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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

10.47 An important measure of the performance of a machine is the mean time between failures(MTBF). A certain printer attached to a word processor was observed for a period of time during which 10 failures were observed. The times between failures averaged 98 working hours, with a standard
10.46 Data from the U.S. Department of Interior–Geological Survey give flow rates for a small river in North Florida. It is of interest to compare flow rates for March and April, two relatively dry months.Thirty-one measurements for March showed a mean flow rate of 6.85 cubic feet per second and
10.45 The t-test for comparing means assumes that the population variances are equal. Is that a valid assumption based on the data of Exercise 10.44?(Test the equality of population variances at the 10% significance level.)
10.44 Biofeedback monitoring devices and techniques in the control of physiologic functions help astronauts control stress. One experiment related to this topic is discussed by G. Rotondo, et al., Acta Astronautica, 1983. Six subjects were placed in a stressful situation (using video games),
10.43 In the presence of reverberation, native as well as nonnative speakers of English have some trouble recognizing consonants. Random samples of 10 natives and 10 nonnatives were given the Modified Rhyme Test, and the percentages of correct responses were recorded. (See Nabelek and Donahue,
10.42 The retention of nitrogen in the soil is an important consideration in cultivation practices, including the cultivation of forests. Two methods for preparing plots for planting pine trees after clear-cutting were compared on the basis of the percentage of labeled nitrogen recovered.Method A
10.41 The strength of concrete depends, to some extent, on the method used for drying.Two different drying methods showed the following results for independently tested specimens (measurements in psi):Do the methods appear to produce concrete with different mean strengths? Use . What assumptions
10.40 A study was conducted by the Florida Game and Fish Commission to assess the amounts of chemical residues found in the brain tissue of brown pelicans. For DDT, random sample of juveniles and nestlings gave the following results (measurements in parts per million):Test the hypothesis that there
10.48 It is claimed that tensile-strength measurements for a 12 mm–diameter steel rod should, on the average, be at least 8 units (N/mm2) higher than the tensile-strength measurements for a 10 mm–diameter steel rod. Independent samples of 50 measurements each for the two sizes of rods gave the
10.49 Alloying is said to reduce the resistance in a standard type of electrical wire. Ten measurements on a standard wire yielded a mean resistance of 0.19 ohm and a standard deviation of 0.03. Ten independent measurements on an alloyed wire yielded a mean resistance of 0.11 ohm and a standard
10.58 Refer to Exercise 10.52. Does site A have significantly more variation among depth measurements than site B? Use .
10.56 Refer to Exercise 10.41. Do the variances of the strength measurements differ for the two drying methods? Test at the 10% significance level.
10.54 The two drying methods for concrete were used on seven different mixes, with each mix of concrete subjected to each drying method. The resulting strength-test measurements (in psi) are given below:Is there evidence of a difference between true average porosity measurements for the two
10.52 The average depth of bedrock at two possible construction sites is to be compared by driving five piles at random locations within each site. The results, with depths in feet, are as follows:Do the average depths of bedrock differ for the two sites at the 10% significance level? What
10.51 Six rockets, nominally with a range of 2,500 meters, were stored for some time and then tested. The ranges found in the tests were 2,490, 2,510, 2,360, 2,410, 2,300, and 2,440. Another group of six rockets, of the same type, was stored for the same length of time but in a different manner.
10.50 Two different machines, A and B, used for torsion tests of steel wire were tested on 12 pairs of different types of wire, with one member of each pair tested on each machine. The results (break angle measurements) were as follows:a Is there evidence at the 5% significance level to suggest
10.57 Refer to Exercise 10.51. Is there sufficient evidence to say that the variances among range measurements differ for the two storage methods?Use a = 0.10.
6.127 Let Y denote a random variable with probability density function given by Find the moment-generating function of Y and use it to find E(Y ).
6.126 The grains comprising polycrystalline metals tend to have weights that follow a lognormal diss tribution. For a certain type of aluminum, gram weights have a lognormal distribution with and (in units of 10–2 gram).a Find the mean and variance of the grain weights.b Find an interval in which
6.125 If X has a lognormal distribution with and, find a P(X 4)b P(X 8)
6.124 Suppose that plants of a certain species are randomly dispersed over a region, with a mean density of plants per unit area. That is, the number of plants in a region of area A has a Poisson distribution with mean . For a randomly selected plant in this region, let R denote the distance to the
6.123 The weekly downtime in hours for a certain production line has a gamma distribution with a = 3 and . Find the probability that downtime for a given week will not exceed 10 hours.
