🚀 go-pugleaf

> Consider the following problem:
>
> A random sample of 40 batches, each containing 5 vouchers is taken from a
> very large population of batches of vouchers and tested to see if the
> vouchers contain an error.
>
> # of Errors   # of batches containing that number of errors
> 0                                 37
>
> 1                                 1
> 4                                 1
> 5                                 1
>
> What is the upper bound on the population error rate one sided at 95%
> confidence? Does anyone know how to calculate this bound exactly? If you
> don't know of an exact solution does anyone know of a better approximation
> than normal-distribution based survey sampling methods provide?
>

Two initial points to be made here are

	- with discrete data like this it is impossible to get an "excact"
	  0.95 confidence interval in the sense of 95% coverage in repeated
	  samples, (except by "randomization", that is by the deliberate
	  injection of further randomness).  All you can get is an interval
	  of values that with this sample would lead to retention of the
	  null hypothesis at the 0.95 level.  Even then it depends on the
	  test you use, which is also to some degree arbitrary.

	- any "exact" property of the procedure you use is likely to be
	  made pretty irrelevant by the inevitable roughness of the
	  modelling assumptions you must make to get it.

Those aside, a simple approach would be as follows.

I don't see that it is a "binomial cluster sample".  A reasonable model
might be that the number of errors on a voucher chosen at random is
Poisson, Po(u).  In which case the total number of errors in the sample of
40 vouchers, T say, is sufficient for  u  and is also Po(40 u).  A one
sided upper bound on u is then got by finding u0 such that

		  if u = u0    then    Pr(T >= 10) = 0.95

(since in this sample there are T0 = 37x0 + 1 + 4 + 5 = 10 errors).  When
you solve this you get

			       u0 = 0.39263,

so you can be 95% confident, in the technical sense, that the mean number
of errors per voucher is no larger than this.

The arbitrariness of the test itself comes in when you need to decide
whether the rejection region should be  T >= T0  or  T > T0.

Even though this is a miniscule sample relative to the rarity of the event,
it seems to me that already there is some evidence that this simple Poisson
model is possibly misleading and that some type of contagious distribution
is likely to be safer.  But you will need much more data to make any
sensible and supportable statement either way, and exact tests are then not
feasible.

E&OE, as always.

--
Bill Venables, Dept. of Statistics, | Email: venables@stats.adelaide.edu.au
Univ. of Adelaide, South Australia. | Tel: +61 8 228 5412  Fax: ...232 5670