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On confidence intervals for the product of three means of three normally distributed populations
Author(s) -
Yfantis E. A.,
Flatman G. T.
Publication year - 1991
Publication title -
journal of chemometrics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.47
H-Index - 92
eISSN - 1099-128X
pISSN - 0886-9383
DOI - 10.1002/cem.1180050317
Subject(s) - confidence interval , statistics , product (mathematics) , random variable , mathematics , variable (mathematics) , robust confidence intervals , variance (accounting) , cdf based nonparametric confidence interval , confidence region , tolerance interval , prediction interval , interval (graph theory) , credible interval , combinatorics , mathematical analysis , geometry , accounting , business
In many environmental applications, such as exposure assessment and risk modelling, the desired estimate is a random variable computed as the product of three independently distributed random variables. These variables may not necessarily have the same mean and variance. The method for finding the 100(1 − α)% confidence interval for the mean of the product random variable has been proposed by some practitioners as the product of the 100(1 − α)% confidence interval of the three means. In this paper we show that the distribution of the product of three independent normal variables is not normal. We find the mean and variance of the product distribution. Further, we show that although the mean of the product is equal to the product of the means, the product of the three confidence intervals is not a good approximation of the confidence intervals for the mean of the product variable. The confidence interval of the mean of the product variable may be estimated by computer simulation. An algorithm for estimating the confidence interval for the mean of the product random variable is given. The program implementing this algorithm is given as an appendix.