The NCAA basketball tournament begins with 64 teams that are apportioned into four regional tournaments, each involving 16 teams. The 16 teams in each region are then ranked (seeded) from 1 to 16.

During the 12-year period from 1991 to 2002, the top-ranked team won its regional tournament 22 times, the second-ranked team won 10 times, the third-ranked team won 5 times, and the remaining 11 regional tournaments were won by teams ranked lower than 3.

Let Pij denote the probability that the team ranked i in its region is victorious in its game against the team ranked j. Once the Pij’s are available, it is possible to compute the probability that any particular seed wins its regional tournament (a complicated calculation because the number of outcomes in the sample space is quite large). The paper “Probability Models for the NCAA Regional Basketball Tournaments” (American Statistician, 1991: 35–38) proposed several different models for the Pij’s.

a. One model postulated with

(from which , etc.). Based on this,

, and . Does this model appear to provide a good fit to the data?

.20834 P(seed #3wins) 5 .15429 P(seed #1wins) 5 .27477, P(seed #2wins) 5 l 5 1/32 P16,1 5 l, P16,2 5 2l Pij 5 .5 2 l(i 2 j)

b. A more sophisticated model has game probabilities , where the z’s are measures of relative strengths related to standard normal percentiles [percentiles for successive highly seeded teams are closer together than is the case for teams seeded lower, and .2813625 ensures that the range of probabilities is the same as for the model in part (a)]. The resulting probabilities of seeds 1, 2, or 3 winning their regional tournaments are .45883, .18813, and .11032, respectively.
Assess the fit of this model.