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.