6.122 There is a relationship between incomplete gamma integrals and sums of Poisson probabilities, given by for integer values of . If Y has a gamma distribution with and , find by using the above equality and Table 3 of the Appendix.
6.121 A builder of houses has to order some supplies that have a waiting time for delivery Y uniformly distributed over the interval 1 to 4 days.Because she can get by without the supplies for 2 days, the cost of the delay is fixed at $100 for any waiting time up to 2 days. However, after 2 days
6.120 The weekly repair cost Y for a certain machine has a probability density function given by with measurements in hundreds of dollars. How much money should be budgeted each week for repair costs so that the actual cost will exceed the budgeted amount only 10% of the time?
6.119 The time (in hours) it takes a manager to interview an applicant has an exponential distribution with 12. Three applicants arrive at 8:00 AM, and interviews begin. A fourth applicant arrives at 8:45 AM. What is the probability that he has to wait before seeing the manager?
6.118 If the lifelength X for a certain type of battery has a Weibull distribution with 2 and 3 (with measurements in years), find the probability that the battery lasts less than four years given that it is now two years old;
6.117 The proportion of time, per day, that all checkout counters in a supermarket are busy is a random variable X having probability density function a Find the value of k that makes this a probability density function.b Find the mean and variance of X.
6.116 An engineer has observed that the gap times between vehicles passing a certain point on a highway have an exponential distribution with a mean of 10 seconds.a Find the probability that the next gap observed will be no longer than 1 minute.b Find the probability density function for the sum of
6.115 The lifetime X of a certain electronic component is a random variable with density function Three of these components operate independently in a piece of equipment. The equipment fails if at least two of the components fail. Find the probability that the equipment operates for at least 200
6.114 The yield force of a steel-reinforcing bar of a certain type is found to be normally distributed with a mean of 8,500 pounds and a standard deviation of 80 pounds. If three such bars are to be used on a certain project, find the probability that all three will have yield forces in excess of
6.113 Find E (Xk ) for the beta random variable. Then find the mean and variance for the beta random variable.
6.112 Let Y have density function a Give the mean and variance for Y.b Give the moment-generating function for Y.c Find the value of c.
6.111 Refer to Exercise 6.110. Suppose five bearings are drawn from production. What is the probability that at least one will be defective?
6.110 A machine operation produces bearings with diameters that are normally distributed with mean and standard deviation equal to 3.0005 and 0.001, respectively. Customer specifications require the bearing diameters to lie in the interval 3.000 0.0020. Those outside the interval are considered
6.109 Refer to Exercise 6.107. Suppose that three students are randomly selected from the student body. What is the probability that all three will possess a grade-point average in excess of 3.07?
6.108 Refer to Exercise 6.107. If students possessing a grade-point average equal to or less than 1.9 are dropped from college, what percentage of the students will be dropped?
6.107 The grade-point averages of a large population of college students are approximately normally distributed with mean equal to 2.4 and standard deviation equal to 0.5. What fraction of the students will possess a grade-point average in excess of 3.0?
6.106 Let Y have the density function given by Answer parts (a) through (f) in Exercise 6.105.
6.105 Let Y have the density function given by a Find c.b Find F (y).c Graph f (y) and F(y).d Use F(y ) in part (b) to find F(–1), F(0), and F(1).e Find P(0 Y 0.5).f Find the mean and variance of Y.
6.104 Let Y possess a density function a Find c.b Find F (y).c Graph f (y) and F(y).d Use F (y) in part (b) to find P(1 Y 2).e Use the geometric figure for f (y) to calculate P(1 Y 2).
6.103 Let Z denote a standard normal random variable.Find the moment-generating function of Z 2.What does the uniqueness property of the moment-generating function tell you about the distribution of Z 2?
6.102 Let Z denote a standard normal random variable.Find the moment-generating function of Z directly from the definition.
