5. Data from Various Sports 49

There were 159 games in the subsets that we are dealing with.

Suppose that we assume the probability that the first team will beat

the spread is .5. What is the probability that in 159 trials, the first

team will actually beat the spread 46% of the time or less, or equiv-

alently, in at most 73 games? The answer is about .17, which a

statistician would not regard as significant.

The data can be pooled by considering all of the subsets (j, k)

with j positive (and j ≥ |k|). There were 1208 games of this type,

and the first team beat the spread 47.9% of the time. We can once

again ask the question: If a fair coin is flipped 1208 times, what is the

probability that we would see no more than 579 (= .479×1208) heads?

We can calculate this exactly, using a computer, or we can find an

accurate approximation, by recalling that the number of heads NH is

a binomially distributed random variable with mean equal to 604 (=

1208×.5) and standard deviation equal to 17.38 (=

√

1208 × .5 × .5).

Thus, the expression

NH − 604

17.38

has, approximately, a standard normal distribution. The value of

this expression in the present case is -1.44. The probability that a

standard normal distribution takes on a value of -1.44 or less is about

.0749, which is thus the p-value of the observation. This observation

is therefore significant at the 10% level, but not at the 5% level.

If one instead takes all of the subsets of the form (j, k) with j ≥ 3

and j ≥ |k|, there are 698 games, and the first team beat the spread

in 318 of these games. This represents 45.6% of the games. What is

the p-value of this observation? It turns out to be about .0095, which

is a very small p-value.

One can also look at the corresponding pooled data for teams

with losing streaks. There were 1140 games in the subsets of the

form (j, k), with j ≤ −1 and |j| ≥ |k|, and the first team beat the

spread in 597 of these games, or 52.4% of the time. The p-value of

this observation is .055. If we restrict our attention to those subsets

for which j ≤ −3 and |j| ≥ |k|, there were 643 games, and the first