We address the issue of reliably discriminating between chaos and noise from a time series. In particular, we are interested in avoiding claims of chaos when simpler models (such as linearly correlated noise) can explain the data. We take a statistical approach, and use a form of bootstrapping to detect nonlinearity by showing that a given linear model is unlikely to have produced the data. Our method requires the careful statement of a null hypothesis which characterizes a candidate linear process, the generation of an ensemble of surrogate'' data sets which are similar to the original time series but consistent with the null hypothesis, and the computation of a discriminating statistic for the original and for each of the surrogate data sets. The idea is to test the original time series against the null hypothesis by checking whether the discriminating statistic computed for the original time series differs significantly from the statistics computed for each of the surrogate sets. We present algorithms for generating surrogate data under various null hypotheses, and we show the results of numerical experiments on artificial data using correlation dimension, Lyapunov exponent, and forecasting error as discriminating statistics. Finally, we consider a number of experimental timemore » series -- including sunspots, electroencephalogram (EEG) signals, and fluid convection -- and evaluate the statistical significance of the evidence for nonlinear structure in each case. 52 refs., 10 figs.« less