Exercises 137be computing some very large numbers, which may lead to substantial rounding error. The
value of .0011 is what my calculator produced. From earlier we know that p(blue) 5 .24)
5.33 This question is not an easy one, and requires putting together material in Chapters 3, 4, and- Suppose we make up a driving test that we have good reason to believe should be passed
by 60% of all drivers. We administer it to 30 drivers, and 22 pass it. Is the result sufficiently
large to cause us to reject (p 5 .60)? This problem is too unwieldy to be approached by
solving the binomial for X 5 22, 23,... , 30. But you do know the mean and variance of
the binomial, and something about its shape. With the aid of a diagram of what the distribu-
tion would look like, you should be able to solve the problem.
5.34 Make up a simple experiment for which a sign test would be appropriate.
a. Create reasonable data and run the test.
b. Draw the appropriate conclusion.
Discussion Questions
5.35 The “law of averages,” or the “gambler’s fallacy,” is the oft-quoted belief that if random
events have come out one way for a number of trials they are “due” to come out the other
way on one of the next few trials. (For example, it is the (mistaken) belief that if a fair coin
has come up heads on 18 out of the last 20 trials, it has a better than 50:50 chance of com-
ing up tails on the next trial to balance things out.) The gambler’s fallacy is just that, a
fallacy—coins have an even worse memory of their past performance than I do. Ann
Watkins, in the Spring 1995 edition of Chancemagazine, reported a number of instances of
people operating as if the “law of averages” were true. One of the examples that Watkins
gave was a letter to Dear Abby in which the writer complained that she and her husband had
just had their eighth child and eighth girl. She criticized fate and said that even her doctor
had told her that the law of averages was in her favor 100 to 1. Watkins also cited another
example in which the writer noted that fewer English than American men were fat, but the
English must be fatter to keep the averages the same. And, finally, she quotes a really re-
markable application of this (non-)law in reference to Marlon Brando: “Brando has had so
many lovers, it would only be surprising if they were all of one gender; the law of averages
alone would make him bisexual.” (Los Angeles Times, 18 September 1994, Book Reviews,
p. 13) What is wrong with each of these examples? What underlying belief system would
seem to lie behind such a law? How might you explain to the woman who wrote to Dear
Abby that she really wasn’t owed a boy to “make up” for all those girls?
5.36 At age 40, 1% of women can be expected to have breast cancer. Of those women with breast
cancer, 80% will have positive mammographies. In addition, 9.6% of women who do not
have breast cancer will have a positive mammography. If a woman in this age group tests
positive for breast cancer, what is the probability that she actually has it. Use Bayes’ theo-
rem to solve this problem. (Hint: Letting BCstand for “breast cancer,” we have p(BC) 5
.01, p( 1 |BC) 5 .80, and p( 1 | ) 5 .096. You want to solve for p(BC| 1 ).)
5.37 The answer that you found in 5.36 is probably much lower than the answer that you ex-
pected, knowing that 80% of women with breast cancer have positive mammographies.
Why is it so low?
5.38 What would happen to the answer to Exercise 5.36 if we were able to refine our test so that
only 5% of women without breast cancer test positive? (In others words, we reduce the rate
of false positives.)BCH 0