Let Y i be a bin(n i , i ) variate for group i, i =

Let Y_i be a bin(n_i, π_i) variate for group i, i = 1, ...... N, with {Yi} independent. Consider the model that π₁ = .... = π_N. Denote that common value by π. For observations {y_i} show that π̂ = (∑y_i)/(∑n_i).

When all n_i = 1, for testing this model’s fit in the N × 2 table, show that X² = n. Thus, goodness-of-fit statistics can be completely uninformative for ungrouped data.