data are from a lognormal distribution. Let and , respectively, denote the population mean and variance for the log-transformed data. If denotes the corresponding random variable, we thus have . We note that is the median air lead level. A confidence interval for can be constructed the usual way, based on the t-distribution; this in turn will provide a confidence interval for the median air lead level. If and denote the sample mean and standard deviation of the log-transformed data for a sample of size n, a 95% confidence interval for is given by , where denotes the quantile of a t-distribution with degrees of freedom. It may also be of interest to derive a 95% upper confidence bound for the median air lead level. Such a bound for is given by . Consequently, a 95% upper confidence bound for the median air lead is given by . Now suppose we want to predict the air lead level at a particular area within the laboratory. A 95% upper prediction limit for the log-transformed lead level is given by . A two-sided prediction interval can be similarly computed. The meaning and interpretation of these intervals are well known. For example, if the confidence interval is computed repeatedly from independent samples, 95% of the intervals so computed will include the true value of , in the long run. In other words, the interval is meant to provide information concerning the parameter only. A prediction interval has a similar interpretation, and is meant to provide information concerning a single lead level only. Now suppose we want to use the sample to conclude whether or not at least 95% of the population lead levels are below a threshold. The confidence interval and prediction interval cannot answer this question, since the confidence interval is only for the median lead level, and the prediction interval is only for a single lead level. What is required is a tolerance interval; more specifically, an upper tolerance limit. The upper tolerance limit is to be computed subject to the condition that at least 95% of the population lead levels is below the limit, with a certain confidence level, say 99%. Calculation One-sided normal tolerance intervals have an exact solution in terms of the sample mean and sample variance based on the noncentral t-distribution. Two-sided normal tolerance intervals can be obtained based on the noncentral chi-squared distribution. See also Engineering