Exam 5: Several Useful Discrete Distributions

The number of defects in a random sample of 200 parts produced by a machine is binomially distributed with p = 0.03. Based on this information, the standard deviation of the number of defects in the sample is 5.82.

Health Care Narrative Students arrive at a health centre, according to a Poisson distribution, at a rate of four every 15 minutes. Let x represent number of students arriving in a 15-minute period. -Refer to Health Care Narrative. What is the probability that no more than three students arrive during a 15-minute period?

Automobile Service Centre Narrative An automobile service centre can take care of eight cars per hour. Assume that the cars arrive at the service centre randomly and independently of each other at a rate of six per hour, on average. -Refer to Automobile Service Centre Narrative. What is the standard deviation of the number of cars that arrive at the centre?

There are only two possible outcomes in a binomial experiment where n is the number of trials and p is the probability of success in each trial.

Which of the following scenarios is an example of a binomial experiment?

Intensive Care Unit Narrative 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. -Refer to Intensive Care Unit Narrative. What is the probability that the number of people entering the intensive care unit on a particular day is two?

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.

As a rule of thumb, if the sample size n is large relative to the population size N, in particular, if n/N > 0.05, then the resulting experiment will NOT be binomial.

Rebuilt ignition systems leave an aircraft repair shop at an average rate of three per hour. The assembly line needs four ignition systems in the next hour. What is the probability they will be available?

Job Applicants Narrative A company has five applicants, three women and two men, for two positions. Suppose that the five applicants are equally qualified and that no preference is given for choosing either gender. Let x equal the number of men chosen to fill the two positions. -A student has decided to rent three movies for a three-day weekend. If there are four action movies and six romance movies that are of equal interest to the student, what is the probability that the student will select one romance movie and two action movies?

A salesperson has found the probability of making a sale on a particular product manufactured by his company is 0.05. If the salesperson contacts 140 potential customers, what is the probability he will sell at least 2 of these products?

Defective Items Narrative A random sample of 4 units is taken from a group of 15 items in which 4 units are known to be defective. Assume that sampling occurs without replacement, and the random variable x represents the number of defective units found in the sample. -Refer to Defective Items Narrative. What is P(x = 0)?

The number of traffic accidents per day on a certain section of highway is thought to be Poisson distributed with a mean to equal 2.19. Which of the following best approximates the probability of no accidents occurring on this section of highway during a one-day period?

Bicycle Repair Shop Narrative The number of people arriving at a bicycle repair shop follows a Poisson distribution, with an average of five arrivals per hour. Let x represent number of people arriving per hour. -Refer to Bicycle Repair Shop Narrative. What is the probability that more than seven people arrive at the bike repair shop in a one-hour period?

In an effort to persuade customers to reduce their electricity consumption during business hours, special discounts have been offered to low-consumption customers. One hydroelectric company estimates that 70% of the local residences have qualified for these discounts. a. If we randomly select four residences, what is the probability exactly two of the four will have qualified for the discounts? b. If, instead, a sample of 20 residences is selected, what is the probability 12 or more will have qualified for the discounts?

To determine the effectiveness of a diet to reduce the amount of serum cholesterol, 20 people were randomly selected to try the diet. After they had been on the diet a sufficient amount of time, their blood was tested to determine the level of serum cholesterol. The clinic running the experiment will approve and endorse the diet only if at least 16 of the 20 people lower their cholesterol significantly. a. What is the random variable of interest? b. What is the probability the clinic will erroneously endorse the diet when, in fact, it is useless in treating high serum cholesterol? [Hint: If the diet is ineffective, a person's cholesterol level is equally likely to go up or down.]

The Scholastic Aptitude Test (SAT) is a standardized test for college admissions in the United States. In 1996, the average combined SAT score (math + verbal) for students in the United States was 1013, and 41% of all high school graduates took this test. Suppose that 500 students are randomly selected throughout the United States. Which of the following random variables has an approximate binomial distribution? In each case, provide a justification for your answer. a. the number of students who took the SAT b. the scores of the 500 students on the SAT c. the number of students who scored above average on the SAT d. the amount of time it took each student to complete the SAT

About 80% of Victoria residents believe that Victoria is a nice or very nice place to live. Suppose that five randomly selected Victoria residents are interviewed. a. Find the probability that all five residents think that Victoria is a nice or very nice place to live. b. What is the probability that at least one resident does not think that Victoria is a nice or very nice place to live? c. What is the probability that exactly one resident does not think that Victoria is a nice or very nice place to live?

Hypergeometric probability distributions are examples of continuous probability distributions.

Job Applicants Narrative A company has five applicants, three women and two men, for two positions. Suppose that the five applicants are equally qualified and that no preference is given for choosing either gender. Let x equal the number of men chosen to fill the two positions. -Refer to Job Applicants Narrative. Construct a probability histogram for x.