Economics 57 Spring 1997 Midterm

1. If you roll 4 standard 6-sided dice and calculate the mean and standard deviation for these 4 numbers, what is
a. the largest possible mean?
b. the smallest possible mean?
c. the largest possible standard deviation?
d. the smallest possible standard deviation?

2. Miller Brewing Company used to advertise a taste test involving 3 glasses of Miller High Life beer: one from a tap, one from a can, and one from a bottle. Of "many beer experts" tested, "not a single person identified all 3 correctly, because all have the famous Miller High Life taste!" If the 3 glasses of beer are indistinguishable, forcing an expert simply to guess, what is the probability of identifying all three correctly?

3. A letter to columnist Marilyn vos Savant began: "As professors of statistics, we found your response to the drug-testing question perplexing and, indeed, incorrect." The question was, "Suppose we assume that 5 percent of people are drug-users. A test is 95 percent accurate, which we'll say means that if a person is a user, the result is positive 95 percent of the time; and if she or he isn't, it's negative 95 percent of the time. A randomly selected person tests positive. Is the individual highly likely to be a drug user?" In fact, vos Savant calculated the probability correctly. What is the correct probability?

4. In the final poll before the 1996 presidential election, the Gallup organization interviewed 2000 likely voters, of whom 800 said they planned to vote for Bob Dole for president. If these 2000 persons were randomly selected from a very large population of whom half were Dole supporters, what is the probability that fewer than 801 of the 2,000 people selected would turn out to be Dole supporters? (Use any correct method.)

5. There are six boxes, five empty and one containing a check for $10,000. Two contestants will alternate choosing boxes until one selects the box with the $10,000 check. After a box has been opened, it is discarded. If you are one of the contestants, would you have a better chance of winning if you choose first or second? Prove it.

6. For each of the following scenarios, state whether it is more likely to happen when a fair coin is flipped 10 billion times or when it is flipped 100 billion times. You do not need to explain your reasoning.
a. getting more than 52 percent heads
b. getting exactly 50 percent heads
c. getting exactly 51 percent heads
d. getting between 49.99 and 50.01 percent heads
e. getting 1000 more heads than tails

7. What is the most misleading feature of this graphic that was used by a water-utility company in its 1994 annual report to show the growth in its customer base?

8. A 1994 article in the American Scientist reported the results of a survey of 17 scientific experts about the chances of the earth experiencing catastrophic global warming. Here, for example, is a summary of their assessments of the probability of a 3-degree-Celsius increase in the global average temperature by 2090. How has this graph been misdrawn?

9. Based solely on your own personal opinion, draw a rough sketch of a histogram of the ages of the Pomona College faculty, using the intervals 20-40, 40-50, 50-60, and 60-80 years. Be sure to explain your reasoning and label the vertical axis clearly.

10. Here is histogram of the chest measurements (in inches) of 5,738 Scottish militiamen in the early 19th century. Based on this figure, make rough estimates of the mean and standard deviation. You must come up with specific numbers and explain how you arrived at them. You will be graded not on whether your estimates are exactly right, but on whether you used a reasonable procedure to obtain these estimates.