post6 - Chi-square Tests Karl Pearson(1900 2 Recall Xi N(i...

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Chi-square Tests Karl Pearson (1900) Recall: X i N ( μ i , σ 2 i ), indep. n X i =1 X i - μ i σ i ! 2 χ 2 ( n ) i.e. (n indep. std normal) 2 χ 2 ( n ) X 1 b ( n, p 1 ) Y n := X 1 - np 1 np 1 (1 - p 1 ) D N (0 , 1) Moreover, Y 2 n D χ 2 (1) i.e. Y n is approximately N (0 , 1) and Y 2 n is ap- proximately χ 2 (1) for large n . 1 Define X 2 = n - X 1 and p 2 = 1 - p 1 Then, Q 1 := Y 2 n = ( X 1 - np 1 ) 2 np 1 + ( X 1 - np 1 ) 2 n (1 - p 1 ) = ( X 1 - np 1 ) 2 np 1 + ( X 2 - np 2 ) 2 np 2 χ 2 (1) H 0 : p 1 = p 10 vs. H 1 : p 1 6 = p 10 Under H 0 , Q 1 = ( X 1 - np 10 ) 2 np 10 + ( X 2 - np 20 ) 2 np 20 χ 2 (1) Rejection region= { Q 1 c } This test is equivalent to the Z -test that we usually use. 2 A 1 , · · · , A k : a partition of sample space A , p i = P (outcome A i ) Repeat the experiment n times and count the frequencies of outcomes in A i X i = frequency of A i , ( X k = n - X 1 - · · · - X k - 1 ) X 1 , · · · , X k - 1 multinomial( n, p 1 , · · · , p k - 1 ) Then, Q k - 1 := k X i =1 ( X i - np i ) 2 np i χ 2 ( k - 1) 3 H 0 : p 1 = p 10 , p 2 = p 20 , · · · , p k - 1 = p k - 1 , 0 Under H 0 , Q k - 1 = k X i =1 ( X i - np i 0 ) 2 np i 0 χ 2 ( k - 1)
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