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# Mathematical Statistics with Applications, Seventh Edition

www.downloadslide.com Exercises 129 3.104 Twenty identical looking packets of white power are such that 15 contain cocaine and 5 do not. Four packets were randomly selected, and the contents were tested and found to contain cocaine. Two additional packets were selected from the remainder and sold by undercover police officers to a single buyer. What is the probability that the 6 packets randomly selected are such that the first 4 all contain cocaine and the 2 sold to the buyer do not? 3.105 In southern California, a growing number of individuals pursuing teaching credentials are choosing paid internships over traditional student teaching programs. A group of eight candidates for three local teaching positions consisted of five who had enrolled in paid internships and three who enrolled in traditional student teaching programs. All eight candidates appear to be equally qualified, so three are randomly selected to fill the open positions. Let Y be the number of internship trained candidates who are hired. a Does Y have a binomial or hypergeometric distribution? Why? b Find the probability that two or more internship trained candidates are hired. c What are the mean and standard deviation of Y ? 3.106 Refer to Exercise 3.103. The company repairs the defective ones at a cost of \$50 each. Find the mean and variance of the total repair cost. 3.107 Seed are often treated with fungicides to protect them in poor draining, wet environments. A small-scale trial, involving five treated and five untreated seeds, was conducted prior to a large-scale experiment to explore how much fungicide to apply. The seeds were planted in wet soil, and the number of emerging plants were counted. If the solution was not effective and four plants actually sprouted, what is the probability that a b c all four plants emerged from treated seeds? three or fewer emerged from treated seeds? at least one emerged from untreated seeds? 3.108 A shipment of 20 cameras includes 3 that are defective. What is the minimum number of cameras that must be selected if we require that P(at least 1 defective) ≥ .8? 3.109 A group of six software packages available to solve a linear programming problem has been ranked from 1 to 6 (best to worst). An engineering firm, unaware of the rankings, randomly selected and then purchased two of the packages. Let Y denote the number of packages purchased by the firm that are ranked 3, 4, 5, or 6. Give the probability distribution for Y. 3.110 A corporation is sampling without replacement for n = 3 firms to determine the one from which to purchase certain supplies. The sample is to be selected from a pool of six firms, of which four are local and two are not local. Let Y denote the number of nonlocal firms among the three selected. a P(Y = 1). b P(Y ≥ 1). c P(Y ≤ 1). 3.111 Specifications call for a thermistor to test out at between 9000 and 10,000 ohms at 25 ◦ Celcius. Ten thermistors are available, and three of these are to be selected for use. Let Y denote the number among the three that do not conform to specifications. Find the probability distributions for Y (in tabular form) under the following conditions: a b Two thermistors do not conform to specifications among the ten that are available. Four thermistors do not conform to specifications among the ten that are available.

www.downloadslide.com 130 Chapter 3 Discrete Random Variables and Their Probability Distributions 3.112 Used photocopy machines are returned to the supplier, cleaned, and then sent back out on lease agreements. Major repairs are not made, however, and as a result, some customers receive malfunctioning machines. Among eight used photocopiers available today, three are malfunctioning. A customer wants to lease four machines immediately. To meet the customer’s deadline, four of the eight machines are randomly selected and, without further checking, shipped to the customer. What is the probability that the customer receives a b no malfunctioning machines? at least one malfunctioning machine? 3.113 A jury of 6 persons was selected from a group of 20 potential jurors, of whom 8 were African American and 12 were white. The jury was supposedly randomly selected, but it contained only 1 African American member. Do you have any reason to doubt the randomness of the selection? 3.114 Refer to Exercise 3.113. If the selection process were really random, what would be the mean and variance of the number of African American members selected for the jury? 3.115 Suppose that a radio contains six transistors, two of which are defective. Three transistors are selected at random, removed from the radio, and inspected. Let Y equal the number of defectives observed, where Y = 0, 1, or 2. Find the probability distribution for Y . Express your results graphically as a probability histogram. 3.116 Simulate the experiment described in Exercise 3.115 by marking six marbles or coins so that two represent defectives and four represent nondefectives. Place the marbles in a hat, mix, draw three, and record Y , the number of defectives observed. Replace the marbles and repeat the process until n = 100 observations of Y have been recorded. Construct a relative frequency histogram for this sample and compare it with the population probability distribution (Exercise 3.115). 3.117 In an assembly-line production of industrial robots, gearbox assemblies can be installed in one minute each if holes have been properly drilled in the boxes and in ten minutes if the holes must be redrilled. Twenty gearboxes are in stock, 2 with improperly drilled holes. Five gearboxes must be selected from the 20 that are available for installation in the next five robots. a b Find the probability that all 5 gearboxes will fit properly. Find the mean, variance, and standard deviation of the time it takes to install these 5 gearboxes. 3.118 Five cards are dealt at random and without replacement from a standard deck of 52 cards. What is the probability that the hand contains all 4 aces if it is known that it contains at least 3 aces? 3.119 Cards are dealt at random and without replacement from a standard 52 card deck. What is the probability that the second king is dealt on the fifth card? *3.120 The sizes of animal populations are often estimated by using a capture–tag–recapture method. In this method k animals are captured, tagged, and then released into the population. Some time later n animals are captured, and Y , the number of tagged animals among the n, is noted. The probabilities associated with Y are a function of N, the number of animals in the population, so the observed value of Y contains information on this unknown N. Suppose that k = 4 animals are tagged and then released. A sample of n = 3 animals is then selected at random from the same population. Find P(Y = 1) as a function of N. What value of N will maximize P(Y = 1)?

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