## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 39, Number 6 (1968), 1946-1952.

### On the Cost of not Knowing the Variance when Making a Fixed-Width Confidence Interval for the Mean

#### Abstract

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 A number of unpublished results on the moments of the sample size are presented.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.

#### Article information

**Source**

Ann. Math. Statist., Volume 39, Number 6 (1968), 1946-1952.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177698024

**Digital Object Identifier**

doi:10.1214/aoms/1177698024

**Mathematical Reviews number (MathSciNet)**

MR239699

**Zentralblatt MATH identifier**

0187.15805

**JSTOR**

links.jstor.org

#### Citation

Simons, Gordon. On the Cost of not Knowing the Variance when Making a Fixed-Width Confidence Interval for the Mean. Ann. Math. Statist. 39 (1968), no. 6, 1946--1952. doi:10.1214/aoms/1177698024. https://projecteuclid.org/euclid.aoms/1177698024