An investigator would like to survey a set of people in order to learn what fraction of them use illegal drugs. The survey involves the standard approach of randomly selecting a subset of individuals to survey from the complete list of population members. The investigator is concerned that potential respondents will be reluctant to answer a question about drug usage truthfully. The investigator will have each respondent flip a coin. If it comes
An investigator would like to survey a set of people in order to learn what fraction of them
use illegal drugs. The survey involves the standard approach of randomly selecting a subset
of individuals to survey from the complete list of population members.
The investigator is concerned that potential respondents will be reluctant to answer a question about drug usage truthfully.
The investigator will have each respondent flip a coin.
If it comes up tails, the respondent will answer 1 (yes) or 0 (no) to the question “Is the first
digit of your social security number even”.
If the coin comes up heads, the respondent will answer 1 (yes) or 0 (no) to the question “do
you use illegal drugs”.
Thus, each respondent will answer 1 or 0 (yes or no) but the investigator does not know
which of the two questions the respondent is actually replying to. The hope is that since the
investigator does not know which question was asked, the respondent will give the correct
answer.
Let Q be the random variable which is 1 if the coin comes up heads (the drugs question is
asked) and 0 if the coin comes up tails (the digit question is asked).
Let R be the random variable representing the answer (1 for yes, 0 for no).
Thus, P(Q = 1) = P(Q = 0) = .5.
The investigator believes that P(R = 1 | Q = 0) = .5.
The investigator would like to know P(R = 1 | Q = 1).
Since we will use P(R = 1 | Q = 1) a lot, to simplify notation let’s also call it p1:
p1 = P(R = 1 | Q = 1).
First, let’s look at things from the point of the respondent.
He might wonder if the investigator can guess what question was asked. For example, if drug
use is very low in the population, then a no answer might suggest it was the drug question
that was asked.
To investigate this, a respondent supposes that prior to collecting the data, the investigator
might believe p1 = .1
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