6.101 Using the moment-generating function for the exponential distribution with mean , find .Use this result to show that V(X ) = u . 2 u E(X2)M(t ) = (1 - bt )-a a b
6.100 Show that a gamma distribution with parameters and ‚ has the moment-generating function
6.95 The velocities of gas particles can be modeled by the Maxwell distribution, with probability density function given by where m is the mass of the particle, K is Boltzmann’s constant, and T is the absolute temperature.a Find the mean velocity of these particles.b The kinetic energy of a
6.94 Maximum wind-gust velocities in summer thunderstorms were found to follow a Weibull g u distribution with 2 and 400 (measurements in feet per second). Engineers designing structures in the areas in which these thunderstorms are found are interested in finding a gust velocity that will be
6.93 Failures to the bleed systems in jet engines were causing some concern at an air base. It was decided to model the failure-time distribution for these systems so that future failures could be anticipated a little better. Ten observed failure times (in operating hours since installation) were
6.92 Resistors being used in the construction of an aircraft guidance system have lifelengths that follow a Weibull distribution with 2 and 10(measurements in thousands of hours).a Find the probability that a randomly selected resistor of this type has a life length that exceeds 5,000 hours.b
6.91 The pressure, in thousand psi, exerted on the tank of a steam boiler has a Weibull distribution with 1.8 and 1.5. The tank is built to withstand pressures of 2,000 psi. Find the probability that this limit will be exceeded at a randomly selected moment.
6.90 The yield strengths of certain steel beams have a Weibull distribution with 2 and 3,600(measurements in pounds per square inch). Two such randomly selected beams are used in a construction project that calls for yield strengths in excess of 70,000 psi. Find the probability that both beams
6.89 The ultimate tensile strength of steel wire used to wrap concrete pipe was found to have a Weibull distribution with g 1.2 and u 270 (measureg ug u g u ments in thousands of pounds). Pressure in the pipe at a certain point may require an ultimate tensile strength of at least 300,000
6.88 The time necessary to achieve proper blending of copper powders before sintering was found to have a Weibull distribution with 1.1 and 2(measurements in minutes). Find the probability that a proper blending in a randomly selected sample takes less than 2 minutes.
6.87 The maximum flood levels, in millions of cubic feet per second, for a certain U.S. river have a Weibull distribution with 1.5 and 0.6. (See Cohen, Whitten, and Ding, J. Quality Technology, 1984, for more details.) Find the probability that the maximum flood level for next year a Will
6.86 Fatigue life (in hundreds of hours) for a certain type of bearing has approximately a Weibull distribution with 2 and 4.a Find the probability that a randomly selected bearing of this type fails in less than 200 hours.b Find the expected fatigue life for this type of bearing.
6.85 The proportion of pure iron in certain ore samples has a beta distribution with 3 and 1.a Find the probability that one of these randomly selected samples will have more than 50% pure iron.b Find the probability that two out of three randomly selected samples will have less than 30% pure iron.
6.84 In blending fine and coarse powders prior to copper sintering, proper blending is necessary for uniformity in the finished product. One way to check the blending is to select many small random samples of the blended powders and measure the weight fractions of the fine particles.These
6.83 Errors in measuring the arrival time of a wave front from an acoustic source can sometimes be modeled by a beta distribution. (See Perruzzi and Hilliard, J. Acoustical Society of America,1984)Suppose these errors have a beta distribution with 1 and 2, with measurements in microseconds.a Find
6.82 To study the disposal of pollutants emerging from a power plant, the prevailing wind direction was measured for a large number of days. The direction is measured on a scale of 0º to 360°, but by dividing each daily direction by 360, the measurements can be rescaled to the interval(0,1).
6.81 The percentage of impurities per batch in a certain type of industrial chemical is a random variable X having the probability density function a Suppose a batch with more than 40% impurities cannot be sold. What is the probability that a randomly selected batch will not be sold?b Suppose the
6.80 During an 8-hour shift, the proportion of time X that a sheet-metal stamping machine is down for maintenance or repairs has a beta distribution with and .That is, The cost (in hundreds of dollars) of this downtime, due to lost production and cost of maintenance and repair, is given by.a Find
6.79 If X has a beta distribution with parameters and , show that
6.78 Suppose X has a probability density function given by a Find the value of k that makes this a probability density function.b Find and .
