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Topic: chi-square in confidence interval for failure rate
Replies: 2   Last Post: Mar 16, 2013 11:44 PM

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Paul

Posts: 263
Registered: 2/23/10
Re: chi-square in confidence interval for failure rate
Posted: Mar 16, 2013 10:41 PM
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On Mar 16, 10:09 pm, Paul <paul.domas...@gmail.com> wrote:
> All the info I found online for estimating failure rate from the
> number of failures over a time period make reference to chi-squre,
> but without actually describing how it comes about. The most
> helpful explanation I found
> ishttp://www.weibull.com/hotwire/issue116/relbasics116.htm.
>
> However, I'm missing some fundamental intuition behind the starting
> point (equation 5). It is assumed that we see r failures over a
> time period T. The probabilities for 0 to r failures are summed up
> and related to the confidence level. Why is this equation true? I
> mean, we do not see 0 to r failures, we only see r failures, so why
> so those lesser failure counts come into the picture at all?
>
> Thanks if anyone can refer to an online explanation. If it is
> offline, I can probably get it eventually, but online would be so
> much more suitable for my timeframe.


I tried to plug in typical numbers to lend a bit of concreteness to
the equations. Assume a confidence level of 95%, so the one-sided
tail is 5%. Assume that we saw 5 failures over T=1 hour. What
equation 5 says is that There is a 5% chance of seeing 0 to 5 failures
in 1 hours. Somehow, that condition is satifisfied by the upper-bound
failure rate. I'm having a hard time seeing why.

This is not quite the same as the typical hypothesis testing problems
that I've seen (though admittedly, I'm relatively new to that
territory as well). In textbook hypothesis testing, one usually has
an H0 that occupies a point on a real line that represents the
possible values of the test statistic e.g. the estimated mean. The
acceptance an rejection region is perfectly obvious.

In the above problem, it seems like the test statistic is the number
of failures r, and it isn't clear to me what H0 is and why the region
[0,r] corresponds to the rejection region. This is all assuming that
I'm not barking up the wrong tree by drawing analogies with hypothesis
testing.



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