I have an electronic weighing-machine, which I believe to be internally very accurate. It will weigh up to 100 kg, but not activate below 10 kg. The digital display reports to one decimal place. The problem is that I don't know whether the reading is rounded (with worst error ± 5050 g) or truncated (with worst error −100 g and expected bias −50 g). I have a great quantity of books and papers that can be stacked on the machine to make any weight within its limits, but nothing of accurately known weight.I guess that any solution must be statistical. A good solution would minimize the number of weighings, given a tolerance probability of a false indication. (Assume a 50/50 prior distribution for rounding/truncation; for illustration, a targeted probability could be 0.1%.)

Question

I have an electronic weighing-machine, which I believe to be internally very accurate. It will weigh up to 100 kg, but not activate below 10 kg. The digital display reports to one decimal place. The problem is that I don&#039;t know whether the reading is rounded (with worst error   ±  5050 g) or truncated (with worst error −100 g and expected bias −50 g). I have a great quantity of books and papers that can be stacked on the machine to make any weight within its limits, but nothing of accurately known weight.I guess that any solution must be statistical. A good solution would minimize the number of weighings, given a tolerance probability of a false indication. (Assume a 50/50 prior distribution for rounding/truncation; for illustration, a targeted probability could be 0.1%.)

Haleigh Vega · Accepted Answer

Hint: Suppose you weigh a batch of individual objects that are a little over 10 kg and record their weights. Now stack them together. What do you expect the result to be in the two cases? If you do this a few times, you will know.

codosse2e · Accepted Answer

Assuming the scales are perfectly accurate then:In the case of truncation, if you choose two random objects with recorded weights p and q, then the weight of them both together will be recorded as either p + q or p + q + 100 g, with equal probabilities.In the case of rounding, if you choose two random objects with recorded weights p and q, then the weight of them both together will be recorded as either p + q - 100 g or p + q, with equal probabilities.So as soon as you see a single +100g or -100g result you will have absolute certainty which is the case, but until then you won&#039;t be able to distinguish between them. So 10 such comparisons will give a probability of less than 0.1% that you still do not know which is the case.

I have an electronic weighing-machine, which I believe to be internally very accurate. It will weigh

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