**Econ 57 Fall 2001 Midterm**

**1**. Legendary Harvard University statistics professor Frederick Mosteller
reported that if you flip a coin in class and ask if anything suspicious is
happening, “hands suddenly go up all over the room” after the fifth
head or tail in a row, “so there is some empirical evidence that the
rarity of events in the neighborhood of .05 begins to set people’s teeth
on edge.” What is the probability that a fairly flipped coin will give
the same result five times in a row (either five heads or five tails)?

**2**. To see who serves first
in their Thursday squash games, Player A spins the racket while Player B guesses
whether the racket logo will stop face up or face down. Player B initially believes
that there is only a 5% chance that A cheats when he spins the racket. But after
A wins the first five times he spins the racket, B isn’t so sure. Assuming
that A will always win if he cheats and has a 50% chance of winning if he doesn’t
cheat, what is B’s revised probability that A cheats?

**3**. The noon temperature on July 4 in two cities, A and B, is normally
distributed with respective means MA and MB and standard deviations SA and SB.
For each of these three cases, identify the city that has the higher probability
of a noon July 4 temperature above 100 degrees.

a. MA = 80, SA = 20, MB = 80, SB = 30

b. MA = 80, SA = 20, MB = 70, SB = 20

c. MA = 80, SA = 20, MB = 70, SB = 30

**4**. What is wrong with this
histogram?

**5**. Mark believes that there
is a 4/5 probability that it will snow in Boston on New Year’s eve 2001;
Mindy believes that the probability is only 2/3. What bet could they make that
would give each a positive expected value? For example, Mark pays Mindy $2 if
it snows and Mindy pays Mark $5 if it doesn’t.

**6**. For each of the following
studies, identify the type of graph (1 histogram, side-by-side boxplots, 1 scatter
diagram, or 1 time series graph) that would be MOST appropriate.

a. Do countries with lots of smokers have lots of lung-cancer deaths?

b. Does the time between eruptions of Old Faithful depend on the duration of the preceding eruption?

c. Is there more variation in annual rainfall in Los Angeles or in New York?

d. Have temperatures in Los Angeles gone up or down over the past 100 years?

e. Could the states that Al Gore won and lost in 2000 have been predicted from how well Bill Clinton did in each state in 1996?

**7**. Answer this question to
Ask Marilyn [Parade, September 10, 2000]: “Say we’re going to toss
a coin repeatedly until we get one of these sequences in order: heads/tails
(HT) or tails/tails (TT). If my sequence comes up first, you pay me a dollar;
but if your sequence comes up first, I pay you a dollar. At that point, we stop
and start all over again. Given that we’re going to play this game repeatedly,
which sequence would you choose?

**8**. Bill is taking a multiple-choice test, where each question has 5 possible
answers. Running out of time, he randomly guesses the answers to the last 5
questions.

a. What is the probability that he get all 5 questions correct?

b. What is the probability that he gets at least 3 questions correct?

**9**. Twenty-four female college
seniors were asked how many biological children they expect to have during their
lifetimes.

Of these four numbers (0, 1, 2, 2.3), one is the mean, one is the median, one is the standard deviation, and one is irrelevant. Identify which number is which.

a. mean

b. median

c. standard deviation

d. irrelevant

**10**. Answer this question to Ask Marilyn: “I recently returned from
a trip to China, where the government is so concerned about population growth
that it has instituted strict laws about family size. In the cities, a couple
is permitted to have only one child. In the countryside, where sons traditionally
have been valued, if the first child is a son, the couple may have no more children.
But if the first child is a daughter, the couple may have another child. Regardless
of the sex of the second child, no more are permitted. How will this policy
affect the mix of males and females?”