Exam 5: Several Useful Discrete Distributions

The number x of people entering the intensive care unit at a particular hospital on any one day has a Poisson probability distribution with mean equal to four persons per day. What is the probability that the number of people entering the intensive care unit on a particular day is two? ______________ What is the probability that the number of people entering the intensive care unit on a particular day is Less than or equal to two? ______________ Is it likely that x will exceed ten? ______________ Explain. ________________________________________________________

Which of the following statements is true with respect to the binomial distribution?

The hypergeometric probability distribution is identical to:

The probability distribution of a Poisson random variable provides a good model for data that represent the number of occurrences of a specified event in a given unit of time or space.

A professor has received a grant to travel to an archaeological dig site. The grant includes funding for five graduate students. If there are five male and four female graduate students eligible and equally qualified, what is the probability that the professor will select three male and two female graduate students to accompany her to the dig site? ______________

The expected number of heads in 200 tosses of an unbiased coin is:

A psychiatrist believes that 90% of all people who visit doctors have problems of a psychosomatic nature. She decides to select 20 patients at random to test her theory. Assuming that the psychiatrist's theory is true, what is the expected value of x, the number of the 20 patients who have psychosomatic problems? ______________ What is the variance of x, assuming that the theory is true? ______________ Find , assuming that the theory is true. ______________ Based on the probability in part (c), if only 14 of the 20 sampled had psychosomatic problems, what conclusions would you make about the psychiatrist's theory? ______________ Explain. ________________________________________________________

In a binomial experiment, the probability of success is the same on every trial.

In a Poisson problem, x represents the number of events occurring in a period of time or space during which an average of such events can be expected to occur.

The taste test for PTC (phenylthiourea) is a common class demonstration in the study of genetics. It is known 70% of the American people are "tasters" and 30% are "non-tasters." Suppose a genetics class of size 20 does the test to see if they match the U.S. percentage of "tasters" and "non-tasters." (Assume the assignment of students to classes constitutes a random process.) For the random variable x, the number of "non-tasters" in the class, Find P(3 < x < 9). ______________ Find the mean of x. ______________ Find the variance of x. ______________

A coin toss experiment represents a binomial experiment only if the coin is balanced, i.e., p = 0.5.

Hypergeometric probability distributions is an example of continuous probability distribution.

The National Rifle Association chapter in a university town claims 60% of the students at the university favor gun control. A social scientist at the university conducts a survey of 20 randomly chosen students and finds 9 who favor gun control. Which of the following is a reasonable conclusion?

A binomial experiment is a sequence of n identical trials such that each trial (1) produces one of two outcomes that are conventionally called success and failure and (2) is independent of any other trial so that the probability of success or failure is constant from trial to trial.

A manufacturer of tennis balls uses a production process that produces 5 percent defective balls. A quality inspector takes samples of a week's output with replacement. Using the cumulative binomial probability table available in your text, the inspector can determine which of the following probabilities?

The number of people arriving at a bicycle repair shop follows a Poisson distribution with an average of 5 arrivals per hour. Let x represent number of people arriving per hour. What is the probability that seven people arrive at the bike repair shop in a one hour period of time? ______________ What is the probability that at most seven people arrive at the bike repair shop in a one hour period of time? ______________ What is the probability that more than seven people arrive at the bike repair shop in a one hour period of time? ______________ What is the probability that between 4 and 9 people, inclusively, arrive at the bike repair shop in a one hour period of time? ______________

It is known that 70% of the customers in a sporting goods store purchase a pair of running shoes. A random sample of 25 customers is selected. Assume that customers' purchases are made independently, and let x represent number of customers who purchase running shoes. What is the probability that exactly 18 customers purchase running shoes? ______________ What is the probability that no more than 19 customers purchase running shoes? ______________ What is the probability that at least 17 customers purchase running shoes? ______________ What is the probability that between 17 and 21 customers, inclusively, purchase running shoes? ______________

The binomial experiment requires that the successes and failure probabilities be constant from one trial to the next and also that these two probabilities be equal to each other.

Which of the following cannot generate a Poisson distribution?

An American Medical Association study showed 20% of all Americans suffer from high blood pressure. Suppose we randomly sample ten Americans to determine the number in the sample who have high blood pressure. What is the probability at most two persons have high blood pressure? ______________ If, instead, we sampled 25 Americans, give a reason why the following statement is true: "The probability between one and nine individuals from a sample of 25 would have high blood pressure is at least 0.75."