Particle Physics Booklet - Particle Data Group - Lawrence Berkeley ...

276 33. Statistics
p-value for test
α for confidence intervals
1.000
0.500
0.200
0.100
0.050
0.020
0.010
0.005
0.002
n = 1
2 3 4 6 8
0.001
1 2 3 4 5 7 10
χ
20 30 40 50 70 100
2
Figure 33.1: One minus the χ 2 cumulative distribution, 1−F (χ 2 ; n),
for n degrees of freedom. This gives the p-value for the χ 2 goodnessof-fit
test as well as one minus the coverage probability for confidence
regions (see Sec. 33.3.2.4).
χ 2 /n
2.5
2.0
1.5
1.0
0.5
10%
50%
90%
99%
0.0
0 10 20 30 40 50
Degrees of freedom n
10
15
25
20
32%
68%
95%
Figure 33.2: The ‘reduced’ χ 2 , equal to χ 2 /n, for n degrees
of freedom. The curves show as a function of n the χ 2 /n that
corresponds to a given p-value.
will not be a consensus about the prior probabilities for the existence
of new phenomena. Nevertheless one can construct a quantity called the
Bayes factor (described below), which can be used to quantify the degree
to which the data prefer one hypothesis over another, and is independent
of their prior probabilities.
Consider two models (hypotheses), Hi and Hj, described by vectors
1%
5%
30
40
50