The NCAA basketball tournament begins with 64 teamsthat are apportioned into four

Chapter 14, Problem 46

(choose chapter or problem)

The NCAA basketball tournament begins with 64 teamsthat are apportioned into four regional tournaments,each involving 16 teams. The 16 teams in each regionare then ranked (seeded) from 1 to 16. During the12-year period from 1991 to 2002, the top-ranked teamwon its regional tournament 22times, the second-rankedteam won 10 times, the third-ranked team won 5 times,and the remaining 11 regional tournaments were won byteams ranked lower than 3. Let Pij denote the probabilitythat the team ranked i in its region is victorious in itsgame against the team ranked j. Once the Pijs are available,it is possible to compute the probability that anyparticular seed wins its regional tournament (a complicatedcalculation because the number of outcomes in thesample space is quite large). The paper ProbabilityModels for the NCAA Regional BasketballTournaments (American Statistician, 1991: 3538)proposed several different models for thePijs.a. One model postulated Pij 5 .5 2 lsi 2 jd withl 5 1y32 (from which P16,1 5 l, P16,2 5 2l, etc.).Based on this, P(seed # 1 wins) 5 .27477, P(seed #2wins) 5 .20834, and P(seed # 3 wins) 5 .15429.Does this model appear to provide a good fit to thedata?b. A more sophisticated model has game probabilitiesPij 5 .5 1 .2813625 szi 2 zjd, where the zs are measuresof relative strengths related to standard normalpercentiles [percentiles for successive highly seededteams are closer together than is the case for teamsseeded lower, and .2813625 ensures that the range ofprobabilities is the same as for the model in part (a)].The resulting probabilities of seeds 1, 2, or 3 winningtheir regional tournaments are .45883, .18813, and.11032, respectively. Assess the fit of this model.SPSS output for Exercise 43Crosstabulation: AREA BY CATEGORY Count Exp ValCATEGORY Row Pct RowAREA Col Pct 1.00 2.00 3.00 4.00 5.00 Total1.00 20 28 23 14 12 97 12.7 24.7 18.0 16.0 25.7 33.3% 20.6% 28.9% 23.7% 14.4% 12.4% 52.6% 37.8% 42.6% 29.2% 15.6%2.00 14 34 21 14 12 95 12.4 24.2 17.6 15.7 25.1 32.6% 14.7% 35.8% 22.1% 14.7% 12.6% 36.8% 45.9% 38.9% 29.2% 15.6%3.00 4 12 10 20 53 9912.9 25.2 18.4 16.3 26.2 34.0% 4.0% 12.1% 10.1% 20.2% 53.5% 10.5% 16.2% 18.5% 41.7% 68.8% Column 38 74 54 48 77 291 Total 13.1% 25.4% 18.6% 16.5% 26.5% 100.0%Chi-Square D.F. Significance Min E.F. Cells with E.F. , 5 70.64156 8 .0000 12.405 None