## The Annals of Statistics

### An Alternative to Student's $t$-Test for Problems with Indifference Zones

#### Abstract

Consider a sample from a normal population with mean, $\mu$, and variance unknown. Suppose it is desired to test $H_0:\mu \leq \mu_0$ versus $H_1:\mu \geq \mu_1$, with the region $H^I_1:\mu_0 < \mu < \mu_1$ being a (nonempty) indifference zone. It is shown that the usual Student's $t$-test is inadmissible for this problem. An alternative test is proposed. The two sided problem with indifference region is also discussed. By contrast with the above result, the usual Student's $t$-test is admissible here. However the two sided version of the alternative test mentioned above does offer some practical advantages relative to the two sided $t$-test. A 3-decision version of the two sided problem is also discussed. Here the $t$-test is inadmissible, and is dominated by the appropriate version of the alternative test. The results concerning tests are also reformulated as results about confidence procedures.

#### Article information

Source
Ann. Statist., Volume 12, Number 2 (1984), 451-469.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176346499

Digital Object Identifier
doi:10.1214/aos/1176346499

Mathematical Reviews number (MathSciNet)
MR740905

Zentralblatt MATH identifier
0544.62023

JSTOR
Brown, Lawrence D.; Sackrowitz, Harold. An Alternative to Student's $t$-Test for Problems with Indifference Zones. Ann. Statist. 12 (1984), no. 2, 451--469. doi:10.1214/aos/1176346499. https://projecteuclid.org/euclid.aos/1176346499