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statistics for engineers and scientists

Probability And Statistics For Engineers 5th Edition Richard L Scheaffer, Madhuri Mulekar, James T McClave, Cecie Starr - Solutions

6.40 One-hour carbon monoxide concentrations in air samples from a large city are found to have an exponential distribution with a mean of 3.6 ppm(Zamurs, Air Pollution Control Association Journal, 34(6), 1984, p. 637).a Find the probability that a concentration will exceed 9 ppm.b A traffic
6.39 The breakdowns of an industrial robot follow a Poisson distribution with an average of C = 100 + 40X + 3X 20.5 breakdowns per 8-hour workday. The robot is placed in service at the beginning of the day.a Find the probability that it will not breakdown during the day.b Find the probability that
6.38 The dial-up connections from remote terminals come into a computing center at the rate of four per minute. The callers follow a Poisson distribution.If a call arrives at the beginning of a 1-minute period, find the probability that a second call will not arrive in the next 20 seconds.
6.37 The life lengths of automobile tires of a certain brand, under average driving conditions, are found to follow an exponential distribution with mean 30 (in thousands of miles). Find the probability that one of these tires bought today will last a Over 30,000 miles b Over 30,000 miles given
6.36 The inter-accident times (times between accidents)for all fatal accidents on scheduled American domestic passenger air flights, 1948–1961, were found to follow an exponential distribution with a mean of approximately 44 days (Pyke, J. Royal Statistical Society(B),27, 1968, p. 426).a If one
6.35 The length of time X to complete a certain key task in house construction is exponentially distributed random variable with a mean of 10 hours. The cost C of completing this task is related to square of the time to completion by the formula a Find the expected value and variance of C.b Would
6.34 Suppose customers arrive at a certain checkout counter at the rate of two every minute.a Find the mean and variance of the waiting time between successive customer arrivals.b If a clerk takes 3 minutes to serve the first customer arriving at the counter, what is the probability that at least
6.33 A pumping station operator observes that the demand for water at a certain hour of the day can be modeled as an exponential random variable with a mean of 100 cfs (cubic feet per second).a Find the probability that the demand will exceed 200 cfs on a randomly selected day.b What is the maximum
6.32 Refer to Exercise 6.31. Of the next 10 earthquakes to strike this region, find the probability that at least one will exceed 5.0 on the Richter scale.
6.31 The magnitudes of earthquakes recorded in a region of North America can be modeled by an exponential distribution with mean 2.4 as measured on the Richter scale. Find the probability that the next earthquake to strike this region will a Exceed 3.0 on the Richter scale b Fall between 2.0 and
6.30 Suppose Y has an exponential density function with mean . Show that P(Y a b |Y a) P(Y b). This is referred to as the “memoryless”property of the exponential distribution.
6.29 The cycle times for trucks hauling concrete to a highway construction site are uniformly distributed over the interval 50 to 70 minutes.a Find the expected value and variance for these cycle times.b How many trucks would you expect to have to assign to this job so that a truckload of concrete
6.28 Suppose three automobiles are used in a test of the type discussed in Exercise 6.27. Find the probability that exactly one of the three travels past the midpoint between a and b.
6.27 In tests of stopping distances for automobiles, those automobiles traveling at 30 miles per hour before the brakes are applied tend to travel distances that appear to be uniformly distributed between two points a andb. Find the probability that one of these automobiles a stops closer to a than
6.26 A customer’s arrival at a counter is uniformly distributed over a 30-minute period. Find the conditional probability that the customer arrived during the last 5 minutes of the 30-minute period given that there were no arrivals during the first 10 minutes of the period.
6.25 Arrivals of customers at a certain chekout counter follow a Poisson distribution. It is known that during a given 30-minute period one customer arrived at the counter. Find the probability that she arrived during the last 5 minutes of the 30-minute period.
6.24 According to Y. Zimmels (AIChE journal, 29(4), 1983, pp. 669–676), the sizes of particles used in sedimentation experiments often have uniform distributions. It is important to study both the mean and variance of particle sizes, since in sedimentation with mixtures of various-size particles
6.23 In the setting of Exercise 6.22 suppose the measurement errors are uniformly distributed from-0.02 to +0.05 microsecond.-0.05 +0.05 a Find the probability that a particular arrivaltime measurement will be in error by less than 0.01 microsecond.b Find the mean and variance of these measurement
6.22 In determining the range of an acoustic source by triangulation, the time at which the spherical wave front arrives at a receiving sensor must be measured accurately. According to an article by J. Perruzzi and E. Hilliard (Journal of the Acoustical Society of America, 75(1), 1984, pp.
