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Journal of Probability and Statistics
Volume 2010, Article ID 139856, 9 pages
Research Article

A Note on Confidence Interval for the Power of the One Sample 𝑡 Test

Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada

Received 26 January 2010; Revised 22 June 2010; Accepted 12 August 2010

Academic Editor: Junbin B. Gao

Copyright © 2010 A. Wong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In introductory statistics texts, the power of the test of a one-sample mean when the variance is known is widely discussed. However, when the variance is unknown, the power of the Student's 𝑡 -test is seldom mentioned. In this note, a general methodology for obtaining inference concerning a scalar parameter of interest of any exponential family model is proposed. The method is then applied to the one-sample mean problem with unknown variance to obtain a ( 1 𝛾 ) 100% confidence interval for the power of the Student's 𝑡 -test that detects the difference ( 𝜇 𝜇 0 ) . The calculations require only the density and the cumulative distribution functions of the standard normal distribution. In addition, the methodology presented can also be applied to determine the required sample size when the effect size and the power of a size 𝛼 test of mean are given.