Problem 3 (RR: Cards Version)

Your colleague, Alice, tells you about a technique to protect the privacy of surveyed subjects in a survey about their views on capital punishment. In this technique, M identical white cards are used. On the face of mA of the cards, the following statement is written: “I support the use of the death penalty as a form of criminal punishment." On the face of the rest of the cards (M −mA), the following statement is written: “I oppose the use of the death penalty as a form of criminal punishment."

Each subject is then asked to pick a card (face down) at random and read the statement to herself or himself in private. The subject is afterwards instructed to return the card to the pile, which is then shuffled in order to lose the subject’s selected card. Subsequently, the subject is asked whether she or he agrees with the statement written on the randomly selected card without revealing what the statement on the card was. The investigator then records the response of the respondent. We will denote the total number of respondents in this survey as n, out of whom n1 respondents respond with “agree."

Alice understands how this technique protects the privacy of the individual subjects in terms of their views on capital punishment, but is perplexed as to how to use the responses to estimate the percentage of the population that supports the death penalty as a form of criminal punishment.

Since you are taking W233, you argue that this is a reduction of the randomized response technique. In this problem, you will prove to Alice that the technique she described is a version of randomized response. You will also devise the maximum likelihood estimate of the percentage of the population that supports capital punishment from M,mA,n1 and n.

(a) You argue that the "cards" mechanism is equivalent to constructing a spinner with two “slices": i) one slice with total area portion p that reads: “I support the death penalty as a form of criminal punishment;" and ii) one slice with total area portion 1− p that reads: “I oppose the death penalty as a form of criminal punishment." Your proposed model is presented in Figure 2.

Calculate the p (in terms of M and mA) that would make the spinner version equivalent to the cards version described by Alice.

Figure 2: Your proposed spinner equivalent of the cards version.

(b) Devise the maximum likelihood estimate of the percentage of the population, πˆ, that supports the death penalty as a form of criminal punishment. Write your answer in terms of n,n1,M and mA only.

(c) What is the variance of the maximum likelihood estimate πˆ? Write your answer in terms of n,n1,M,mA and π (the true percentage of the population that supports the death penalty) only.

(d) Alice now brings you the data from the actual survey conducted using the “cards" variation of the study. 100 total cards were used in the study, 11 out of them were “support" cards, and the rest were “oppose" cards. There were a total of 1520 participants in the survey; 499 out of them reported “agree."

What is the maximum likelihood estimate of the percentage of the population that supports the death penalty as a form of criminal punishment? 

(e) A year later, Alice asks you to design the new survey to study the percentage of the population that supports the death penalty as a form of criminal punishment. She says that you have a budget to survey n = 1200 respondents. Furthermore, you have 100 blank cards to use in the study.

Under these assumption, you are tasked to find how many cards (mA) you will use to write the statement “I support the death penalty as a form of criminal punishment" under the following conditions:

(a) mA ≤ 50;

(b) The worst-case 99.7% confidence interval of the maximum likelihood estimate has width of at most 16 percentage points; and

(c) You provide the highest level of privacy to the respondents.

Note that mA has to be an integer value

Hint: 99.7% confidence interval for normal distributions can be approximated by 3 standard deviations around the mean, from each side (i.e., the width of the 99.7% confidence interval is 6 standard deviations).

Problem 4 (Acknowledgments)

List all individuals and sources that you consulted with while working on this homework.