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abstract: 'We propose an elicitation method for quantifying an expert’s opinion about an uncertain population mean and variance. The method involves eliciting judgements directly about the population mean or median, and eliciting judgements about the population proportion with a particular characteristic, as a means of inferring the expert’s beliefs about the variance. The method can be used for a range of two-parameter parametric families of distributions, assuming a finite mean and variance. An illustration is given involving an expert’s beliefs about the distribution of times taken to translate pages of text. The method can be implemented in R using the package `SHELF`.'
author:
- 'Ziyad A. Alhussain^1^, Jeremy E. Oakley^2^'
title: Eliciting judgements about uncertain population means and variances
---

[^2^ School of Mathematics and Statistics, University of Sheffield, UK\
[j.oakley@sheffield.ac.uk]{}]{}

**Keywords**: Prior elicitation, expert judgement, population distributions.

Introduction
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We consider the problem of eliciting an expert’s opinion (or a group of experts’ opinions) about an uncertain mean and variance for a population of independent and identically distributed random variables $X_1,X_2,\ldots$. We suppose that either the population distribution is normal, or that a transformation can be applied so that the expert is willing to accept a normal distribution 