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226 Chapter 4 Special Distributions (Case Study 4.2.1 continued) P(X â¥ 8) = 1 â P(X â¤ 7) = Ë 1â 7 eâ1.75 (1.75)k k=0 k! = 1 â 0.99953 = 0.00047 How close can we expect 0.00047 to be to the âtrueâ 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 conï¬dent 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 difï¬culty 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 ï¬nd quantitative methods capable of detecting clustering in space and time for diseases having low epidemicity. Several techniques were ultimately put forth, but the inherent ânoiseâ in the dataâvariations in population densities, ethnicities, risk factors, and medical practicesâoften 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 waysâ ï¬rst, 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 ï¬rst annual âI was Hit by Lightingâ Club. Approximate the probability that at most one of the ï¬ve hundred was born on Poissonâs 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 âone in ten thousandâ estimate is too low? [Hint: Calculate P(X â¥ k) = 1 â P(X â¤ k â 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?