How to Ask an Embarrassing Question

I despise telemarketers and hate filling questionnaires. Usually there are too many questions, most either irrelevant (in my view) and rather personal. I may not mind answering a single question, though, especially if that requires a simple Yes/No answer. Provided, of course, it does not touch on a private matter.

And what if it does?

What if a question XXX is of the potentially embarrassing sort that most of people would not care to answer unless assured of complete anonymity? But how complete anonymity may be assured?

Here's how: let each person from a random population toss a (fair) coin. If it shows tails ask the person to truthfully answer the XXX question, Yes or No. If the coin shows heads, instruct the person to record the result of the second toss, with Yes for, say, heads and No for the tails. Ask the person to report the final Yes/No result.

As the Yes/No answers are collected there is no way to tell whether these relate to the XXX question or to the second toss of a coin. However, there is a way to estimate the proportion of the respondents that answered the XXX question one way or the other. How is this done?

Answer

References

E. B. Burger, M. Starbird, The Heart of Mathematics, Key College Publishing, 2000, Ch 7.6
P. Nahin, Duelling Idiots and Other Probability Puzzlers, Princeton University Press, 2000, pp 15-16

Assume the sample population counted M people of which N gave the Yes answer. Assuming the coins used were all fair, as was requested, M/2 answered the question XXX while the remaining M/2 simply reported the result of the second coin toss. A half of these, we may expect, gave the Yes answer. So out of N such answers, N - M/4 relate to the XXX question. Which means that from a sample populations of (approximately) M/2 people the Yes answer was given by (approximately) N - M/4 making the proportion:

(N - M/4) / M/2 = 2 (N/M - 1/4),

which is a reasonable approach to gathering some private data anonymously. The clever idea has been presented in the paper by Stanley L. Warner, "Randomized Response: A Servey Technique for Eliminating Evasive Answer Bias," Journal of the American Statistical Association, vol. 60, March 1965, pp. 63-69.

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