8.3.2.5 Confidence Limits and
Intervals
Previously, we discussed methods of obtaining point estimates of
reliability parameters, e.g., R(t), l, MTBF, etc. For most practical applications, we are
interested in the accuracy of the point estimate and the confidence which we
can attach to it. We know that statistical estimates are more likely to be
closer to the true value as the sample size increases. Only the impossible
situation of having an infinitely large number of samples to test could give
us 100 percent confidence or certainty that a measured value of a parameter
coincides with the true value. For any practical situation, therefore, we must
establish confidence intervals or ranges of values between which we know, with
a probability determined by the finite sample size, that the true value of the
parameter lies.
Confidence intervals around point
estimates are defined in terms of a lower confidence limit, L, and an upper
confidence limit, U. If, for example, we calculate the confidence limits for a
probability of, say, 95 percent, this means that in repeated sampling, 95
percent of the calculated intervals will contain the true value of the
reliability parameter. If we want to be 99 percent sure that the true value
lies within certain limits for a given sample size, we must widen the interval
or test a larger number of samples if we wish to maintain the same interval
width. The problem, then, is reduced to one of either determining the interval
within which the true parametric value lies with a given probability for a
given sample size, or determining the sample size required to assure us with a
specified probability that true parametric value lies within a specific
interval.
Thus, we would like to be able to
make assertions such as
where q is some unknown population parameter, qL and qU are
estimators associated with a random sample and h
is a probability value such as 0.99, 0.95,
0.90, etc. If, for instance, h = 0.95 we refer to the interval
for particular values of _{L} and qˆ_{U}
as a 95% confidence interval. In this case we are willing to accept a
5% probability (risk) that our assertion is not, in fact, true.
Or, we may also want to make statements such as

P [q > _{L}
] = h 
(8.10) 
in which case we make statements like,
“we are 90% confident that the true MTBF is greater than some lower confidence
limit (or measured value).” Eq. (8.10) is the case of the onesided confidence
limit, versus Eq. (8.9) which is a twosided confidence limit, or confidence
interval.
To help clarify the concept of a
confidence interval we can look at the situation in a geometrical way. Suppose
we draw repeated samples (x_{1},
x_{2}) from a population, one of
whose parameters we desire to bracket with a confidence interval. We construct
a threedimensional space with the vertical axis corresponding to q and with the two horizontal
axes corresponding to values of X_{1} and
X_{2} (see Figure 8.314). The actual value of the population
parameter q is
marked on the vertical axis and a horizontal plane is passed through this
point. Now we take a random sample (X_{1}, X_{2})
from which we calculate the values _{U }and _{L} at, say, the 95% confidence level. The interval defined
by _{U} and _{L} is plotted on the figure.
Next, we take a second sample (X’_{1}
, X’_{2}) from which we calculate the value ''_{U}
and ''_{L} at the 95% level. This
interval is plotted on the figure. A third sample (X''_{1} ,
X''_{2}) yields the values ''_{U} and ''_{L} , etc. In this way we
can generate a large family of confidence intervals. The confidence intervals
depend only on the sample values (X_{1} , X_{2}),
(X’_{1} , X’_{2}), etc., and hence we can calculate these
intervals without knowledge of the true value of q. If the confidence intervals
are all calculated on the basis of 95% confidence and if we have a very large
family of these intervals, then 95% of them will cut the horizontal plane
through q (and
thus include q)
and 5% of them will not.
The process of taking a random sample and
computing from it a confidence interval is equivalent to the process of
reaching into a bag containing thousands of confidence intervals and grabbing
one at random. If they are all 95% intervals, our chance of choosing one that
does indeed include q will be 95%. In contrast, 5% of the time we will be
unlucky and select one that does not include q
(like the interval (''_{U} and ''_{L}) in Figure 8.314. If a risk of 5% is judged too high,
we can go to 99% intervals, for which the risk is only 1%. As we go to higher
confidence levels (and lower risks) the lengths of the intervals increase
until for 100% confidence levels (and lower risks) the interval includes every
conceivable value of q (I am 100% confident that the number of defective items
in a population of 10,000 is somewhere between 0 and 10,000). For this reason
100% confidence intervals are of little interest.
Let us now look at some simple examples of how
these concepts are applied to analyze reliability for some of the more
commonlyused distributions.