  High resolution experiments have recently lead to a complete identification
(energy, spin, and parity) of 151 nuclear levels up to an excitation Energy of
Ex= 6.20 MeV in 208Pb. We present a thorough study of the fluctuation
properties in the energy spectra of the unprecedented set of nuclear bound
states. In a first approach we grouped states with the same spin and parity
into 14 subspectra, analyzed standard statistical measures for short- and
long-range correlations and then computed their ensemble average. Their
comparison with a random matrix ensemble which interpolates between Poisson
statistics expected for regular systems and the Gaussian Orthogonal Ensemble
(GOE) predicted for chaotic systems shows that the data are well described by
the GOE. In a second approach, following an idea of Rosenzweig and Porter we
considered the complete spectrum composed of the independent subspectra. We
analyzed their fluctuation properties using the method of Bayesian inference
involving a quantitative measure, called the chaoticity parameter f, which also
interpolates between Poisson (f=0) and GOE statistics (f=1). It turns out to be
f~0.9. This is so far the closest agreement with GOE observed in spectra of
bound states in a nucleus. The same analysis has also been performed with
spectra computed on the basis of shell model calculations with different
interactions (SDI, KB, M3Y). While the simple SDI exhibits features typical for
nuclear many-body systems with regular dynamics, the other, more realistic
interactions yield chaoticity parameters f close to the experimental values.