6.77 Because the use of CD-ROM and CD-R for storing data has increased considerably, successful retrieving of data from them becomes an important issue for their manufacturers. Eastman Kodak Company was interested in developing Kodak Writable CD and Kodak Photo CD with exceptional data life for CDs
6.76 In a large-scale study to understand root causes of systems failures, Los Alamos National Laboratory collected data about 23,000 failures of different high-performance computing systems.Schroeder and Gibson (Technical Report CMU-PDL-05-112, 2005) showed that even for different systems, the
6.75 One study of cancer patients showed that the survival times (in weeks) of cancer (Mesothelioma)patients since detection are distributed lognormal with and .a What percent of cancer patients are expected to survive for 6 months after detection?b What percent of patients are expected to survive
6.74 In automobiles, rubber boot seals are used to protect velocity joints and steering mechanisms.Lifetimes of these boot seals depend on how much they come in contact with the metal shaft.The lognormal distribution has been used to describe the lifetime of boot seals (Guerin and Hambli, J.
6.73 The lifetime distributions of high-speed recordable optical disks is assumed to follow lognormal distribution(Irine and Okino, IEEE Trans. on Magnetics, 2007).Suppose and weeks.a Compute the mean and variance of this distribution.b What percent of optical disks are expected to last over 20
6.72 The cumulative proportions of age groups of U.S. residents are shown in table below for years 1900 and 2000.a Construct a Q–Q plot for each year. Use these plots as a basis for discussing key differences between the two age distributions.b Each of the Q–Q plots should show some departures
6.71 The cumulative percent distribution of population(20–29 years) by height and sex (1988–1994) is given in table below.a Produce a Q–Q plot for male and female heights.b Discuss their goodness of fit for a normal distribution.c Approximate the mean and standard deviation of these heights
6.70 Another contributor to air pollution is sulfur dioxide.Shown below are 12 peak sulfurdioxide measurements (in ppm) for 1990 from randomly selected U.S. sites:0.003, 0.001, 0.014, 0.024, 0.024, 0.032, 0.038, 0.042, 0.043, 0.044, 0.047, 0.061 a Should the normal distribution be used as a model
6.69 The histogram in Figure 6.37 depicts the total points scored per game for every NCAAtournament basketball game between 1939 and 1992 (n 1,521). The mean is 143 and the standard deviation is 26.a Does it appear that total points can be modeled as a normally distributed random variable?b Show
6.68 Refer to Exercise 6.67. Suppose the standard deviation s is not known but can be fixed at certain levels by carefully adjusting the machine. What is the largest value of that will allow the actual value dispensed to be within 1 ounce of the mean with probability at least 0.95?
6.67 A machine for filling cereal boxes has a standard deviation of 1 ounce of fill per box. What setting of the mean ounces of fill per box will allow 16-ounce boxes to overflow only 1% of the time?Assume that the ounces of fill per box are normally distributed.
6.66 The times of first failure of a unit of a brand of ink jet printers are approximately normally distributed with a mean of 1,500 hours and a standard deviation of 200 hours.a What fraction of these printers will fail before 1,000 hours?b What should be the guarantee time for these printers if
6.65 Sick-leave time used by employees of a firm in one month has approximately a normal distribution with a mean of 200 hours and a variance of 400.a Find the probability that total sick leave for next month will be less than 150 hours.b In planning schedules for next month, how much time should
6.64 A type of capacitor has resistances that vary according to a normal distribution with a mean of 800 megohms and a standard deviation of 200 megohms (see Nelson, Industrial Quality Control, 1967, pp. 261–268, for a more thorough discussion). A certain application specifies capacitors with
6.63 A vehicle driver gauges the relative speed of a preceding vehicle by the speed with which the image of the width of that vehicle varies. This speed is proportional to the speed X of variation of the angle at which the eye subtends this width.According to Ferrani and others (Transportation
6.62 At a temperature of C, the resistances of a type of thermistor are normally distributed with a mean of 10,000 ohms and a standard deviation of 4,000 ohms. The thermistors are to be sorted, with those having resistances between 8,000 and 15,000 ohms being shipped to a vendor.What fraction of
6.61 Wires manufactured for use in a certain computer system are specified to have resistances between 0.12 and 0.14 ohm. The actual mea-P(-z0 … Z … z0) = 0.95 P(-z0 … Z … z0) = 0.90 P(Z Ú z0) = 0.617 P(Z Ú z0) = 0.117 P(Z … z0) = 0.8749 P(Z … z0) = 0.5 z0 P(-200 … Z …
6.60 Refer to Exercise 6.59. What should be the mean diameter of the shafts produced to minimize the fraction not meeting specifications?
