Originally Posted by Geof
You're right...the data is not obtainable and that includes the true lamp failure rate. Now we can presume
all lamps in the 2 or more failure category are bad but it's just that...a presumption. We could also presume they count just once. So, depending on how you want to spin it you can come up with several answers. In reality there is only 1 truth but there is not enough data to conclude what that is.
Well, there is actually more than one truth, because there is more than one possible question. If you change your question a bit, you can make it less difficult to obtain a reasonable answer.
The most useful question about this problem -- where useful includes the concept that it is possible to obtain an approximate answer given available resources -- is something like:
What is the probability that a randomly selected projector will have a lamp failure on its first (factory) lamp before the lamp reaches 500 hours?
This question has the disadvantage that it could include lamp failures due to faulty projectors (with good lamps) as well as failures due to bad lamps. But it has the advantage that given a reasonably selected sample of projectors, you have a chance at getting a good estimate of the true answer by asking owners whether they had their first lamp fail within 500 hours. And I think it is a question to which many prospective owners might like to know the answer.
If you want to glean more information from the sample, you could go back to the people who had their first lamp fail within 500 hours, and ask them the same question about their second lamp (same projector). If you find that the probability estimate is higher than the original one (first lamp failure) by a statistically significant amount, then you can conclude that some projectors do significantly increase the chance of having a lamp failure. Unfortunately, you may have trouble getting a large enough sample size for this second question. Well, unless there are A LOT of failures in your original sample. Which I suppose could be the case for the 2010 JVC projectors.
Or, you might find that almost all of the first lamp failures also have second lamp failures, which would obviously be a strong indicator that most lamp failures are due to faulty projectors rather than faulty lamps. Which suggests a possible poll question: Only answer if you had a factory lamp failure with less than 500 hours and replaced the lamp in the same projector. The second lamp: (a) failed with less than 500 hours (b) is still working fine after more than 500 hours (c) is working fine but with less than 500 hours so far