## The Annals of Statistics

- Ann. Statist.
- Volume 5, Number 4 (1977), 803-807.

### One-Sample $t$-Test When Sampling from a Mixture of Normal Distributions

Austin F. S. Lee and John Gurland

#### Abstract

For the one-sample $t$-test a new form of the exact distribution of the test statistic $t^2$ is obtained when sampling from a distribution which is a mixture of two normal distributions. A numerical example is provided to show that the size of the test can differ greatly when sampling from distributions having the same skewness and kurtosis. Contours of equal size are plotted for a particular case in a certain cross section of the parameter space.

#### Article information

**Source**

Ann. Statist., Volume 5, Number 4 (1977), 803-807.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343904

**Digital Object Identifier**

doi:10.1214/aos/1176343904

**Mathematical Reviews number (MathSciNet)**

MR501510

**Zentralblatt MATH identifier**

0378.62015

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E15: Exact distribution theory

Secondary: 62F05: Asymptotic properties of tests

**Keywords**

Mixture of two normal distributions one-sample $t$-test distribution of $t$ effect of nonnormality equal probability contours

#### Citation

Lee, Austin F. S.; Gurland, John. One-Sample $t$-Test When Sampling from a Mixture of Normal Distributions. Ann. Statist. 5 (1977), no. 4, 803--807. doi:10.1214/aos/1176343904. https://projecteuclid.org/euclid.aos/1176343904