(1 point) Determining the prevalence of illicit substance in society can be very

Question

(1 point) Determining the prevalence of illicit substance 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 which 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 4664 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? 13/52 (b) What is the probability of not getting a club? 39/52 (c) What is the expected probability of a person's phone number ending in an odd digit? 1/2 (d) What is the expected probability of a person's phone number not ending in an odd digit? 1/2 (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 4664 people are surveyed and 2008 of them say "yes". Use this information to fill in the entire table drew a club did not draw a club Total 1506 1328 3498 response yes 502 2008 response = no | 1328 2656 Total 1166 4664 (f) Based on the table, what is a point estimate for the percent of people in the Helena community that have used marijuana in the last month? Estimate= 0.50

Explanation / Answer

A deck of 52 cards has 13 cards each of clubs, spades, diamonds and hearts.

(a)

We need to find the probability that a card picked from the deck is a club.

Total number of favourable outcomes = 13 (since there are 13 club cards)

Total number of possible outcomes = 52.

So, the probability that a card picked from the deck is a club will be 13/52.

(b)

By the law of total probability we know that,

Probability of a event occurring + probability of the same event not occurring = 1

So, probability of not getting a club = 1 - probability of getting a club

= 1 - 13/52 = 39/52

(c)

We needn't he probability that the contact number of the person ends in an odd digit.

The sample space = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

favourable outcomes = { 1, 3, 5, 7, 9}

So, total number of outcomes = 10

And, total number of favourable outcomes = 5

And the required probability = 5/101= 1/2

(d)

Similarly, by law of total probability,

Probability that the number does not end in an odd digit = 1 - 1/2 = 1/2.

(e)

Since 2008 people say yes.

And the probability that they drew a club card is 13/52

The expected number of people who drew a club and said yes

= 2008 × 13/52

= 502.

So the expected number of people who said yes but did not get a club card = 2008 - 502 = 1506

Also, 2008/4664 said yes, so, 4664 - 2008 = 2656 people said no.

Since probability that a person's number ends in even = 1/2

And he will say no only if the number ends in even digit.

Also, he will be asked this question only if he doesn't get a club.

expected number of people who said no and did not get a club card

=2656 × 1/2

= 1328

Similarly, people who got a club and said no

=total people who said no - people who didn't get a club and said no

=2656 - 1328

=1328

Thus, expected number of people who drew club = 4664 × 13/52 = 1166

And, expected number of people who didn't get clubc= 4664 × 39/52 = 3498

Related Questions

Navigate

• Browse (All)
• Browse #
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.