For one-sided tests about the mean of a skewed distribution, thettest is asymptotically robust for validity; however, it can be quite inaccurate and inefficient with small sample sizes. Results presented here confirm that a procedure due to Johnson should be preferred to thettest when the parent distribution is asymmetrical, because it reduces the probability of type I error in cases where thettest has an inflated type I error rate and it is more powerful in other situations. But if the skewness is severe and the sample size is small, then Johnson's test can also be appreciably inaccurate. For such situations, computer-intensive test procedures using bootstrap resampling are proposed, and with an extensive Monte Carlo study it is shown that these procedures are remarkably robust and can result in reduced probabilities of type I and type II errors compared to Johnson's test.