ECN 422
PS 3

Due in class on Oct 17th

Binomial Distribution

(Question 1) NBC News reported on May 2, 2013, that 1 in 20 children in the United States have a food allergy of some sort. Consider selecting a random sample of 25 children and let X be the number in the sample who have a food allergy. Then X ~ Bin(25, .05).

a) Determine both P (X ≤ 3) and P(X < 3).

b) Determine P (X ≥ 4).

c) Determine P (1 ≤ X ≤ 3).

d) What are E [X] and V ar [X]

(Question 2) A particular type of tennis racket comes in a midsizd version and an oversize version. Sixty percent of all customers at a certain store want the oversize version.

a) Among ten randomly selected customers who want this type of racket, what is the probability that at least six want the oversize version?

b) Among ten randomly selected customers, what is the probability that the number who want the oversize version is within 1 standard deviation of the mean value?

c) The store currently has seven rackets of each version. What is the prob-ability that all of the next ten customers who want this racket can get the version they want from current stock?

(Question 3) The College Board reports that 2% of the 2 million high school students who take the SAT each year receive special accommodations be-cause of documented disabilities (Los Angeles Times, July 16, 2002). Con-sider a random sample of 25 students who have recently taken the test.

a) What is the probability that exactly 1 received a special accommodation?

b) What is the probability that at least 1 received a special accommodation?

c) What is the probability that at least 2 received a special accommodation?

d) What is the probability that the number among the 25 who received a special accommodation is within 2 standard deviations of the number you would expect to be accommodated?

e) Suppose that a student who does not receive a special accommodation is allowed 3 hours for the exam, whereas an accommodated student is allowed 4.5 hours. What would you expect the average time allowed the 25 selected students to be?

Hypergeometric Distribution and Poisson Distribution

(Question 4) Eighteen individuals are scheduled to take a driving test at a particular DMV office on a certain day, eight of whom will be taking the test for the first time. Suppose that six of these individuals are randomly assigned to a particular examiner, and let X be the number among the six who are taking the test for the first time.

a) What kind of a distribution does X have (name and values of all para-meters)?

b) Compute P (X = 2); P (X ≤ 2); and P (X ≥ 2):

c) Calculate the mean value and standard deviation of X.

(Question 5) Suppose small aircraft arrive at a certain airport according to a Poisson process with rate a 5 8 per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter μ = 8t.

a) What is the probability that exactly 6 small aircraft arrive during a 1-hour period? At least 6? At least 10?

b) What are the expected value and standard deviation of the number of small aircraft that arrive during a 90-min period?

c) What is the probability that at least 20 small aircraft arrive during a 2.5-hour period? That at most 10 arrive during this period?