6.21 The number of defective circuit boards among those coming out of a soldering machine follows a possion distribution. On a particular 8-hour workday, one defective board is found.a Find the probability that it was produced during the first hour of operation of that day.b Find the probability
6.20 Beginning at 12:00 midnight, a computer center is up for 1 hour and down for 2 hours on a regular cycle. A person who doesn’t know the schedule dials the center at a random time between 12:00 midnight and 5:00 AM. what is the probability that the center will be operating when he dials in?
6.19 A telephone call arrived at a switchboard at a random time within a 1-minute interval. The switchboard was fully busy for 15 seconds into this 1-minute period. Find the probability that the call arrived when the switchboard was not fully occupied.
6.18 A bomb is to be dropped along a mile-long line that stretches across a practice target. The target center is at the midpoint of the line: The target will be destroyed if the bomb falls within a tenth of a mile on either side of the center. Find the probability that the target is destroyed if
6.17 If a point is randomly located in an interval (a, b)and if X denotes its distance froma, then X will be assumed to have a uniform distribution over.A plant efficiency expert randomly picks a spot along a 500-foot assembly line from which to observe work habits. Find the probability that she is
6.16 Upon studying low bids for shipping contracts, a microcomputer manufacturing firm finds that intrastate contracts have low bids that are uniformly distributed between 20 and 25, in units of thousands of dollars.a Find the probability that the low bid on the next intrastate shipping contract is
6.15 Suppose X has a uniform distribution over the interval (a, b).a Find F(x).b Find for some point c between a and b.c If , find
6.14 A retail grocer has a daily demand X for a certain food sold by the pound, such that X (measured in hundreds of pounds) has probability density function(He cannot stock over 100 pounds.) The grocer wants to order 100k of food on a certain day. He buys the food at 6¢ per pound and sells it at
6.13 The pH of water samples from a specific lake is a random variable X with probability density function a Find E(X ) and V(X ).b Find an interval shorter than (5, 7) in which at least 3/4 of the pH measurements must lie.c Would you expect to see a pH measurement blow 5.5 very often? Why?
6.12 Weekly CPU time used by an accounting firm has probability density function (measured in hours)f (x) =L 332(x - 2)(6 - x) 2 … x … 6 0 elsewhere Y = 200X - 60 f (x) = e 2x 0 … x … 1 0 elsewhere f (x) =L 12 59 … x … 61 0 elsewhere a Find the expected value and variance of weekly CPU
6.11 Daily total solar radiation for a certain location in Florida in October has probability density function with measurements in hundreds of calories. Find the expected daily solar radiation for October.
6.10 The proportion of time X that an industrial robot is in operation during a 40-hour work week is a random variable with probability density function a Find E(X ) and V(X ).b For the robot under study, the profit Y for a week is given by Find E(Y ) and V(Y ).c Find an interval in which the
6.9 The temperature X at which a thermostatically controlled switch turns on has probability density function Find E(X ) and V(X ).
6.8 The proportion of impurities X in certain copper ore samples is a random variable having probability density function If four such samples are independently selected, find the probability that a Exactly one has a proportion of impurities exceeding 0.5.b At least one has a proportion of
6.7 The proportion of time, during a 40-hour workweek, that an industrial robot was in operation was measured for a large number of weeks, and the measurements can be modeled by the probability density function If X denotes the proportion of time this robot will be in operation during a coming
6.6 The “on” temperature of a thermostatically controlled switch for an air conditioning system is set at 60°F, but the actual temperature X at which the switch turns on is a random variable having probability density function a Find the probability that it takes a temperature in excess of
6.5 The pH, a measure of the acidity of water, is important in studies of acid rain. For a certain Florida lake, baseline measurements on acidity are made so any changes caused by acid rain can be noted.The pH of water samples from a lake is a random variable X with probability density function f
6.4 An accounting firm that does not have its own computing facilities rents time from a consulting company.The firm must plan its computing budget carefully and hence has studied the weekly use of CPU time quite thoroughly.The weekly use of CPU time approximately follows the probability density
6.3 The effectiveness of solar-energy heating units depends on the amount of radiation available from the sun. For a typical October, daily total solar radiation in Tampa, Florida, approximately follows the probability density function given below (units are hundreds of calories):a Find the
6.2 Suppose a random variable X has a probability density function given by f (x) = e kx (1 - x) 0 … x … 1 0 elsewhere a Find the value of k that makes this a probability density function.b Find P (0.4 X 1).c Find P (X 0.4 |X 0.8).d Find F (b) P (Xb) and sketch the graph of this function.
