

MILHDBK338B: Electronic Reliability Design Handbook 
 

8.3.2.5.1 Confidence Limits  Normal Distribution
8.3.2.5.1 Confidence Limits 
Normal Distribution
When the lives of n components are known from a wearout test and
we compute their mean, , and their standard deviation, s, and when n is large so
that we can assume that s » s, the upper and lower confidence limits can be readily
evaluated from Table 8.39 for the more commonlyused confidence
levels.
Strictly speaking, this procedure
of assigning confidence intervals to an estimate is correct only when the true
standard deviation, s, of component wearout is known and used instead of s in
Table 8.39. However, it can be applied in reliability work as an
approximation whenever the estimate s, of s, was obtained from a large
sample, i.e., when the number of failures is at least 25, and preferably,
more. In fact, it can be shown for samples of 20, k_{a/2} (at the 95% confidence level) is 2.09 vs. a value of
1.96 for an infinite number of samples. a
is equal to 100(1  confidence
level)%.
TABLE 8.39: CONFIDENCE
LIMITS  NORMAL DISTRIBUTION
Figure 8.315 graphically illustrates what is
being done. Since the normal distribution is symmetrical, we are computing the
confidence interval as the area (1  a) under the curve, leaving an area a/2 in each of the left and
right hand tails which is outside of the confidence interval (CI). For
example, using the calculated values of (or ) and s obtained from the data in Table 8.310, the CI at the 95%
level is
"1">
In other words, we can be 95%
confident that the true value of the mean life (M) lies between 1566.6 and
2343.8 hours.
Actually, in reliability work, we are
usually more interested in the lower confidence limit L of the mean wearout
life than in the upper limit. Given a measured value of , we would like to make some
statement about our confidence that the true value of M exceeds some minimum
value.
When only the lower confidence limit,
L, is of interest, we apply the procedure of socalled “onesided” confidence
limits, as opposed to the twosided CI of the preceding example. The problem
is to assure ourselves (or our customer) that the true mean life, M, is equal
to or larger than some specified minimum value with a probability of (1 
a).
FIGURE 8.315: TWOSIDED CONFIDENCE
INTERVAL AND LIMITS
Whereas in the case of the twosided confidence limits, we had
an area of a/2
under the left tail of the normal curve (Figure 8.315), we now have an area
a to the left of
L and an area (1  a) to the right.
Therefore, the estimate of mean
life obtained from the data should be:
If this equation is not satisfied, the
requirement that the true M must be at least L at the specified 100 (1 
a) percent
confidence level has not been fulfilled.
Table 8.310, in which the assumption s
» s is made,
allows a quick check as to whether an estimate, ^ M , obtained from a sample
of size n fulfills the requirement that the true M must not be smaller than
the specified minimum L. Only the more commonlyused confidence levels are
given.
TABLE 8.310: CONFIDENCE
INTERVAL
Once again, using the data and calculated values of and s from Table
8.310, assume that we would like to be 95% confident that the true M
³ 1500 hours. The
equation from Table 8.310 is
"1">
Since the inequality is
satisfied, the requirement has been met.
As previously mentioned,
the above procedure can be applied if the sample size n is at least 25.
However, similar procedures also apply to smaller sample sizes except that now
we cannot assume that s » s, and we must use another set of equations based on
Student’s t distribution. Actually, all we do is replace the normal percentage
points Ka/2
and Ka
in the previously developed equations by
the tabulated percentage points t_{a/2;n1}
and t_{a;n1}
of the t distribution, where n1 is called
the degrees of freedom and n is the number of failures. Student’s t tables are
available in most standard statistical texts.
For example, for the
twosided CI example using the data from Table 8.310 and calculated values
of and s,
"1">
which is a slightly wider
CI than the case where it was assumed the s »
s.




 
 