Exam 5: Several Useful Discrete Distributions

Toll Station Narrative It is known that between 8 and 10 a.m. on Saturdays, cars arrive at a certain toll station at a rate of 60 per hour. Assume that a Poisson process is occurring and that the random variable x represents the number of cars arriving at the station between 9:00 and 9:05 a.m. -Refer to Toll Station Narrative. Find P(x = 2).

If x is a binomial random variable with n = 20, and p = 0.5, then P(x = 20) = 1.0.

If the random variable x is binomially distributed with n = 10 and p = 0.05, what is P(x = 2)?

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. Find P(x = 2).

Scholarship Narrative A college has seven applicants for three scholarships: four females and three males. Suppose the seven applicants are equally qualified and no preference is given by the selection committee for choosing either gender. Let x equal the number of female students chosen for the three scholarships. -Refer to Scholarship Narrative. What is the variance of the distribution of x?

A telephone survey of Canadian families is conducted to determine the number of children in the average Canadian family. Past experience has shown that 30% of the families who are telephoned will refuse to respond to the survey. Which of the following data could NOT be classified as a binomial random variable?

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 between four and eight students, inclusively, arrive in a 15-minute period?

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 local Ford dealership claims 40% of its customers are return buyers, meaning that the customers have previously purchased a Ford vehicle. Let the random variable x be the number of return buyers in a random sample of 10 recent customers. a. Find P(1 < x < 5). b. Find P(x > 6). c. What would you conclude about the accuracy of the dealership's claim if you found nine repeat buyers in the sample? Explain your reasoning.

What is the expected number of heads in 200 tosses of an unbiased coin?

From past experience, it is known 90% of one-year-old children can distinguish their mother's voice from the voice of a similar-sounding female. A random sample of 20 one-year-olds are given this voice recognition test. a. Find the probability that at least three children do not recognize their mother's voice. b. Find the probability that all 20 children recognize their mother's voice. c. Let the random variable x denote the number of children who do not recognize their mother's voice. Find the mean of x. d. Let the random variable x denote the number of children who do not recognize their mother's voice. Find the variance of x. e. Find the probability that, at most, four children do not recognize their mother's voice.

A warehouse contains six parts made by company A and ten parts made by company B. If four parts are selected at random from the warehouse, the probability that one of the four parts is from company B is approximately 0.1099.

A politician claims 55% of the voters will vote for her in an upcoming election. An independent watchdog organization queried a random sample of 500 likely voters, of whom only 249 said they would vote for the politician in question. Do the results of this sample support the politician's claim? [Hint: Binomial distributions with p near 50% are quite mound-shaped.]

Telephone Switchboard Narrative The number of telephone calls coming into a business's switchboard averages four calls per minute. Let x be the number of calls received. -Refer to Telephone Switchboard Narrative. What is the probability there will be at least one call in a given one-minute 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. -It was estimated that 2% of a particular 1997 model minivan had incorrectly installed brake lines. Suppose 300 minivans of this model are selected at random. Let x represent the number of minivans with incorrectly installed brake lines. What is the probability that nine have incorrectly installed brake lines?

Scholarship Narrative A college has seven applicants for three scholarships: four females and three males. Suppose the seven applicants are equally qualified and no preference is given by the selection committee for choosing either gender. Let x equal the number of female students chosen for the three scholarships. -Refer to Scholarship Narrative. What is the probability that none of the four females will receive a scholarship?

Supporters of gun control in a university town claim that 60% of the students are in favour of stronger gun control in Canada. A social scientist at the university conducts a survey of 20 randomly chosen students and finds that 9 of the 20 favour stronger gun control. Given this information, which of the following is a reasonable conclusion?

Where n = 100 is the number of trials and p = 0.04 is the probability of success in each trial, the mean of a binomial experiment is .

Which of the following correctly describes a Poisson random variable?

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. What are the mean, variance, and standard deviation of this distribution?