6.59 A machining operation produces steel shafts having diameters that are normally distributed with a mean of 1.005 inches and a standard deviation of 0.01 inch. Specifications call fordiameters to fall within the interval 1.00 ±0.02 inches. What percentage of the output of this operation will
6.58 In the setting of Exercise 6.57, how much should be budgeted weekly for maintenance and repairs so that the budgeted amount will be exceeded with probability only 0.1?
6.57 The weekly amount spent for maintenance and repairs in a certain company has approximately a normal distribution with a mean of $400 and a standard deviation of $20. If $450 is budgeted to cover repairs for next week, what is the probability that the actual costs will exceed the budgeted
6.56 For a standard normal random variable Z, use Table 4 of the Appendix to find a number such that ab cd ef
6.55 Use Table 4 of the Appendix to find the following probabilities for a standard normal random variable Z.a bc de
6.54 The response times at an online terminal have approximately a gamma distribution with a mean of 4 seconds and a variance of 8.Write the probability density function for these response times.
6.53 If service times at a teller window of a bank are exponentially distributed with a mean of 3.2 minutes, find the probability distribution, mean, and variance of the time taken to serve three waiting customers.
6.52 The time intervals between dial-up connections to a computer center from remote terminals are exponentially distributed with a mean of 15 seconds.Find the mean, variance, and probabilitydistribution of the waiting time from the opening of the computer center until the fourth dial-up connection
6.51 A 40-year history of maximum river flows for a certain small river in the United States shows a relative frequency histogram that can be modeled by a gamma density function with and . (Measurements are in cubic feet per second.)a Find the mean and standard deviation of the annual maximum river
6.50 The total sustained load on the concrete footing of a planned building is the sum of the deadload plus the occupancy load. Suppose the dead load X1 has a gamma distribution with and , while the occupancy load X2 has a gamma distribution with and .(Units are in kips, or thousands of pounds.)a
6.49 Suppose two houses are to be built and each will involve the completion of a certain key task.The task has an exponentially distributed time to completion with a mean of 10 hours.Assuming the completion times are independent for the two houses, find the expected value and variance of L = 30Y +
6.48 Customers arrive at a checkout counter according to a Poisson process with a rate of two perminute.Find the mean, variance, and probability density function of the waiting time between the opening of the counter and a The arrival of the second customer b The arrival of the third customer
6.47 The weekly downtime Y (in hours) for a certain industrial machine has approximately a gamma distribution with and . The loss, in dollars, to the industrial operation as a result of this downtime is given by a Find the expected value and variance of L.b Find an interval that will contain L on
6.46 Annual incomes for engineers in a certain industry have approximately a gamma distribution with and .a Find the mean and variance of these incomes.b Would you expect to find many engineers in this industry with an annual income exceeding$35,000?
6.45 Four-week summer rainfall totals in a certain section of the midwestern United States have a relative frequency histogram that appears to fit closely to a gamma distribution with and a Find the mean and variance of this distribution of 4-week rainfall totals.b Find an interval that will
6.44 Explosive devices used in a mining operation cause nearly circular craters to form in a rockysurface.The radii of these craters are exponentially distributed with a mean of 10 feet. Find the mean and variance of the area covered by such a crater.
6.43 In deciding how many customer service representatives to hire and in planning their schedules, it is important for a firm marketing electronic typewriters to study repair times for the machines.Such a study revealed that repair times have approximately an exponential distribution with a mean
6.42 The service times at teller windows in a bank were found to follow an exponential distribution with a mean of 3.2 minutes. A customer arrives at a window at 4:00 PM.a Find the probability that he will still be there at 4:02 PM.b Find the probability that he will still be there at 4:04 PM given
6.41 The weekly rainfall totals for a section of the midwestern United States follow an exponential distribution with a mean of 1.6 inches.a Find the probability that a weekly rainfall total in this section will exceed 2 inches.b Find the probability that the weekly rainfall totals will not exceed

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