6.1 For each of the following situations, define an appropriate random variable and state whether it is continuous or discrete.a An environmental engineer is looking at 10 field plots to determine whether they contain a certain type of insect.b A quality-control technician samples a continuously
5.115 Four possible winning numbers for a lottery—AB-4536, NH-7812, SQ-7855, and ZY-3221—are given to you. You will win a prize if 1 of your numbers matches one of the winning numbers.You are told that one first prize of $100,000, two second prizes of $50,000 each, and 10 third prizes of $1,000
5.114 It is known that 5% of a population have disease A, which can be detected by a blood test.Suppose that N (a large number) people are to be tested. This can be done in two ways: Either(1) each person is tested separately or (2) the blood samples of k people are pooled together and analyzed.
5.113 A merchant stocks a certain perishable item. She knows that on any given day she will have a demand for either two, three, or four of these items with probabilities 0.1, 0.4, and 0.5, respectively. She buys the items for $1.00 each and sells them for $1.20 each.Any that are left at the end of
5.112 For simplicity, let us assume that there are two kinds of drivers. The safe drivers, which comprise 70% of the population, have a probability of 0.1 of causing an accident in a year. The rest of the population consists of accident makers, who have a probability of 0.5 of causing an accident
5.111 A lot of industrial products contains 40 defectives. Let Y be the number of defectives in a random sample of size 20. Find p(10) by using a The hypergeometric probability distribution.b The binomial probability distribution.c Is N large enough so that the binomial probability function is a
5.110 Show that the hypergeometric probability function approaches the binomial in the limit as and remains constant. That is, show that for constant and .
5.109 The manufacturer of a low-calorie dairy drink wishes to compare the taste appeal of a new formula(B) with that of the standard formula (A).Each of four judges is given three glasses in random order, two containing formula A and the other containing formula B. Each judge is asked to state
5.108 a Consider a binomial experiment for n 20, p 0.05. Use Table 2 of the Appendix to calculate the binomial probabilities for Y 0, 1, 2, 3, 4.b Calculate the same probabilities as in (a), using the Poisson approximation with np.Compare your results.
5.107 Suppose that 10% of a brand of microcomputers will fail before their guarantee has expired. If 1,000 computers are sold this month, find the expected value and variance of Y, the number that have not failed during the guarantee period.Within what limit would Y be expected to fall?[Hint: Use
5.106 The mean number of automobiles entering a mountain tunnel per 2-minute period is one. An excessive number of cars entering the tunnel during a brief period of time produces a hazardous situation.a Find the probability that the number of autos n entering the tunnel during a 2-minute period
5.105 Sixty percent of a population of customers is reputed to prefer Brand A toothpaste. If a group of consumers is interviewed, what is the probability the exactly five people must be interviewed before a consumer is encountered who prefers Brand A? At least five people?
5.104 The probability of a customer arrival at a grocery service counter in any 1-second interval is equal to 0.1. Assume that customers arrive in a random stream and hence that the arrival at any one second is independent of any other.a Find the probability that the fist arrival will occur during
5.103 Refer to Exercise 5.102. If the supply office also stocks three welding units that are not Brand A, find the probability that exactly one of these will be left immediately after the third Brand A unit is requested.
5.102 The supply office for a large construction firm has three welding units of Brand A in stock. If a welding unit is requested, the probability is 0.7 that the request will be for this particular brand.On a typical day, five requests for welding units come to the office. Find the probability
5.101 For any probability function p(y), if the sum is taken over all possible values y that the random variable in question can assume.Show that this is true for the following:a The binomial distribution b The geometric distribution c The Poisson distribution
5.100 Refer to Exercise 5.99. Find the probability that six cars must arrive at the intersection while the light is red to fill up the left-turn lane.
5.99 The probability that any one vehicle will turn left at a particular intersection is 0.2. The left-turn lane at this intersection has room for three vehicles. If five vehicles arrive at this intersection while the light is red, find the probability that the left-turn lane will hold all the
5.98 A certain type of bacteria cell divides at a constant rate over time. (That is, the probability that a cell will divide in a small interval of time t is approximately t.) Given that a population starts out at time zero with k cells of this type and that cell divisions are independent of one
5.97 A production line produces a variable number N of times each day. Suppose each item produced has the same probability p of not conforming to manufacturing standards. If N has a Poisson distribution with mean , then the number of nonconforming items in one day’s production Y has a Poisson
5.96 The number of vehicles passing a specified point on a highway averages 10 per minute.a Find the probability that at least 15 vehicles pass this point in the next minute.b Find the probability that at least 15 vehicles pass this point in the next 2 minutes.
