Assessment of Sample Size Inflation for Accurate Estimation of Population Means

Abstract

Goal: The required sampled size is determined for estimating a population mean within a given margin of error based on a preliminary sample. An inflation factor is needed to prevent confidence intervals from being anti-conservative. Methodology: When estimating a population mean \(\mu\) within margin of error m, a preliminary sample of size n is taken from a Normal (\(\mu\) , \(\sigma\)2) distribution to produce a preliminary sample variance s2, which is then used to determine the required sample size (zs/m)2, where z is the Normal critical value for a given level of confidence, and the distribution of s2 is known to be related to a chi-squared distribution for Normally-distributed data. Evaluation: Upon taking a new sample based on the required sample size, the coverage probabilities on \(\mu\) are determined exactly for various values of n and z. These coverage probabilities of \(P(~|\bar X-\mu|\leq m~)\) are simulated for non-Normal distributions as well, where -\(\bar X\)  is the sample mean using the required sample size. Findings: The coverage probabilities tend to be somewhat smaller than their nominal values, which would result in anti-conservative confidence intervals, especially when the non-Normal distribution is heavy-tailed. Conclusion: To compensate for the confidence intervals being anti-conservative, an inflation factor on the required sample size is introduced.