At a medium-sized airport, mechanics need to replace tires on some of the airplanes each week because the tread on the tires is below the safe limit. You collected 36 weeks of data and observed that the numbers followed a Poisson distribution and that there was an average of 2 tires replaced per week. 1) What is the lambda value for the Poisson distribution? Select one: a. 0 b. 1/2 c. 1 d. 2 e. √(1/2) 2) The time between incoming customer service calls to a computer-repair hotline follows an exponential distribution with an expectation of 2 minutes between calls. What is the probability that the time between calls will be less than 1 minute for a randomly selected period? Select one: a. .44 b. .31 c. .39 d. .61 e. .07 3) The time between customer service calls to a computer-repair hotline follows an exponential distribution with an expectation of 2 minutes between calls. We collect calls for 40 minutes at random times during the month (a sample of size n=40). What is the probability that the mean of our sampling distribution will be greater than 2.2 minutes? Select one: a. .17 b. .26 c. .37 d. .68 e. .78
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In one town, the number of burglaries in a week has a Poisson distribution with parameter λ = 3.1. Find the probability that in a randomly selected week the number of burglaries is at least three. A) 0.401 B) 0.224 C) 0.599 D) 0.375 The number of calls received by a car towing service in an hour has a Poisson distribution with parameter λ = 2.49. Let X denote the number of calls received by the service in a randomly selected hour. Find the mean of X. A) 6.2 B) 1.578 C) 2.49 D) 1.245 The number of calls received by a car towing service in an hour has a Poisson distribution with parameter λ = 2.100. Let X denote the number of calls received by the service in a randomly selected hour. Find the standard deviation of X. A) 1.050 B) 2.100 C) 4.410 D) 1.449
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The number of calls to a customer service center is Poisson with an average of 5.7 calls per 10 minutes. a) What is the standard deviation of the number of calls in 10 minutes? Round to two decimal places. b) What is the probability of more than 3 but less than 9 calls in 10 minutes? Round to four decimal places. c) What is the expected number of calls in one hour? Round to one decimal place. d) What is the probability of less than 35 calls in one hour? Round to four decimal places. 2. Scores on a standardized math test are normally distributed with a mean of 500 and a standard deviation of 105. a) What is the probability that a score on the test is greater than 390? Round your answer to four decimal places. b) What is the probability that a score on the test is between 280 and 440? Round your answer to four decimal places. c) What is the probability that a score on the test is less than 790? Round your answer to four decimal places. d) What is the test score such that 29% of all test scores are smaller? Round your answer to two decimal places. e) What test score is smaller than 31% of all test scores? Round your answer to two decimal places.
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Another important discrete probability distribution is the Poisson distribution, named in honor of the French mathematician and physicist Simeon Poisson (1781-1840). This probability distribution is often used to model the frequency with which a specified event occurs during a particular period of time. The Poisson probability formula is $$P(X=x)=e^{-\lambda} \frac{\lambda^{x}}{x !},$$ where $X$ is the number of times the event occurs and $\lambda$ is a parameter equal to the mean of $X .$ The number $e$ is the base of natural logarithms and is approximately equal to 2.7183. To illustrate, consider the following problem: Desert Samaritan Hospital, located in Mesa, Arizona, keeps records of emergency room traffic. Those records reveal that the number of patients who arrive between 6: 00 P.M. and 7: 00 P.M. has a Poisson distribution with parameter $\lambda=6.9 .$ Determine the probability that, on a given day, the number of patients who arrive at the emergency room between 6: 00 PM. and 7: 00 PM. will be a. exactly 4. b. at most 2. c. between 4 and $10,$ inclusive.
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