I used to be and industrial chemist and I have a degree in applied
chemistry but I readily admit that I am no mathematician and certainly
no statistician.  I would like to know how the results of an
experiment could be interpreted concerning the degree of protection
from the weather by means of a layer of a protective lacquer that has
been applied to test panels that have been exposed to the rain and sun
over a long period of time, the panels first having been painted with
a dark green paint.

The test panels are all the same size and that is 100cm x100cm and to
be awarded a pass, there had to be no blisters on the painted test
panels after the period of the test (one year).


They are:
39 out of 78 lacquered test panels passed

32 out of 78 unlacquered test panels passed

Now, my question is this: what would the expected time on average be
in order for a given panel that was not lacquered to have any blisters
on it?  Is the sample big enough for the results of the experiment to
be meaningful?

---

Clarification of Question by yaffle-ga on 19 Jul 2005 00:42 PDT

I omitted to say, when I posted my question yesterday, that it was
about a thought experiment that has not actually been done.  The
question represents a more important thing that is unfortunately
rather controversial and therefore I thought that this representation
would be the best way of finding the answer. The failure figures are
however the real thing.
I am sorry if you feel misled but the answer to this question is very
important for me to know.  Your time has not been wasted.

It is anybody's guess. In order to answer this, you have to know the
time dependence of the failure distribution. The test had many
failures, so you would be able to estimate this if you knew exactly
when each of the panels failed. Since you are simply reporting the
number of failures at the end of one year, you know very little. A
common assumption is that the failure rate is independent of time.
This is very unlikely in your case, since the blistering is
undoubtedly driven by cumulative exposure to the elements, thus the
failure rate should get larger as time elapses.

Let's analyze two extreme scenarios:

Constant failure rate
=====================
There were 46 failures of the unlacquered panels in one year. Thus,
our best estimate of the failure rate is (46/78) per year. From this
we can easily estimate the expectation value of the failure time,
using an exponential distribution for the number of remaining panels
after a given time. The math is trivial, but I will leave it to you.
It should come out to be a bit less than a year.

Tight Wearout Distribution
==========================
Here we assume that all the panels tend to fail at about the same time
after wearing out at the same rate. This is another extreme
assumption. In this case, since about half of the panels have already
failed, they must have failed very recently and the rest will fail
very soon. In this case the expected failure time is about one year.

Conclusion
==========
Under either of these scenarios, the panels only last about a year, on
average. We could dream up other scenarios in which the expected time
could be much longer than a year, but these don't seem very realistic.
Of course, the lifetime will depend on how typical the exposure was
during the year of the test.

I hope this is helpful.

I'd like to ammend my comment by adding one more scenario:

Infant Mortality
================
It could be that most of the failures occurred very early because some
fraction of the panels have imperfections that cause them to fail
early. To consider another extreme scenario, assume that all of the
failures happened very early in the test and the remaining panels have
a very long life. In this case the average lifetime could be much
larger than one year. This type of distribution is very common and
cannot be ruled out in your case.

This brings me back to my first comment: It is anybody's guess, since
you know nothing about the time dependence of the failure rate.

nice comment, kottekoe

Thank you kottekoe, I genuinely think that your comments are indeed
useful.  I wondered myself if people were reading too much into the
results with regards to the expected life.