We now make several remarks concerning the foregoing definition.
R emark 1: as we see from Equation (9.126), confidence limits are functions of
a given sample. The confidence interval thus will generally vary in position
and width from sample to sample.
Remark 2: for a given sample, confidence limits are not unique. In other
words, many pairs of statistics L 1 and L 2 exist that satisfy Equation (9.128).
For example, in addition to the pair ( 1.96, 1.96), there are many other pairs
of values (not symmetric about zero) that could give the probability 0.95 in
Equation (9.125). H owever, it is easy to see that this particular pair gives the
minimum-width interval.
Remark 3: in view of the above, it is thus desirable to define a set of quality
criteria for interval estimators so that the ‘best’ interval can be obtained.
Intuitively, the ‘best’ interval is the shortest interval. Moreover, since interval
width L L 2 L 1 is a random variable, we may like to choose ‘minimum
expected interval width’ as a good criterion. Unfortunately, there may not
exist statistics L 1 and L 2 that give rise to an expected interval width that is
minimum for all values of.
Remark 4: just as in point estimation, sufficient statistics also play an
important role in interval estimation, as Theorem 9.5 demonstrates.
Theorem 9. 5: let L 1 and L 2 be two statistics based on a sample X 1 ,...,Xn
from a population X with pdf f(x; ) such that Let
be a sufficient statistic. Then there exist two functions R 1
and R 2 of Y such that and such that two interval
widths L L 2 L 1 and have the same distribution.
This theorem shows that, if a minimum interval width exists, it can be
obtained by using functions of sufficient statistics as confidence limits.
The construction of confidence intervals for some important cases will be carried
out in the following sections. The method consists essentially of finding an appro-
pria te ra ndom varia ble for which values can be calcula ted on t he basis of observed
sample values and the parameter value but for which the distribution does not
depend on the parameter. More general methods for obtaining confidence inter-
vals are discussed in M ood (1950, chapter 11) and Wilks (1962, chapter 12).
9.3.2.1 Confidence Interval formin N(m^2 )with Known^2
The confidence interval given by Equation (9.126) is designed to estimate the
mean of a normal population with known variance. In general terms, the
procedure shows that we first determine a (symmetric) interval in U to achieve
a confidence coefficient of 1. Writing u/2 for the value of U above which
the area under fU (u) is /2, that is, (see Figure 9.6), we have
296 Fundamentals of Probability and Statistics for Engineers
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PL 1 <<L 2 ) 1 .
YhX 1 ,...,Xn)
PR 1 <<R 2 ) 1
RR 2 R 1
s s (^)
PU>u /2) /2