Statistics in a Modern World 800
Homework 6 NAME:_________________________________

Homework 6 Exercises Assigned from Chapters 16-18 (40 pts.) due Fri., Oct. 30 in Lecture.

CHAPTER 16

#1 (2 pts.) Recall that there are two interpretations of probability: relative frequency and personal probability.

  1. Which interpretation applies to this statement: "The probability that I will get the flu this winter is 30%"?
  2. Assume it is known that the proportion of adults who get the flu each winter remains at about 0.30. Which interpretation applies to this statement: "The probability that a randomly selected adult in America will get the flu this winter is 30%"?

#2c (1 pt.) The probability that a birth would result in twins was reported to be 0.0231, and the probability that a birth would result in triplets or more was 0.0008. What was the probability that a birth resulted in a single child only?

#4 (1 pt.) The probability of being born on a Friday the 13th was reported to be 1/214. What is the probability of not being born on a Friday the 13th?

#11 (3 pts.) The probability that a randomly selected American adult belongs to the American Automobile Association (AAA) is 0.10, and the probability that a randomly selected American adult belongs to the American Association of Retired Persons (AARP) is 0.11).

  1. Rule 3 p. 303 states: "If two events do not influence each other, and if knowledge about one doesn't help with knowledge of the probability of the other, the events are said to be independent of each other. If two events are independent, the probability that they both heppen if found by multiplying their individual probabilities." What assumption would we have to make in order to use this rule to conclude that the probability that a person belongs to both is (0.10)*(0.11)=0.011?
  2. Do you think that assumption holds in this case?
  3. Let's assume that the assumption in part (a) does hold. What would be the probability of a randomly selected American adult belonging to AAA or AARP?

#12 (1 pt.) A study by Kahneman and Tversky described the following situation: "Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in antinuclear demonstrations."

Circle the more likely alternative:

  1. Linda is a bank teller.
  2. Linda is a bank teller and is active in the feminist movement.

#17 (4 pts.) Refer again to Rule 3 in #11 above. State whether each of the following pairs of events is likely to be independent:

  1. Event A is that it snows tomorrow; Event B is that the high temperature tomorrow is at least 60 degrees Fahrenheit.
  2. You buy a lottery ticket, betting the same numbers two weeks in a row. Event A is that you win in the first week; Event B is that you win in the second week.
  3. You pick 2 cards with replacement from an ordinary deck. Event A is that the first card is an ace. Event B is that the second card is an ace.
  4. You pick 2 cards without replacement from an ordinary deck. Event A is that the first card is an ace. Event B is that the second card is an ace.

    5.  

#18 (4 pts.) Suppose you routinely check coin-return slots in vending machines to see if they have any money in them. The overall proportion of times that you find money is 0.10.

  1. What is the probability that you do not find money next time you check?
  2. What is the probabilty that the next time you will find money is on the third try? (Show your work.)

    3.  

  3. What is the probability that you will have found money by the third try? Show your work.

    4.  

  4. Given that you find money, the chance of finding at least a quarter is 0.8. When you check a slot, what is the chance of finding money and it is at least a quarter? Show your work.

    5.  

#21 (2 pts.) Suppose you have to cross a train track on your commute. The probability that you will have to wait for a train is 0.20. If you don't have to wait, the commute takes 15 minutes, but if you have to wait, it takes 20 minutes.

  1. What is the expected value of the time it takes you to commute?
  2. Is the expected value ever the actual commute time?

#25 (2 pts.) A report stated that 72% of children in the U.S. were living with both parents, 22% with mother only, 3% with father only, and 3% with neither parent.

  1. What was the expected value for number of parents a randomly selected child was living with? Show your work.

    2.  

  2. Would "mean" make more sense to describe your answer to (a) than "expected value"?

CHAPTER 17

#4 (3 pts.) Suppose a defense attorney is trying to convince the jury that his client's wallet, found at the scene of the crime, was actually planted there by his client's gardener. Here are two possible ways he might present this to the jury:

Statement A: The gardener dropped the wallet when no one was looking.

Statement B: The gardener hid the wallet in his sock and when no one was looking he quickly reached down and pulled off his sock, allowing the wallet to drop to the ground.

