# Sample size approximation for estimating the arithmetic mean of a lognormally distributed random variable with and without Type I censoring

## Description

This research presents the approximate sample size needed to estimate the true arithmetic mean of a log normally distributed random variable to within a specified accuracy (100$\pi$) percent difference from the true arithmetic mean with geometric standard deviations (GSDs) up to 4.0 and with a specified level of confidence Exact minimum required sample sizes are based on the confidence interval width of Land's exact interval. For the non censored case, sample size tables and nomograms are presented. Box-Cox transformations were used to derive formulae for approximating these exact sample sizes. In addition, new formulae, adjusting the classic central limit approach, were derived. Each of these formulas as well as other existing formulas (the classical central limit approach and Hewett's (1995) formula) were compared to the exact sample size to determine under which conditions they perform optimally These comparisons lead to the following recommendations for the 95% confidence level: The Box-Cox transformation formula is recommended for GSD = 1.5 and 100$\pi>$ 20% levels; for GSDs of 2, 2.5 and 3 and 100$\pi>$ 20% levels; for GSD = 3 and 100$\pi>$ 30% levels; and for GSD = 4 and 100$\pi>$ 40% levels. The adjusted classical formula is recommended for GSD = 1.1 and all 100$\pi$ levels; for GSD = 1.5 and 100$\pi>$ 25% levels; and for GSDs of 2, 2.5 and 100$\pi\leq$ 20% levels. The classical formula is recommended for GSD = 3 and 100$\pi\leq$ 20% levels; for GSD = 3.5 and 100$\pi\leq$ 30% levels and for GSD = 4 and 100$\pi\leq$ 40% levels Exact sample size requirements are presented for samples where 10%, 20% and 50% are Type I left censored. These sample sizes are based on Land's exact confidence interval width of the bias corrected maximum likelihood estimates The accuracy of the confidence intervals based on these proposed sample sizes are evaluated using Monte Carlo simulations for both the non censored and censored case. These simulations showed conservative results regarding targeted significant levels