Problem 2 (Analyze the Results)

A survey was conducted with randomized response as a privacy paradigm. The survey asks subjects about their views on the same-sex marriage. The spinner’s p was set to 0.95 to land on the group “I oppose same-sex marriage."

(a) The percentage of people answering “no" to the spinner question is 78.55%. Based on these results, what is the maximum likelihood estimate of the true proportion of the population that supports samesex marriage?

(b) For this part only, you may assume that the total number of respondents to the survey were 21 people. Moreover, for this part only, you may assume that the value of πˆ is 18%.

You tell your friend Bob about this survey and the result πˆ = 18%. Your friend Bob is concerned about the accuracy of the estimate. He claims that you don’t have a large enough sample to be confident about your maximum likelihood estimate. He claims that the probability that you get an estimate of πˆ = 18% or smaller, even if the true percentage of the population supporting same-sex marriage is 50%, is larger than 0.13.

Is Bob right or wrong? Justify your answer!

From Part (c) onward, you may assume that the true portion of the population that opposes same-sex marriage is π = 0.18.

(c) Bob is now concerned about the level of privacy that this survey provides its participants. His claim is that p = 0.95 means that 95% of the subjects will get the spinner on the statement “I oppose samesex marriage." Because of that, Bob claims, the investigators (i.e., you) are able to identify which of those individuals are indeed opposed to same-sex marriage with a high probability. Bob suggests that having p set at 0.98 reduces the chances that the investigators can identify which of the individuals answering “yes" on the spinner question are indeed opposed to same-sex marriage.

Compare the probabilities P(Gi = “oppose”|Xi = “yes”; p = 0.95) (identifying a subject in group “oppose" assuming that the subject answered “yes", when p = 0.95) and P(Gi = “oppose”|Xi = “yes”; p = 0.98) (identifying a subject in group “oppose" assuming that the subject answered “yes", when p = 0.98) to prove to Bob that his argument is incorrect. \

(d) Since increasing p didn’t help in the previous part, Bob now wants to decrease p to 0.05. Bob uses your calculation from the previous part to show that the risk of identifying an individual who answered “yes" as an “opposer" to same-sex marriage now drops as opposed to the case when p = 1. From this, Bob argues that p = 0.05 is better for the privacy of the “opposers."

You note that Bob’s logic is still flawed because Bob is not making a fair comparison in this case. Under Bob’s proposal, p = 0.05 means that 95% of the subjects will now get the statement “I support same-sex marriage," making it the most common statement shown to subjects. This means that answering “no" is now the “sensitive" answer since it is the one that reveals “opposition" to same-sex marriage.

So, to figure out whether Bob is right or wrong, you now calculate the probability P(Gi = “oppose”|Xi = “no”; p = 0.05) (identifying a subject in group “oppose" assuming that the subject answered “no", when p = 0.05) and compare it to P(Gi = “oppose”|Xi = “yes”; p = 0.95) (identifying a subject in group “oppose" assuming that the subject answered “yes", when p = 0.95) from the previous question. Is Bob right this time?

(e) Bob finally understands that he needs to take W233 to become an expert on randomized response. He stops making arguments and instead asks you which value p provides the most level of privacy to subjects. Show Bob that p = 0.5 provides the highest level of privacy to subjects by proving that whenever p = 0.5, the subject’s answer to the spinner question Xi becomes statistically independent from the group that the subject actually belongs to Gi .

Hint: show that P(Gi |Xi ; p = 0.5) = P(Gi) (for all combination of values for Gi and Xi).