5.95 In checking river water samples for bacteria, water is placed in a culture medium so that certain bacteria colonies can grow if those bacteria are present. The number of colonies per dish averages 12 for water samples from a certain river.a Find the probability that the next dish observed will
5.94 For a certain section of a pine forest, the number of diseased trees per acre Y has a Poisson distribution with mean 10. The diseased trees are sprayed with an insecticide at a cost of $3 per tree, plus a fixed overhead cost for equipment rental of $50. Letting C denote the total spraying
5.93 A quality-control engineer wants to study the alternative sampling plans n 5, a 1 and n 25, a 5. On a sheet of graph paper, construct the operating characteristic curves for both plans; use acceptance probabilities at p 0.05, p 0.10, p 0.20, p 0.30, and p 0.40 in each case.a
5.92 Refer to Exercise 5.91. Use Table 2 of the Appendix to construct the operating characteristic curve for a sampling plan with a n 10, a 0 b n 10, a 1 c n 10, a 2 For each plan, calculate P(lot acceptance) for p 0, 0.05, 0.1, 0.3, 0.5, and 1.0. Our intuition suggests that sampling
5.91 Sampling for defectives from large lots of a manufactured product yields a number of defectives Y that follows a binomial probability distribution.A sampling plan involves specifying the number of items to be included in a sample n and an acceptance numbera. The lot is accepted if Y a and
5.90 Suppose that the four engines of a commercial aircraft were arranged to operate independently and that the probability of in-flight failure of a single engine is 0.01. What is the probability that, on a given flight, a No failures are observed?b No more than one failure is observed?
5.89 The probability that a single radar set will detect an airplane is 0.9. If we have five radar sets, what is the probability that exactly four sets will detect the plane? At least one set? (Assume that the sets operate independently of each other.)
5.88 Use Table 2 of the Appendix to construct a probability histogram for the binomial probability distribution for n 20 and p 0.5. Note that almost all the probability falls in the interval 5 y 15.
5.87 Construct probability histograms for the binomial probability distribution for n 5 and p 0.1, 0.5, and 0.9. (Table 2 of the Appendix will reduce the amount of calculation.) Note the symmetry for p 0.5 and the direction of skewness for p 0.1 and 0.9.
5.86 Use the result of Exercise 5.85 to show that E(Y) = aE(X) + b and V(Y) = a2V(X)
5.85 If X is a random variable with moment-generating function M(t ), and Y is a function of X given by Y aX b, show that the moment-generating function for Y is etb M(at ).
5.84 Show that the moment-generating function for the Poisson random variable with mean is given by M(t ) = el(et-1)M(t ) = [pet + (1 - p)]n Use this result to derive the mean and variance for the Poisson distribution.
5.83 Show that the moment-generating function for the binomial random variable is given by Use this result to derive the mean and variance for the binomial distribution.
5.82 Find the moment-generating function for the Bernoulli random variable.
5.81 Two assembly lines (I and II) have the same rate of defectives in their production of voltage regulators.Five regulators are sampled from each line and tested. Among the total of 10 tested regulators, there were 4 defectives. Find the probability that exactly two of the defectives came from
5.80 Given the setting and terminology of Exercise 5.79, answer parts (a) through (e) if c 2.
5.79 Lot acceptance sampling procedures for an electronics manufacturing firm call for sampling n items from a lot of N items and accepting the lot if Yc, where Y is the number of nonconforming items in the sample. From an incoming lot of 20 printer covers, 5 are to be sampled. Find the probability
5.78 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 selects two of these packages for purchase without looking at the ratings. Let Y denote the number of packages purchased by the firm that are ranked
5.77 An auditor checking the accounting practices of a firm samples three accounts from an accounts receivable list of eight accounts. Find the probability that the auditor sees at least one past-due account if there are a Two such accounts among the eight.b Four such accounts among the eight.c
5.76 The “worst-case” requirements are defined in the design objectives for a brand of computer terminal.A quick preliminary test indicates that 4 out of a lot of 10 such terminals failed the “worst-case” requirements.Five of the 10 are randomly selected for further testing. Let Y denote
5.75 An eight-cylinder automobile engine has two misfiring spark plugs. If all four plugs are removed from one side of the engine, what is the probability that the two misfiring ones are among them?
