It is shown that the mean of a normal distribution with unknown variance $\sigma^2$ may be estimated to lie within an interval of given fixed width at a prescribed confidence level using a procedure which overcomes ignorance about $\sigma^2$ with no more than a finite number of observations. That is, the expected sample size exceeds the (fixed) sample size one would use if $\sigma^2$ were known by a finite amount, the difference depending on the confidence level $\alpha$ but not depending on the values of the mean $\mu$, the variance $\sigma^2$ and the interval width $2d$. A number of unpublished results on the moments of the sample size are presented. Some do not depend on an assumption of normality.
Gordon Simons. "On the Cost of not Knowing the Variance when Making a Fixed-Width Confidence Interval for the Mean." Ann. Math. Statist. 39 (6) 1946 - 1952, December, 1968. https://doi.org/10.1214/aoms/1177698024