A word of caution is in order. The normal curve is a good approximation to the binomial distribution when n 3 p is greater than 10 and n 3 (1 2 p) is greater than 10. If either of these conditions does not hold, the binomial distri- bution should be used. In addition, the normal curve is continuous: It tells you the probability that r or fewer events occurred, or that r or more events occurred.
It does not tell you the probability of exactly r events occurring. Th is probability can be determined only by using the (discrete) binomial distribution.
Chapter Summary
Th e binomial probability distribution can be used whenever the outcome of any trial can be classifi ed into one of two mutually exclusive and jointly exhaustive outcomes, one outcome is termed a success and the other a failure and whenever each trial is independent of the other trials. To determine a probability by using the binomial probability distribution, you must know the number of trials (n), the number of successes (r), the probability of a success in any one trial (p), and the probability of failure (q 5 1 2 p). Th e probability of an event for the binomial probability distribution can be calculated by using the formula
Crnprq1n2r2
Problems
9.1 Th e state legislative council has been charged with sex discrimination in hiring.
Last year it hired only 3 women out of 12 new employees. Th e civil service lists show that women comprise 40% of the qualifi ed applicants for these jobs. What is the probability of hiring three or fewer women if the legislative council does not discriminate? Show all calculations.
9.2 Th e Armenian Navy gives all persons who want to join the navy an intelligence test. Past tests had a mean of 90 and a standard deviation of 10.
(a) What percentage of applicants scored between 82 and 104?
(b) What percentage of applicants scored below 75?
(c) What is the probability that a person taking the test will score 109 or more?
9.3 The foreign service exam gives a passing grade to only the top 10% of those taking the exam. Th e mean score is 84, with a standard deviation of 8. What should be the minimum passing grade?
9.4 In a grand jury case, a bookstore was indicted in Oklahoma County on several counts of selling an obscene book. Th e grand jury was composed of 22 Baptists and 8 other people. The defendant feels that Baptists are biased against free speech. What is the probability that 22 or more Baptists are selected on a jury if Oklahoma County is 40% Baptist?
9.5 Mary Doyle ran against Bernie Hobson for state senate. In one precinct, one of the two voting machines did not work, and 182 votes were cast on the broken machine. Mary needed to receive 177 of the 182 votes to win the election. On the other machine in the precinct, Mary received 51% of the votes. Assignment where Crn is the number of combinations of n things taken r at a time. Th e for- mula for a combination is
Crn5 n!
r!1n2r2!
In circumstances in which n 3 p . 10 and n 3 (1 2 p) . 10, the normal distribution can be used to approximate the binomial probability distribution.
As explained in Chapter 8, the z score is used to fi nd probabilities in the normal distribution. Th e formula for the z score is
z5 X2 m s
where X is a score of interest (here, the number of successes), m is the mean or expected value, and s is the standard deviation.
For the binomial probability distribution, the mean (m) is equal to n 3 p, and the standard deviation (s) is equal to the square root of n 3 p 3 (1 2 p). In the formula, X is the number of successes (r).
of voters to ballot boxes is independent. What is the probability that Mary won the election?
9.6 Traditionally, one-half of all cities that apply get job training grants. Four southern cities apply for grants, but none receive them. Assume all the cities were equally qualifi ed. What is the probability that no southern city gets a grant?
9.7 Seaman David Brady is one of 16 seamen in Petty Offi cer Rickels’s unit. Every day four seamen are assigned to chip paint, and the others are assigned to screen movies to see if they are suitable for viewing. Seaman Brady believes that Rickels does not like him because he has been assigned to the paint detail 16 times in the past 20 days. What can you tell Seaman Brady?
9.8 Th e Procurement Bureau runs tests on 30 brand X teletype machines. It fi nds that an average of three machines fail in any 1-day period. Against the bureau’s advice, the Public Aff airs Department purchased 140 of these machines for all state offi ces. On the fi rst day, 26 machines fail. What is the probability that this would happen if the true failure rate were 10%?
9.9 Th is year, 620 persons are nominated to participate in the president’s Manage- ment Internship Program. After screening, 212 are selected for the program.
Maxwell George University nominates fi ve persons, and all fi ve receive awards.
What is the probability that this event would occur if the events were indepen- dent and all nominees are equally qualifi ed?
9.10 Refer to Problem 9.9. Of the 620 nominees, 211 are women; and of these, 91 are selected. Is there any preference with regard to sex?
9.11 Past experience has shown that 60% of all captains are promoted to major. Th e 819th Infantry Division has 48 captains who are eligible for promotion. Nine of these captains are West Point graduates. Eight of the nine West Pointers are promoted. Is there any reason to suspect that West Point graduates are given preferential treatment? Why? Show your work.
9.12 If one-third of the University of Wisconsin teaching assistants (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?
9.13 The area supervisor of the Occupational Safety and Health Administra- tion (OSHA) has heard a story that a certain inspector is not enforcing safety regulations. Past statistics reveal that inspectors fi nd safety violations in 91% of all inspections. Th e supervisor pulls the fi les for the eight most recent inspec- tions for the inspector whose behavior is questioned. Th ese fi les reveal that safety violations were cited in two of the eight cases. Analyze these data and present a statistical conclusion.
9.14 Th e BFOQ Job Training Corporation believes that it has a new program that will increase job placements. Essentially, the corporation thinks that if it runs its trainees through a simulated interview before sending them out on a job
interview, their likelihood of getting the job increases. BFOQ randomly selects 36 individuals of relatively equal skills on a matched-pair basis. Eighteen of these individuals are run through the simulation. BFOQ then sends one pair, one per- son who went through the training and one person who did not, to interview for one of 18 diff erent jobs. (Th ese individuals are the only ones to interview for the jobs.) Th irteen of the eighteen persons who went through the simulation get the jobs. What is the probability that 13 of the 18 would get jobs if there were no diff erence between the two sets of 18 persons? Show your work.
9.15 Th e Bureau of Paperwork wants to know whether agency personnel prefer to use health maintenance organizations (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 all bureau employees favor use of HMOs, what is the probability of obtaining a sample in which 29 out of 100 personnel are in favor?]
9.16 Th e Bluefi eld Regional Employment Service needs to place fi ve individuals in jobs this week to meet its yearly quota. During the past several years, the service’s track record is that every person sent to interview for a job has a .6 probability of getting the job. Th e placements appear to be independent of each other. Th e service decides to send seven individuals for interviews this week. Based on what you know, what is the probability that the service will make its yearly quota this week?
9.17 Th e Department of the Treasury is concerned because one member of Congress has charged that by normal accounting standards, one-third of all savings and loans (S&Ls) in the country are bankrupt. To refute this claim, chief economist Tom Holbrook takes a sample of six savings and loans and fi nds that only one is insolvent (his staff had intended to gather a larger sample, but they were too busy processing S&L failures). If the true proportion of insolvencies is one-third, what is the probability that the Treasury Department would get the results that it did?