Let niyi be a bin(ni

, ????i) variate for group i, i = 1,…,N, with {yi} independent. Consider the null model, for which ????1 = ⋯ = ????N. Show that

̂???? = (

∑

i ni yi

)∕(

∑

i ni). When all ni = 1, for testing goodness of fit of the null model in the N × 2 table, show that X2 = N.