  1. The second statement cannot have a higher probability of being true than the first because of (circle one: ) Rule 1 Rule 2 Rule 3 Rule 4

    2. Rule 1: If there are only two possible outcomes in an uncertain situation, then their probabilities must add to 1.

    Rule 2: If two outcomes cannot happen simultaneously, they are said to be mutually exclusive. The probability of one or the other of two mutually exclusive outcomes happening is the sum of their individual probabilities.

    Rule 3: "If two events do not influence each other, and if knowledge about one doesn't help with knowledge of the probability of the other, the events are said to be independent of each other. If two events are independent, the probability that they both heppen if found by multiplying their individual probabilities."

    Rule 4: If the ways in which one event can occur are a subset of those in which another event can occur, then the probabiity of the subset event cannot be higher than the probability of the one for which it is a subset.

  2. Based on the material in this chapter, to which statement are members of the jury likely to assign higher personal probabilities? (i) statement A (ii) statement B
  3. Which concept applies here? (i) anchoring (ii) availability (iii) representativeness

    4. (iv) forgotten base rates (v) optimism (vi) overconfidence

#6 (2 pts.) A telephone solicitor recently contacted the author to ask for money for a charity in which typical contributions are in the range of $25 to $50. The solicitor said, "We are asking for as much as you can give, up to $300.00"

  1. Which concept applies here? (i) anchoring (ii) availability (iii) representativeness

    2. (iv) forgotten base rates (v) optimism (vi) overconfidence

  2. A second way of asking would have been, "We typically get $25 to $50, but give as much as you can." Which way is likely to solicit more money, the first or second?

#13 (4 pts.)Which concept applies in each of the following scenarios?Choose

ANchoring, AVailability, Representativeness, Forgotten Base Rates, OPtimism, or OVerconfidence

  1. Most people rate death by shark attacks to be much more likely than death by falling airplane parts, yet the chances of dying from the latter are actually 30 times greater (Plous, 1993, p. 121)

    AN AV R FBR OP OV

  2. You are a juror on a case involving an accusation that a student cheated on an exam. The jury is asked to assess the likelihood of the statement, "Even though he knew it was wrong, the student copied from the person sitting next to him because he desperately wants to get into medical school." The other jurors give the statement a high probability assessment although they know nothing about the accused student.

    AN AV R FBR OP OV

  3. Research by Tversky and Kahneman (1982b) has shown that people think that words beginning with the letter k are more likely to appear in a typical segment of text in English than words with k as the third letter. In fact, there are about twice as many words with k as the third letter than words that begin with k.

    AN AV R FBR OP OV

  4. A 45-year-old man dies of a heart attack and does not leave a will.

    AN AV R FBR OP OV

CHAPTER 18

#1 (3 pts.) Suppose the probability of having a male child (M) is equal to the probability of having a female child (F). A couple has four children.

  1. Which is more likely? (i) FFFF (ii) MFFM (iii) both equally likely
  2. According to the belief in the law of small numbers, which sequence would people tend to say is more likely? (i) FFFF (ii) MFFM
  3. If a couple has 4 children, which is more likely? (i) 4 girls (ii) 2 boys, 2 girls in any order (iii)same

#10 (6 pts.) Suppose the proportion of people with a certain rare disease is 0.0001. A test for the disease has sensitivity of 0.95 and specificity of 0.90.

  1. On the bottom of this page, draw a tree diagram for this situation, labeling branches for having or not having the disease and testing positive or negative, along with their probabilities.
  2. What is the probability of having the disease and testing positive?
  3. What is the overall probability of testing positive?
  4. Given that test results are positive, what is the probability of having the disease?

#11 (1 pt.) You are at a casino with a friend, playing a game in which dice are involved. Your friend has just lost six times in a row. She is convinced that she will win on the next bet because she claims that, by the law of averages, it's her turn to win. She explains to you that the probability of winning this game is 40%, and because she has lost six times, she has to win four times to make the odds work out. What do we call this misconception?

#21 (1 pt.) It is a time for the end-of-summer sales. One store is offering bathing suits at 50% of their usual cost, and another store is offering to sell you two for the price of one.

Assuming the suits originally all cost the same amount, which store is offering a better deal? (i) first (ii) second (iii) both the same

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