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Let's say I have one collection of machines (made by manufacturer 1). These machines run over a few days and fail at a certain rate. Let's say they run for a total of $h_1$ machine hours and the number of failures are $f_1$. If we assume an exponential distribution, the failure rate can be estimates as $\lambda_1 = f_1/h_1$ failures per hour. Also, the number of failures in an interval $t$ hours is Poisson distributed with rate $\lambda_1 t$.

Now, there is another collection of machines. These run for $h_2$ machine hours and fail $f_2$ times. The failure rate for these is $\lambda_2=f_2/h_2$

I discover that $\lambda_2 > \lambda_1$. But, I want to do a hypothesis test and say how certain I am that the second collection of machines fail at a higher rate than the first. What is the best way to construct such a test?

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    $\begingroup$ Maybe i see this a bit to simple, but why not calculating the f/h value for every machine and then using a between sample test for collection 1 vs 2? This only works if the error-rate is not dependent on the total opperating time of a machine (does a machine produce as many errors during hour 1 as during hour 10 as during hour 1000)? Another approach could be finding "bad" machines in collection 1 and 2. Maybe the standard-machine of collection 1 produces just as many errors as of collection 2, but collection 2 has 5% of bad machines and collection 1 only 3%? $\endgroup$
    – TinglTanglBob
    Commented Aug 16, 2018 at 11:07

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