A researcher wants to estimate the percentage of people in some population who have

used illegal drugs, by conducting a survey. Concerned that a lot of people would lie

when asked a sensitive question like "Have you ever used illegal drugs?", the researcher

uses a method known as randomized response. A hat is filled with slips of paper, each

of which says either "I have used illegal drugs" or "I have not used illegal drugs". Let p

be the proportion of slips of paper that say "I have used illegal drugs" (p is chosen by

the researcher in advance).

Each participant chooses a random slip of paper from the hat and answers (truthfully)

"yes" or "no" to whether the statement on that slip is true. The slip is then returned

to the hat. The researcher does not know which type of slip the participant had. Let

y be the probability that a participant will say "yes", and d be the probability that a

participant has used illegal drugs.

(a) Find y, in terms of d and p.

(b) What would be the worst possible choice of p that the researcher could make in

designing the survey? Explain.

(c) Now consider the following alternative system. Suppose that proportion p of the slips

of paper say "I have used illegal drugs", but that now the remaining 1

(a) There are two crimson jars (labeled C1 and C2) and two mauve jars (labeled M1 and

M2). Each jar contains a mixture of green gummi bears and red gummi bears. Show by

example that it is possible that C1 has a much higher percentage of green gummi bears

than M1, and C2 has a much higher percentage of green gummi bears than M2, yet if

the contents of C1 and C2 are merged into a new jar and likewise for M1 and M2, then

the combination of C1 and C2 has a lower percentage of green gummi bears than the

combination of M1 and M2.

(b) Explain how (a) relates to Simpson's paradox, both intuitively and by explicitly

defining events A, B, C as in the statement of Simpson's paradox.

52. As explained in this chapter, Simpson's paradox says that it is possible to have events

A, B, C such that P(A|B,C)

P(A|B) > P(A|Bc).

(a) Can Simpson's paradox occur if A and B are independent? If so, give a concrete

example (with both numbers and an interpretation); if not, prove that it is impossible.

(b) Can Simpson's paradox occur if A and C are independent? If so, give a concrete

example (with both numbers and an interpretation); if not, prove that it is impossible.

(c) Can Simpson's paradox occur if B and C are independent? If so, give a concrete

example (with both numbers and an interpretation); if not, prove that it is impossible.

53.

Oriental Reproductions, Inc. is a company that produces handmade carpets with oriental designs. The production records show that the monthly production has ranged from 1 to 5 carpets. The production levels and their respective probabilities are shown below. Production Per Month Probability x x} 1 0.01 2 0.04 3 0.10 4 0.80 5 0.05 The standard deviation for production is: (Please express answer to three decimal places in the following form: 2.552 or 2.000). 3. Two video products distributors supply video tape boxes to a video production company. Company A sold 100 boxes of which 5 were defective. Company B sold 300 boxes of which 21 were defective. If a box was defective, find the probability that it came from Company B. (Bayes' Theorem)Suppose you own a publishing company. The daily production of books/magazines approximately follows normal distribution with a mean of 6000 books/magazines per day and standard deviation of 950 books/magazines per day. What is the probability that your publishing company produces less than 8200 books / magazines in a day? [ Select ] What is the probability that your publishing company produces atmost 3700 books / magazines in a day ? [ Select ] Which of the following intermediate step is involved in finding probability that the company produces less th 2700 books/magazines in a day [ Select ] [ Select ] PIZ> 3,47) 1 Plz-3.47)on Daminov 4. A die is rolled 3 times; find the probability of getting exactly one five. 6 5 5 [ 3573 5. A telemarketing company gets on average 6 orders per 1000 calls. If a company calls 500 people, find the probability of getting 2 orders (Poisson random variable)