A number of tests and estimates for mean life and other parameters derived under the exponential distribution assumption are studied under the alternative condition that the distribution has an increasing (decreasing) failure rate. The tests and estimates considered are, for the most part, based on censored and truncated samples. It is shown that the usual acceptance sampling procedures based on mean life generally favor the producer (consumer) in the IFR (DFR) case. However, acceptance sampling procedures based on theqth quantile tend to favor the consumer in the IFR case whenqis small. The usual estimates for the mean based on the exponential assumption are positively (negatively) biased when the distribution is IFR (DFR). Many of these results hold under the weaker assumption that the distribution has an increasing (decreasing) failure rate on the average.