Determining the prevalence of illicit substances in society can be very difficult. Simply looking at police records doesn’t give the whole picture, and any surveys designed to determine usage are subject to all sorts of biases. In particular, if responding “yes” to a particular question carries with it a stigma or implication of guilt, then many respondents will lie. To alleviate this issue, we can use a survey design that is based on conditional probability. Suppose we want to estimate the proportion of the Helena population who have used marijuana within the last month. We select 4744 participants and ask them each to first privately draw a card from a standard 52-card deck. * If they get a “club” they are asked to respond truthfully to the question “Have you used marijuana within the last month?” * If they get any other suit they are asked to respond to the question “Does your phone number end in an odd digit?” The study is blinded in the sense that the researcher doesn’t know which question the participant is answering. Hence, participants are guaranteed anonymity and are more likely to be truthful in their responses. The researchers only get a “yes” or “no” answer from each respondent and don’t know the actual results of the card draw. (a) What is the probability of getting a club? ____ (b) What is the probability of not getting a club? _____ (c) What is the expected probability of a person’s phone number ending in an odd digit? ______ (d) What is the expected probability of a person’s phone number not ending in an odd digit? ______ (e) We don’t know the probability that a Helena resident has used marijuana in the last month (that is what we want!), but from this study we know the probability of answering the marijuana question and the percent of participants that answered “yes”. Assume that 4744 people are surveyed and 2072 of them say “yes”. Use this information to fill in the entire table. drew a club did not draw a club Total response = yes 2072 response = no Total 4744 (f) Based on the table, what is a point estimate for the proportion of people in the Helena community that have used marijuana in the last month? Estimate = ___________
Determining the prevalence of illicit substances in society can be very difficult. Simply looking at police records doesn’t give the whole picture, and any surveys designed to determine usage are subject to all sorts of biases. In particular, if responding “yes” to a particular question carries with it a stigma or implication of guilt, then many respondents will lie. To alleviate this issue, we can use a survey design that is based on conditional
Suppose we want to estimate the proportion of the Helena population who have used marijuana within the last month. We select 4744 participants and ask them each to first privately draw a card from a standard 52-card deck.
* If they get a “club” they are asked to respond truthfully to the question “Have you used marijuana within the last month?”
* If they get any other suit they are asked to respond to the question “Does your phone number end in an odd digit?”
The study is blinded in the sense that the researcher doesn’t know which question the participant is answering. Hence, participants are guaranteed anonymity and are more likely to be truthful in their responses. The researchers only get a “yes” or “no” answer from each respondent and don’t know the actual results of the card draw.
(a) What is the probability of getting a club? ____
(b) What is the probability of not getting a club? _____
(c) What is the expected probability of a person’s phone number ending in an odd digit? ______
(d) What is the expected probability of a person’s phone number not ending in an odd digit? ______
(e) We don’t know the probability that a Helena resident has used marijuana in the last month (that is what we want!), but from this study we know the probability of answering the marijuana question and the percent of participants that answered “yes”. Assume that 4744 people are surveyed and 2072 of them say “yes”. Use this information to fill in the entire table.
| drew a club | did not draw a club | Total | |
| response = yes | 2072 | ||
| response = no | |||
| Total | 4744 |
(f) Based on the table, what is a point estimate for the proportion of people in the Helena community that have used marijuana in the last month?
Estimate = ___________
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