Page 183 - Statistics for Environmental Engineers

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The value 0.0143 can be looked up in a table of the standard normal distribution. It is the area under
the tail that lies beyond z = 2.19.
A two-sided confidence interval for p = Prob(y ≤ L) has the form:

[h 1−α/2;−K,n , h 1−α/2;K,n ]

where K = (y – L)/s. A one-sided upper confidence bound is [h 1−α;K,n ]. The h factors are found in Table 7
of Odeh and Owen (1980).
For 1 − α = 0.95, n = 27, and K = [4.01 − ln(300)]/0.773 = −2.2, the factor is h = 0.94380 and the
upper 95% confidence bound for p is 1 − 0.9438 = 0.0562. Thus, we are 95% confident that the probability
of a reading exceeding 300 ppm is less than 5.6%. This 5.6% probability of getting a future value above
L = 300 may be disappointing given that all of the previous 27 observations have been below the limit.
Had the normal distribution been incorrectly assumed, the upper 95% confidence limit obtained would
have been 0.015%, the contrast between 0.00015 and 0.05620 (a ratio of about 375). This shows that
confidence bounds on probabilities in the tail of a distribution are badly wrong when the incorrect
distribution is assumed.

Comments
In summary, a confidence interval contains the unknown value of a parameter (a mean), a tolerance
interval contains a proportion of the population, and a prediction interval contains one or more future
observations from a previously sampled population.
The lognormal distribution is frequently used in environmental assessments. The logarithm of a variable
with a lognormal distribution has a normal distribution. Thus, the methods for computing statistical
intervals for the normal distribution can be used for the lognormal distribution. Tolerance limits, confi-
dence limits for distribution percentiles, and prediction limits are calculated on the logarithms of the
data, and then are converted back to the scale of the original data.
Intervals based on the Poisson distribution can be determined for the number of occurrences. Intervals
based on the binomial distribution can be determined for proportions and percentages.
All the examples in this chapter were based on assuming the normal or lognormal distribution.
Tolerance and prediction intervals can be computed by distribution-free methods (nonparametric meth-
ods). Using the distribution gives a more precise bound on the desired probability than the distribution-
free methods (Hahn and Meeker, 1991).

References
ASTM (1998). Standard Practice for Derivation of Decision Point and Confidence Limit Testing of Mean
Concentrations in Waste Management Decisions, D 6250, Washington, D.C., U.S. Government Printing
Office.
Hahn, G. J. (1970). “Statistical Intervals for a Normal Population. Part I. Tables, Examples, and Applications,”
J. Qual. Tech., 3, 18–22.
Hahn, G. J. and W. Q. Meeker (1991). Statistical Intervals: A Guide for Practitioners, New York, John
Wiley.
Gibbons, R. D. (1994). Statistical Methods for Groundwater Monitoring, New York, John Wiley.
Johnson, R. A. (2000). Probability and Statistics for Engineers, 6th ed., Englewood Cliffs, NJ, Prentice-Hall.
Odeh, R. E. and D. B. Owen (1980). Tables for Normal Tolerance Limits, Sampling Plans, and Screening,
New York, Marcel Dekker.
Owen, D. B. (1962). Handbook of Statistical Tables, Palo Alto, CA, Addison-Wesley.

© 2002 By CRC Press LLC