suppose that the population mean is known to be $\hat{m}$, her median value from the previous step. This is simply to enable the expert to consider uncertainty about the population variance in isolation; she is not required to update her beliefs (in any case, we are assuming independence between $\mu$ and $\sigma^2$). There remains a concern when providing hypothetical data: the expert may think differently if she had *actually observed* $\mu=\hat{m}$, but as $\hat{m}$ could be thought of as her ‘best guess’, she at least is not having to condition on a value of $\mu$ she considers implausible.

The expert is now asked to consider the proportion $\theta$ (or percentage if she prefers) of the population $X_1,X_2,\ldots$ that would lie in the interval $[\hat{m}, \hat{m}+c]$, for some $c>0$ (we consider the choice of $c$ in the next section). The expert will be uncertain about this proportion, because she is uncertain about $\sigma^2$. To illustrate this idea, suppose we wish to elicit an expert’s beliefs about end-of-year students’ marks for an undergraduate statistics module, for a large population of students. Suppose the marks are normally distributed with a mean of 60. Then there is a true proportion of students who will 