The following data on X = number of corrosion defects per segment is consistent with data on one of the pipelines described in the article “The Negative Binomial Distribution as a Model for External Corrosion Defect Counts in Buried Pipelines” (Corr. Sci. 2015: 114–131):

Based on these n = 103 randomly selected segments, the authors estimate the negative binomial parameters to be r̂ = 1.272 and p̂ = .258. Test the hypothesis that the data is consistent with a generalized negative binomial distribution at the .10 significance level. To ensure that all expected counts are > 5, define “cells” by x = 0, 1, …, 6, 7–8, and > 9.

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0 1 2 Freq. 17 15 17 3 4 19 9 55 X 9 10 11 12 13 14 15 Freq. 0 30 1 1 2 1 k(r,x)= for x ≥ 1 6 4 75 5 The authors propose a generalized negative binomial model for X, x! 83 16 17 0 1 pmf nb(x; r,p) =k(r,x) x p' (1-P)* for x = 0, 1, 2, ..., where k(r, 0) = 1 and r(r+1) X, which has (r +x-1)