You may have noticed that the Poisson probability tables give values only for l less than or equal to 20. Poisson tables for larger l values are not presented because the normal distribution table provides fairly accurate probabilities when l is greater than 20. Th e normal distribution of this magnitude has a mean of l and a standard deviation equal to the square root of l. For a l value of 30, there- fore, you would use the normal table with a mean of 30 and a standard deviation of 5.5 1!3055.52.
Problems
10.1 One hundred cities (60 are northern cities) apply for Comprehensive Employ- ment and Training Act (CETA) grants to train the unemployed. Fifty grants are awarded, 40 to northern cities. Is there any anti–Sun Belt bias?
10.2 Th e mean number of persons shot by the Metro City Police Department (MPD) in an hour is .042. What is the probability that the MPD will shoot no one in a 24-hour period? What is the probability that the MPD will shoot one person?
Two persons? More than two persons? Explain your work.
10.3 Police chief Lenny Lawnorder has heard through a normally reliable source that 30% of his offi cers are not using regulation handguns. To see whether this is true, Lenny randomly inspects 35 of his 200 offi cers. Seventeen have nonregulation handguns. If the true proportion is .3, what is the probability that this event will occur?
10.4 Th e mean number of major fi res in Flandreau, South Dakota, is .057 on any given day. What is the probability of no major fi res occurring in any given week? What is the probability of one major fi re? Two major fi res? Th ree or more major fi res?
10.5 In the First Army, 150 fi rst lieutenants were up for promotion to captain (35 of these are West Point graduates). Of these, 120 were promoted, including 24 West Pointers. Does the Army discriminate in promotions?
10.6 Th e mean number of teletype machines in the Bureau of Communications that break down is .625 in any given hour. If repairs take 8 hours, how many repair people should the Bureau of Communications staff ?
10.7 Congressman Lester Asperin has charged that 50% of all IRS agents would fail a simple test on tax law. IRS Commissioner John Tight believes no more than 15%
would fail. Of the 1,000 agents, 200 are randomly selected and given the test, and 60 fail. What is the probability that Asperin is correct? What is the probabil- ity that Tight is correct?
10.8 Th e mean number of murders reported in Metro City during the 8-hour grave- yard shift is .5 for any hour. It takes two people 2 hours to investigate a homicide call. How many offi cers should be assigned to the homicide squad during these hours?
10.9 Refer to Problem 9.10 (Chapter 9), where 91 of the 211 women nominated for the president’s Management Internship Program are selected and 212 of the 620 total nominees are accepted. Use what you have learned in this chapter to deter- mine the probability of this happening if all persons are equally qualifi ed. What can you say if you do not assume equal qualifi cations?
10.10 Th e mean number of contracts that consulting fi rms receive with the Department of Human Services is .4 for any given year. Ecosystems Inc., a heavy contributor to political candidates, receives three contracts in 1 year. What is the probability that this event would occur by chance? What is the probability that Ecosystems will receive six contracts in 2 years?
10.11 Th e Madison Fire Department’s fi les reveal that an average of six fi refi ghters a year suff er heart attacks while on duty. Th e personnel offi ce is concerned about this because it must budget funds for disabilities. Th ere are some concerns that the physical condition of fi refi ghters is getting worse. Th is year, nine fi refi ghters suff ered heart attacks. What is the probability that an event of this severity will occur? Given past history, what is the probability that no heart attacks will occur?
Exactly one? Exactly two? Th ree or fewer?
10.12 Th e mean number of water main breaks is .4 per hour. In a 10-hour night shift, four mains break. What is the probability of exactly four mains breaking? Of four or more breaking? Which probability tells you the most about the chance of this situation happening?
10.13 The Nuclear Regulatory Commission estimates that the probability that a Westinghouse X-27 nuclear plant’s warning system will fail in a year’s time is .1.
New York Power and Light operates six of the X-27 plants; last year one failed.
What is the probability that one or more plants’ warning systems will fail?
10.14 Th e mean number of fi res per hour in Smallsville, Utah, is .1042. What is the probability that Smallsville will experience no fi res on any given day? Five or more fi res on any given day?
10.15 Th e Bureau of Paperwork has 1,000 employees, and the personnel division wants to know whether agency personnel prefer to use HMOs for their health care benefi ts. If more than 25% favor using HMOs, then the bureau will begin to set up procedures for this type of health care. A sample of 100 of the bureau’s personnel reveals that 29 favor the use of HMOs. What is the probability that less than 25% of the bureau’s employees favor the use of HMOs? [Hint: If 25%
(mean) or less of bureau personnel favor use of HMOs, what is the probability of obtaining a sample in which 29 out of 100 are in favor?]
10.16 Th e University of Wisconsin employs 200 teaching assistants (TAs). If one-third of the TAs sign a petition calling for a collective bargaining election, an election will be held. A survey of 50 TAs indicates that 40% will sign the petition. What is the probability that a sample such as this could have occurred if one-third or fewer of the TAs in the population will sign such a petition?