Champaign.

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Stat400lec36 - Test statistic 2 2 2 1 10 2 20 1 10 20 k k k k y np y np y np q np np np-= Critical region P-value= Example Ping Ma Lecture 36 Fall

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