ConfidencepnLogExp
pppCcFor
ConfidenceppC
nxnx
c
x
n
x
xnx
c
x
n
x
1)1(
)1()1(0.
1)1(
0
0
Taking Logarithms on both sides, noticing that p = 1 – R, and
after some algebra, we obtain:
)(
)(
))1(
)1(
RLog
Log
pLog
ConfidenceLog
n
For example, applying this formula to the immediately
preceding example, we obtain:
4589.44
2227.0
0.1
)95.0(
)1.0(
Log
Log
n
The results, obtained using the Binomial and the Logarithm
formula, are close because both methods are totally equivalent.
However, the second result (formula) is easier and faster to
obtain than the first one (trial and error).
Summarizing, we first establish the problem requirements
regarding the desired (1-α) confidence and acceptable reliability
R. Then, we calculate the sample size n that satisfies these
requirements. Such sample size n can then be used to estimate
the reliability R, with the desired confidence. The life test must
be of length equal to Mission Time T.
Conclusions
The theory for determining the sample size that meets a testing
or estimation requirement is extensive and complex. Such
theory is driven by the type of parameter we want to estimate or
test (i.e. location, scale, or shape) and by the distribution of the
sampling statistic we use to implement the hypothesis test or to
obtain the estimation.
In this START Sheet, we have discussed the problem of
estimating and testing some location parameters (mean,
proportion) for the Normal, Exponential, and Weibull
distributions, and for distribution-free (nonparametric)
situations. Our objective has been to illustrate the logic and the
statistical thinking behind the derivation of such sample sizes.
A better understanding of this logic may help practicing
engineers to better implement such procedures.
We have only discussed a few of the most widely used cases.
There are many other situations of interest. For a more
extensive and in-depth treatment of this subject, the reader is
referred to Chapter 13, pages 699 to 776, of Reference 1.
An assessment of the complexity of these derivations may be
provided by the fact that the referred Chapter 13 is the last one
of this extensive, two-volume reliability handbook. However,
the manifold advantages that deriving an adequate sample size
for our problem provides in terms of savings in time and effort,
far outweighs its theoretical complexities.
Further Reading
1. Reliability and Life Testing Handbook. Kececioglu, D..
Volume 2. Prentice Hall, NJ. 1993.
2. Empirical Assessment of the Weibull Distribution. Romeu,
J. L. RAC START. Volume 10, Number3.
http://rac.alionscience.com/pdf/WEIBULL.pdf
3. The Anderson-Darling Goodness of Fit Test. Romeu, J. L.
RAC START. Volume 10, Number 5.
http://rac.alionscience.com/pdf/A_DTest.pdf
4. Reliability Estimations for the Exponential Life. Romeu, J.
L. RAC START. Volume 10, Number 7.
http://rac.alionscience.com/pdf/R_EXP.pdf
5. OC Functions and Acceptance Sampling Plans. Romeu, J.
L. RAC START. Volume 12, Number 1.
http://rac.alionscience.com/pdf/OC_Curves.pdf
6. Quality Toolkit. Coppola, A. RAC, 2001.
7. Practical Statistical Tools for Reliability Engineers.
Coppola, A. RAC, 2000.
8. Mechanical Applications in Reliability Engineering. Sadlon,
R. RAC 2000.
9. Statistical Analysis of Materials Data. Romeu, J. L. and C.
Grethlein. AMPTIAC, 2000.
10. Probability and Statistics for Engineers and Scientists.
Walpole, R., R. Myers, S. Myers. Prentice Hall. NJ 1998.
11. Introduction to Statistical Analysis (3
rd
Ed). Dixon, W. J.
and F. J. Massey McGraw-Hill. NY 1969.
12. An Introduction to Reliability and Maintainability
Engineering. Ebeling, C. E. Waveland Press. IL. 1997.
About the Author
Dr. Jorge Luis Romeu is the founder and director of the Juarez
Lincoln Marti International Education Project, that provides
books and faculty workshops to Iberoamerican institutions of
higher education. JLM Project is the sponsor of the QR&CII
Project. Romeu has over 30 years of statistics and operations
research experience in consulting, research, and teaching.
As a consultant, Romeu has worked for both manufacturing and
agriculture. He has worked in simulation modeling and data
analysis, in software and hardware reliability, in software
engineering, and ecological problems.
Romeu has taught both undergraduate and graduate industrial
statistics, operations research, and computer science in several
American and foreign universities. He is currently a Research
Professor and an Adjunct Professor of Industrial Statistics and
Quality Engineering, with Syracuse University.