Abstract
The rationale and methodology for estimating a mean with a fixed width confidence interval through sampling in three stages are extended to cover the additional problem of testing hypotheses concerning shifts in the mean with controlled Type II error. The coverage probability and operating characteristic function of the confidence interval based on the integrated approach are derived and compared with those of the usual triple sampling confidence interval. The extended methodology leads to better coverage probability and uniformly better Type II error probabilities. Achieving the additional objective of controlling Type II error inevitably implies a two- to threefold increase in the required optimal sample size. Some suggestions for dealing with this apparent limitation are discussed from a practical viewpoint. It is recommended that an integrated approach to estimation and testing based on confidence intervals be incorporated in the design stage for credible inferences.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Anscombe, F. J. (1953). Sequential estimation, J. Roy. Statist. Soc. Ser. B, 15, 1–29.
Bristol, D. R. (1989). Sample sizes for constructing confidence intervals and testing hypotheses, Statistics in Medicine, 8, 808–811.
Brownlee, K. A. (1965). Statistical Theory and Methodology in Science and Engineering, 2/e, Wiley, New York.
Chow, Y. S. and Robbins, H. (1965). On the asymptotic theory of fixed width sequential confidence intervals for the mean, Ann. Math. Statist., 36, 457–462.
Costanza, M. C., Hamdy, H. I., Haugh, L. D. and Son, M. S. (1995). Type II error performance of triple sampling fixed precision confidence intervals for the normal mean, Metron, 53, 69–82.
Ghosh, M. and Mukhopadhyay, N. (1981). Consistency and asymptotic efficiency of two-stage and sequential procedures, Sankhya Ser. A, 43, 220–227.
Hall, P. (1981). Asymptotic theory of triple sampling for sequential estimation of a mean, Ann. Statist., 9, 1229–1238.
Hamdy, H. I. (1988). Remarks on the asymptotic theory of triple stage estimation of the normal mean, Scand. J. Statist., 15, 303–310.
Hamdy, H. I., Mukhopadhyay, N., Costanza, M. C. and Son, M. S. (1988). Triple stage point estimation for the exponential location parameter, Ann. Inst. Statist. Math., 40, 785–797.
Kupper, L. L. and Hafner, K. B. (1989). How appropriate are popular sample size formulas?, Amer. Statist., 43, 101–105.
Lehmann, E. L. (1986). Testing Statistical Hypotheses, 2/e, Wiley, New York.
Montgomery, D. C. (1982). The economic design of control charts: a review and literature survey, Journal of Quality Technology, 14, 40–43.
Rahim, M. A. (1993). Economic design of x control charts assuming Weibull in-control times, J. Qual. Tech., 25, 296–305.
Stein, C. (1945). A two-sample test for a linear hypothesis whose power is independent of the variance, Ann. Math. Statist., 16, 243–258.
Tukey, J. W. (1991). The philosophy of multiple comparisons, Statist. Sci., 6, 100–116.
Author information
Authors and Affiliations
About this article
Cite this article
Son, M.S., Haugh, L.D., Hamdy, H.I. et al. Controlling Type II Error While Constructing Triple Sampling Fixed Precision Confidence Intervals for the Normal Mean. Annals of the Institute of Statistical Mathematics 49, 681–692 (1997). https://doi.org/10.1023/A:1003266326065
Issue Date:
DOI: https://doi.org/10.1023/A:1003266326065