USES AND ABUSES Statistics in the Real World Uses and Abuses 223 EXERCISES In Exercises 1–3, assume the fire department guidelines are correct and that the department responds to an average of 11 incidents per day. Use the graph of the Poisson distribution and technology to answer the questions. Explain your reasoning. 1. On a random day, what is more likely, 11 incidents or at least 16 incidents? 2. On a random day, what is more likely, 10 to 12 incidents or fewer than 10 incidents? 3. On the 4th of July, the fire department responds to 17 incidents. Is there reason to believe the guidelines should be adjusted for this holiday? Uses There are countless occurrences of Poisson probability distributions in business, sociology, computer science, and many other fields. For instance, suppose you work for the fire department in the city of Erie, Pennsylvania. You have to make sure the department has enough personnel and vehicles on hand to respond to fires, medical emergencies, and other situations where they provide aid. The fire department’s records show that it responds to an average of 11 incidents per day, but one day the department responds to 15 incidents. Is this an unusual event? If so, the department may need to update the guidelines so that it is prepared to respond to more incidents. Knowing the characteristics of the Poisson distribution will help you answer this type of question. By the time you have completed this course, you will be able make educated decisions about the reasonableness of the fire department’s guidelines. Abuses A common misuse of the Poisson distribution is to think that the “most likely” outcome is the outcome that will occur most of the time. For instance, suppose you are planning a typical day of responding to emergencies for the fire department. The most likely number of incidents the department will need to respond to is 11. Although this is the most likely outcome, the probability that it will occur is only about 0.119. There is about a 0.202 chance the department will respond to 12 or 13 incidents, and about a 0.219 chance of 14 or more incidents. So, it would be a mistake to simply plan for 11 incidents every day, thinking that days with fewer incidents and days with more incidents will balance out over time. Citizens’ safety and even lives can depend on the fire department, so it is important to be ready for any likely scenario. So, the fire department should be ready to respond to any number of incidents for which the cumulative probability of that many or more incidents is greater than 0.05. Number of incidents Probability x 1 3 5 7 9 111315171921 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 P(x)
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