5. Data from Various Sports 43
team is streaky is .79 (= .30/(.08 + .30)). This language corresponds
to a Bayesian approach for comparing the two models.
We will proceed in a different direction with the parameter Black.
We are trying to determine whether the Bernoulli model does a good
job of fitting the win-loss sequences in our data set. For each of the
390 teams in this data set, we can calculate a p-value for the observed
value of Black, under the null hypothesis of the Bernoulli model. For
each team, we use the winning percentage of that team, rounded to
the nearest .01, to compute a simulated distribution for Black. If there
are more streaky teams than can be explained well by the Bernoulli
model, we should see more small p-values than expected, i.e. the set
of p-values should not be uniformly distributed across the interval
[0, 1]. (The reason that the p-values of streaky teams are small is
because the parameter Black is very large for such teams, and large
values of Black correspond to small p-values.) Figure 24 shows this
set of p-values. The set has been sorted. If the set were perfectly
uniformly distributed across [0, 1], the graph would be the straight
line shown in the figure. One sees that the set of p-values is very
close to uniform, and in addition, there are certainly no more small
p-values than expected under the Bernoulli model. In fact, we see
that there are slightly more teams with large p-values than might be
expected. Since large p-values correspond to small values of Black,
this indicates that the number of very consistent teams is slightly
more than expected.
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