Far East Journal of Mathematical Education

The Far East Journal of Mathematical Education is a peer-reviewed journal focused on mathematical education. It publishes research papers that enhance understanding of mathematical concepts and encourages the use of technology, statistics, algorithms, and simulations in mathematics learning.

Submit Article

USING A SIMPLE LINEAR MODEL TO PROVE THAT THE CLASSICAL $t$-TEST IS THE UNIFORMLY MOST POWERFUL (UMP) TEST FOR COMPARING TWO MEANS

Authors

  • David Sotres-Ramos

Keywords:

t-test, linear model, uniformly most powerful test.

DOI:

https://doi.org/10.17654/0973563125008

Abstract

The classical t-test for comparing two means plays an important role in statistical applications and is widely presented in most introductory statistics textbooks. Despite this, a formal demonstration of its optimality in hypothesis testing is often overlooked. In this article, we show that the classical t-test is the Uniformly Most Powerful (UMP) test among all invariant tests for comparing two means at a given significance level  To achieve this, we use a simple linear model. This approach allows us to establish the theoretical foundations of the t-test’s optimality, reinforcing its validity and importance in statistical inference.

Received: April 22, 2025
Accepted: May 8, 2025

References

A. Agresti, C. A. Franklin and B. Klingenberg, Statistics: The Art and Science of Learning from Data, 4th ed., Pearson Prentice Hall, Upper Saddle River, NJ, 2018.

C. H. Brase and C. P. Brase, Understandable Statistics: Concepts and Methods 12th ed., Cengage Learning, Stanford, CT, 2018.

R. A. Johnson and G. K. Bhattacharyya, Statistics: Principles and Methods, 8th ed., Wiley, 2019.

R. Larson and B. Farber, Elementary Statistics: Picturing the World, 7th ed., Addison-Wesley, 2018.

D. S. Moore, W. I. Notz and M. A. Fligner, The Basic Practice of Statistics, 8th ed., W. H. Freeman and Company, New York, 2018.

J. Utts and R. F. Heckard, Mind on Statistics, 6th ed., Cengage Learning, Melbourne, Australia, 2021.

S. R. Searle and M. H. J. Gruber, Linear Models, John Wiley and Sons, 2nd ed., 2017.

S. G. Wang and S. C. Chow, Advanced Linear Models: Theory and Applications, CRC Press, Taylor and Francis Group, Boca Raton FL, 1994.

P. L. Hsu, Canonical reduction of the general regression problem, Ann. Eugenics 11 (1941a), 42-46.

P. L. Hsu, Analysis of variance from the power function standpoint, Biometrika 32 (1941b), 62-69.

Downloads

Published

05-06-2025

Issue

Vol. 27 No. 1 (2025)

Section

Articles

License

Copyright (c) 2025 PUSHPA PUBLISHING HOUSE, PRAYAGRAJ, INDIA

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

_________________________

Licensing Terms:

Attribution: Credit Pushpa Publishing House as the original publisher, including title and author(s) if applicable.

No Derivatives: Modifying or creating derivative works not allowed without written permission.

Contact Puspha Publishing House for more info or permissions.

How to Cite

USING A SIMPLE LINEAR MODEL TO PROVE THAT THE CLASSICAL $t$-TEST IS THE UNIFORMLY MOST POWERFUL (UMP) TEST FOR COMPARING TWO MEANS. (2025). Far East Journal of Mathematical Education, 27(1), 63-69. https://doi.org/10.17654/0973563125008

Most read articles by the same author(s)