Page 332 - Fundamentals of Probability and Statistics for Engineers

10

Model Verification

The parameter estimation procedures developed in Chapter 9 presume a dis-
tribution for the population. The validity of the model-building process based
on this approach thus hinges on the substantiability of the hypothesized dis-
tribution. Indeed, if the hypothesized distribution is off the mark, the resulting
probabilistic model with parameters estimated by any, however elegant, proced-
ure would, at best, still give a poor representation of the underlying physical or
natural phenomenon.
In this chapter, we wish to develop methods of testing or verifying a hypothe-
sized distribution for a population on the basis of a sample taken from the
population. Some aspects of this problem were addressed in Chapter 8, in
which, by means of histograms and frequency diagrams, a graphical compar-
ison between the hypothesized distribution and observed data was made. In the
chemical yield example, for instance, a comparison between the shape of a
normal distribution and the frequency diagram constructed from the data, as
shown in Figure 8.1, suggested that the normal model is reasonable in that case.
However, the graphical procedure described above is clearly subjective and
nonquantitative. On a more objective and quantitative basis, the problem of
model verification on the basis of sample information falls within the frame-
work of testing of hypotheses. Some basic concepts in this area of statistical
inference are now introduced.

10.1 PRELIMINARIES

In our development, statistical hypotheses concern functional forms of the
assumed distributions; these distributions may be specified completely with
prespecified values for their parameters or they may be specified with para-
meters yet to be estimated from the sample.
Let X 1 , X 2 ,..., X n be an independent sample of size n from a population X
q
with a hypothesized probability density function (pdf) f (x; ) or probability

Fundamentals of Probability and Statistics for Engineers T.T . Soong  2004 John Wiley & Sons, Ltd
ISBNs: 0-470-86813-9 (HB) 0-470-86814-7 (PB)

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