## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 38, Number 5 (1967), 1365-1375.

### A Sequential Three Hypothesis Test for Determining the Mean of a Normal Population with Known Variance

#### Abstract

This paper examines a sequential testing procedure for choosing one of three simple hypotheses concerning the unknown mean $\mu$ of the normal distribution when the variance is known. The test is conducted by plotting $S_n$, the sum of the observations, versus $n$, the current sample size, until the point $(n, S_n)$ is contained within one of three triangular regions. When this occurs, sampling is terminated and the region containing $(n, S_n)$ determines which state of nature is accepted. Although we shall formally view the problem as one with only three states of nature $(\mu = \mu_1, \mu_2$ or $\mu_3)$, we shall proceed with the usual understanding that the performance of the test procedure should be evaluated for a wider class of states $(- \infty < \mu < \infty)$. The test is approximated by a corresponding exact test for the Wiener process. Formulas are derived which approximate the operating characteristics (OC) and the average sample size (ASN) for all values of $\mu$. The ASN function is compared with theoretical lower bounds. The testing procedure is compared with a modification of a three hypothesis testing procedure proposed by Sobel and Wald [5].

#### Article information

**Source**

Ann. Math. Statist., Volume 38, Number 5 (1967), 1365-1375.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177698692

**Digital Object Identifier**

doi:10.1214/aoms/1177698692

**Mathematical Reviews number (MathSciNet)**

MR217949

**Zentralblatt MATH identifier**

0161.38601

**JSTOR**

links.jstor.org

#### Citation

Simons, Gordon. A Sequential Three Hypothesis Test for Determining the Mean of a Normal Population with Known Variance. Ann. Math. Statist. 38 (1967), no. 5, 1365--1375. doi:10.1214/aoms/1177698692. https://projecteuclid.org/euclid.aoms/1177698692