Thank you to everyone who took the time to respond to this query (reproduced at the bottom of this note). There were 6 responses and nearly everyone made the same point, namely that it depends on the sensitivity of the instrumental test. If the test has a certain detection threshold, then it is possible to get a positive result with 100g whilst getting no positive results if one could test 10x10g with the Salmonella evenly distributed across each 10g sample such that each one is below the threshold. I have been advised by the microbiologists that the test is extremely sensitive to the extent that a single cell of salmonella in 100g would be detectable. In that case the confidence interval would remain unchanged.
One interesting reference was an exact calculation for the confidence interval using Wilson’s method. The upper bound of the 95% interval is apparently 3.84/(3.84+n) rather the 3/n that I used. The following link contains more information on this calculation.
http://www.uwcm.ac.uk/study/medicine/epidemiology_statistics/research/statistics/newcombe.htm
ORIGINAL QUERY
I have a question regarding the following sampling scenario.
An agricultural cargo (such as rice or coconut) is tested for microbiological contamination such as Salmonella. The cargo consists of 300 bags of the product and a 10g sample is taken from each bag giving 300 x 10g in total. Suppose it were possible to test each 10g sample in turn for Salmonella and zero positive results were obtained. Then a 95% confidence interval for the contamination rate for the cargo would [ 0%, 1% ].
However, the instrument used to detect Salmonella requires a minimum of 100g for the test to be effective. Consequently, the 300 x 10g samples are combined into 30 x 100g samples. This also reduces the time taken to do all tests as only 30 tests have to be done rather than 300.
My question is what is the confidence interval for the contamination rate if zero positives occur. Is it still [ 0%, 1% ] or is it now [ 0%, 10% ] because only 30 tests were done? I feel the confidence interval is still unchanged since even if all samples were combined into a single test, the fact that the test is negative must imply that all of 10g samples were negative. Obviously if there is a positive test then one would have to base the confidence on 30 tests rather than 300 tests.
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