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226 Chapter 4 Special Distributions
(Case Study 4.2.1 continued)
P(X ≥ 8) = 1 − P(X ≤ 7)
=
˙ 1−
7
e−1.75 (1.75)k
k=0
k!
= 1 − 0.99953
= 0.00047
How close can we expect 0.00047 to be to the “true” binomial sum? Very
close. Considering the accuracy of the Poisson limit when n is as small as one
hundred (recall Table 4.2.2), we should feel very confident here, where n is 7076.
Interpreting the 0.00047 probability is not nearly as easy as assessing its
accuracy. The fact that the probability is so very small tends to denigrate the
hypothesis that leukemia in Niles occurred at random. On the other hand, rare
events, such as clusters, do happen by chance. The basic difficulty of putting
the probability associated with a given cluster into any meaningful perspective
is not knowing in how many similar communities leukemia did not exhibit a
tendency to cluster. That there is no obvious way to do this is one reason the
leukemia controversy is still with us.
About the Data Publication of the Niles cluster led to a number of research efforts
on the part of biostatisticians to find quantitative methods capable of detecting
clustering in space and time for diseases having low epidemicity. Several techniques were ultimately put forth, but the inherent “noise” in the data—variations in
population densities, ethnicities, risk factors, and medical practices—often proved
impossible to overcome.
Questions
4.2.1. If a typist averages one misspelling in every 3250
words, what are the chances that a 6000-word report is
free of all such errors? Answer the question two ways—
first, by using an exact binomial analysis, and second, by
using a Poisson approximation. Does the similarity (or
dissimilarity) of the two answers surprise you? Explain.
4.2.2. A medical study recently documented that 905 mistakes were made among the 289,411 prescriptions written
during one year at a large metropolitan teaching hospital. Suppose a patient is admitted with a condition serious
enough to warrant 10 different prescriptions. Approximate the probability that at least one will contain an
error.
4.2.3. Five hundred people are attending the first annual
“I was Hit by Lighting” Club. Approximate the probability that at most one of the five hundred was born on
Poisson’s birthday.
4.2.4. A chromosome mutation linked with colorblindness is known to occur, on the average, once in every ten
thousand births.
(a) Approximate the probability that exactly three of
the next twenty thousand babies born will have the
mutation.
(b) How many babies out of the next twenty thousand would have to be born with the mutation
to convince you that the “one in ten thousand”
estimate is too low? [Hint: Calculate P(X ≥ k) =
1 − P(X ≤ k − 1) for various k. (Recall Case
Study 4.2.1.)]
4.2.5. Suppose that 1% of all items in a supermarket are
not priced properly. A customer buys ten items. What is
the probability that she will be delayed by the cashier
because one or more of her items require a price check?