5.74 Used photocopying machines are returned to the supplier, cleaned, and then sent back out on lease agreements. Major repairs are not made, and, as a result, some customers receive malfunctioning machines. Among eight used photocopiers in supply today, three are malfunctioning. A customer wants
5.73 Specifications call for a type of thermistor to test out at between 9,000 and 10,000 ohms at 25°C. From ten thermistors available, three are to be selected for use. Let Y denote the number among the three that do not conform to specifications. Find the probability distribution for Y (tabular
5.72 A foreman has 10 employees from whom he must select 4 to perform a certain undesirable task.Among the 10 employees, 3 belong to a minority ethnic group. The foreman selected all three minority employees (plus one other) to perform the undesirable task. The members of the minority group then
5.71 A corporation has a pool of six firms, four of which are local, from which it can purchase certain supplies.If three firms are randomly selected without replacement, find the probability that a At least one selected firm is not local.b All three selected firms are local.
5.70 Refer to Exercise 5.69. The company purchasing the machines returns the defective ones for repair. If it costs $50 to repair each machine, find the mean and variance of the total repair cost. In what interval would you expect the repair costs on these five machines to lie? [Hint: Use
5.69 A warehouse contains 10 printing machines, 4 of which are defective. A company randomly selects five of the machines for purchase. What is the probability that all five of the machines are nondefective?
5.68 From a box containing four white and three red balls, two balls are selected at random without replacement. Find the probability that a Exactly one white ball is selected.b At least one white ball is selected.c Two white balls are selected given that at least one white ball is selected.d The
5.67 The number of cars entering a parking lot is a random variable having a Poisson distribution with a mean of 4 per hour. The lot holds only 12 cars.a Find the probability that the lot fills up in the first hour. (Assume all cars stay in the lot longer than 1 hour.)b Find the probability that
5.66 A food manufacturer uses an extruder (a machine that produces bite-size foods such as cookies and many snack foods) that produces revenue for the firm at the rate of $200 per hour when in operation. However, the extruder breaks down an average of two times 10 hours of operation.If Y denotes
5.65 Let Y have a Poisson distribution with mean .Find E [Y (Y 1)] and use the result to show that V(Y ) .
5.64 The number of bacteria colonies of a certain type in samples of polluted water has a Poisson distribution with a mean of two per cubic centimeter.a If four 1-cubic-centimeter samples are independently selected from this water, find the probability that at least one sample will contain one or
5.63 Refer to Exercise 5.62. The cost of repairing the imperfections in the weave is $10 per imperfection.Find the mean and standard deviation of the repair costs for an 8-square-yard bolt of the textile in question.
5.62 The number of imperfections in the weave of a certain textile has a Poisson distribution with a mean of four per square yard.a Find the probability that a 1-square-yard sample will contain at least one imperfection.b Find the probability that a 3-square-yard sample will contain at least one
5.61 Refer to Exercise 5.59. Find the probability that exactly two customers arrive in the 2-hour period of time a Between 2:00 PM and 4:00 PM (one continuous 2-hour period).b Between 1:00 PM and 2:00 PM and between 3:00 PM and 4:00 PM (two separate 1-hour periods for a total of 2 hours).
5.60 Refer to Exercise 5.59. If it takes approximately 10 minutes to service each customer, find the mean and variance of the total service time connected to the customer arrivals for 1 hour.(Assume that an unlimited number of servers are available, so no customer has to wait for service.)Is it
5.59 Customer arrivals at a checkout counter in a department store have a Poisson distribution with an average of eight per hour. For a given hour, find the probability that a Exactly eight customers arrive.b No more than three customers arrive.c At least two customers arrive.
5.58 Rebuilt ignition systems leave an aircraft rework facility at the rate of three per hour on the average.The assembly line needs four ignition systems in the next hour. What is the probability that they will be available?
5.57 In a time-sharing computer system, the number of teleport inquiries averages 0.2 per millisecond and follows a Poisson distribution.a Find the probability that no inquiries are made during the next millisecond.b Find the probability that no inquiries are made during the next 3 milliseconds.
5.56 In the article cited in Exercise 5.55, the projected fatality rate for 1975 if the NMSL had not been in effect was 25 per 109 vehicle miles. Under these conditions,a Find the probability of at most 15 fatalities occurring in 109 vehicle miles.b Find the probability of at least 20 fatalities

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