Betsymccall.net Index



Stat 2470, Homework #2, Fall 2014Name ______________________________________Instructions: Show work or give calculator commands used to solve each problem. You may use Excel or other software for any graphs. Be sure to answer all parts of each problem as completely as possible, and attach work to this cover sheet with a staple.The table shown gives information on the type of coffee selected by someone purchasing a single cup at a particular airport kiosk. Consider randomly selecting such a purchaser.SmallMediumLargeRegular14%20%26%Decaf20%10%10%What is the probability that the individual purchased a small cup? A cup of decaf coffee?If we learn that the selected individual purchased a small cup, what now is the probability that he/she chose decaf coffee, and how would you interpret this probability?If we learn that the selected individual purchased decaf, what now is the probability that a small size was selected, and how does this compare to the corresponding probability of (a)?Seventy percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, 60% have an emergency locator, whereas 90% of the aircraft not discovered do not have such a locator. Suppose a light aircraft has disappeared.Build a table similar to the one in question #1 to describe the data. Assume that the data was collected from 1000 aircraft.If it has an emergency locator, what is the probability that it will not be discovered?If it does not have emergency locator, what is the probability that it will be discovered?Describe the scenarios in (b) and (c) using conditional probability notation. Be sure to say what your events are that are being described.For customers purchasing a refrigerator at a certain appliance store, let A be the event that the refrigerator was manufactured in the US, B be the event that the refrigerator had an icemaker, and C be the event that the customer purchased an extended warranty. The relevant probabilities are: PA=0.75, PAB=0.9, PBA'=0.8, PCA∩B=0.8, PCA∩B'=0.6, PCA'∩B=0.7, PCA'∩B'=0.3.Construct a tree diagram consisting of first-, second- and third-generation branches and place an event label and appropriate probability next to each pute P(A∩B∩C).Compute P(B∩C).Compute P(C).Compute PAB∩C, the probability of a US purchase given that an icemaker and extended warranty are also purchased.Seventy percent of all vehicles examined at a certain emissions inspection station pass the inspection. Assuming that successive vehicles pass or fail independently of one another, calculate the following probabilities.The probability that all of the next three vehicles fail.The probability that at least one of the next three vehicles fails.The probability that exactly ones of the next three vehicles passes.The probability that at most one of the next three vehicles passes.Given that at least one of the next three vehicles passes, what is the probability that all three pass?One percent of all individuals in a certain population are carriers of a particular disease. A diagnostic test for this disease has a 90% detection rate for carriers and a 5% detection rate for non-carriers. Suppose the test is applied independently to two different blood samples from the same randomly selected individual. [Hint: It may help to draw a tree diagram of the two tests.]What is the probability that both tests yield the same result?If both tests are positive, what is the probability that the selected individual is a carrier?Given three examples of Bernoulli random variables (other than those in the text), that you might encounter as an engineer or scientist?For each random variable defined here, describe the set of possible values for the variable, and state whether the variable is discrete.X is the number of unbroken eggs in a randomly chosen standard egg carton.Y is the number of students on a class list for a particular course who are absent on the first day of class.U is the number of times a golfer has to swing at a golf ball before hitting it.V is the length of a randomly selected rattlesnake.W is the amount of royalties earned from the same of a first edition of $10,000 textbooks.Z is the pH of a randomly chosen soil sample.T is the tension (psi) at which a randomly selected tennis racket has been strung.S is the total number of coins tosses required for three individuals to obtain a match.Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable Y as the number of ticketed passengers who actually show up for the flight. The probability mass function is shown in the table below.y4546474849505152535455p(y)0.050.100.120.140.250.170.060.050.030.020.01What is the probability that the flight will accommodate all ticketed passengers who show up?What is the probability that not all ticketed passengers who show up can be accommodated?If you are the first person on the standby list, what is the probability that you will be able to take the flight? What is this probability if you are the third person on the standby list?Two fair six-sided dice are tossed independently. Let M be the maximum of the two tosses, so M(1,5)=5, and M(3,3)=3, etc.What is the probability mass function of M? Create a table to display the distribution.Determine the cumulative distribution function of M and graph it.Benford’s law is used by the IRS to detect fraud in financial reporting. The law says that the probability of the random variable X representing the leading digit of numbers reported are given by px=log10x+1x, x=1,2,3,…,9.Without computing individual probabilities from this formula, show that it specifies a legitimate probability pass function. [Hint: calculate the sum of the probabilities by using log properties and summation properties.]Now compute the individual probabilities and compare to the corresponding discrete uniform distribution (if, in fact, all leading digits of numbers were genuinely random).Obtain the cumulative distribution function of X.What the result in part (c), what is the probability that the leading digit is at most 3? At least 5?A consumer organization that evaluates new automobiles customarily reports the number of major defects in each car examined. Let X denote the number of major defects in a randomly selected car of a certain type. The cumulative distribution function of X is as follows:Fx=0, x<00.06, 0≤x<10.19, 1≤x<20.39, 2≤x<30.67, 3≤x<40.92, 4≤x<50.97, 5≤x<61, 6≤xCalculate the following probabilities.p(2), that is P(X=2).PX>3P(2≤x≤5)P(2<x<5) ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download