Applied Statistics And Probability For Engineers 6th Edition Douglas C. Montgomery, George C. Runger - Solutions

A steel plate contains 20 bolts. Assume that 5 bolts are not torqued to the proper limit. 4 bolts are selected at random, without replacement, to be checked for torque. If an operator checks a bolt, the probability that an incorrectly torqued bolt is identified is 0.95. If a checked bolt is
In an acid-base titration, the milliliters of base that are needed to reach equivalence are measured to the nearest milliliter between 0.1 and 0.15 liters (inclusive).
The number of mutations in a nucleotide sequence of length 40,000 in a DNA strand after exposure to radiation is measured. Each nucleotide may be mutated.
Consider the hospital patients in Example 2-8. Two patients are selected randomly, with replacement, from the total patients at Hospital. What is the probability mass function of the number of patients in the sample who are admitted?Data from 2-8 Hospital 2 Total 6991 4329 22,252 Total 5292
Orders arrive at a Web site according to a Poisson process with a mean of 12 per hour. Determine the following:(a) Probability of no orders in five minutes.(b) Probability of 3 or more orders in five minutes.(c) Length of a time interval such that the probability of no orders in an interval of this
A total of 12 cells are replicated. Freshly synthesized DNA cannot be replicated again until mitosis is completed. Two control mechanisms have been identified—one positive and one negative—that are used with equal probability. Assume that each cell independently uses a control mechanism.
The distribution of X is approximated with a triangular probability density function f (x) = 0.025x − 0.0375 for 30 < x < 50 and f (x) = −0.025x + 0.0875 for 50 < x < 70.Determine the following:(a) P(X ≤ 40)(b) P(40 < X ≤ 60)(c) Value x exceeded with probability 0.99.
The waiting time for service at a hospital emergency department (in hours) follows a distribution with probability density function f (x) = 0.5exp(−0.5x) for 0 < x. Determine the following:(a) P(X < 0.5)(b) P(X > 2)(c) Value x (in hours) exceeded with probability 0.05.
The probability density function of the time you arrive at a terminal (in minutes after 8:00 a.m.) is f (x) = 0.1 exp(− 0.1x) for 0 < x. Determine the probability that(a) You arrive by 9:00 a.m.(b) You arrive between 8:15 a.m. and 8:30 a.m.(c) You arrive before 8:40 a.m. on two or more days of
The probability density function of the weight of packages delivered by a post office is f (x) = 70 / (69x2) for 1 < x < 70 pounds.(a) Determine the mean and variance of weight.(b) If the shipping cost is $2.50 per pound, what is the average shipping cost of a package?(c) Determine the
An adult can lose or gain two pounds of water in the course of a day. Assume that the changes in water weight are uniformly distributed between minus two and plus two pounds in a day. What is the standard deviation of a person’s weight over a day?
A show is scheduled to start at 9:00 a.m., 9:30 a.m., and 10:00 a.m. Once the show starts, the gate will be closed. A visitor will arrive at the gate at a time uniformly distributed between 8:30 a.m. and 10:00 a.m. Determine the following:(a) Cumulative distribution function of the time (in
The volume of a shampoo filled into a container is uniformly distributed between 374 and 380 milliliters.(a) What are the mean and standard deviation of the volume of shampoo?(b) What is the probability that the container is filled with less than the advertised target of 375 milliliters?(c) What is
An e-mail message will arrive at a time uniformly distributed between 9:00 a.m. and 11:00 a.m. You check e-mail at 9:15 a.m. and every 30 minutes afterward.(a) What is the standard deviation of arrival time (in minutes)?(b) What is the probability that the message arrives less than 10 minutes
Measurement error that is continuous and uniformly distributed from –3 to +3 millivolts is added to a circuit’s true voltage. Then the measurement is rounded to the nearest millivolt so that it becomes discrete. Suppose that the true voltage is 250 millivolts.(a) What is the probability mass
A beacon transmits a signal every 10 minutes (such as 8:20, 8:30, etc.). The time at which a receiver is tuned to detect the beacon is a continuous uniform distribution from 8:00 a.m. to 9:00 a.m. Consider the waiting time until the next signal from the beacon is received.(a) Is it reasonable to
An electron emitter produces electron beams with changing kinetic energy that is uniformly distributed between three and seven joules. Suppose that it is possible to adjust the upper limit of the kinetic energy (currently set to seven joules).(a) What is the mean kinetic energy?(b) What is the
Cholesterol is a fatty substance that is an important part of the outer lining (membrane) of cells in the body of animals. Its normal range for an adult is 120–240 mg/dl. The Food and Nutrition Institute of the Philippines found that the total cholesterol level for Filipino adults has a mean of
In 2002, the average height of a woman aged 20–74 years was 64 inches with an increase of approximately 1 inch from 1960 (http://usgovinfo.about.com/od/healthcare). Suppose the height of a woman is normally distributed with a standard deviation of two inches.(a) What is the probability that a
In an accelerator center, an experiment needs a 1.41-cmthick aluminum cylinder (http://puhep1.princeton.edu/mumu/ target/Solenoid_Coil.pdf). Suppose that the thickness of a cylinder has a normal distribution with a mean of 1.41 cm and a standard deviation of 0.01 cm.(a) What is the probability that
An article in Atmospheric Chemistry and Physics “Relationship Between Particulate Matter and Childhood Asthma—Basis of a Future Warning System for Central Phoenix” (2012, Vol. 12, pp. 2479–2490)] reported the use of PM10 (particulate matter < 10 μm diameter) air quality data measured
A signal in a communication channel is detected when the voltage is higher than 1.5 volts in absolute value. Assume that the voltage is normally distributed with a mean of 0. What is the standard deviation of voltage such that the probability of a false signal is 0.005?
An article in International Journal of Electrical Power & Energy Systems [“Stochastic Optimal Load Flow Using a Combined Quasi–Newton and Conjugate Gradient Technique” (1989, Vol.11(2), pp. 85–93)] considered the problem of optimal power flow in electric power systems and included the
A test instrument needs to be calibrated periodically to prevent measurement errors. After some time of use without calibration, it is known that the probability density function of the measurement error is f (x) = 1− 0.5x for 0 < x < 2 millimeters.(a) If the measurement error within 0.5
An article in Electric Power Systems Research [€œModeling Real-Time Balancing Power Demands in Wind Power Systems Using Stochastic Differential Equations€ (2010, Vol. 80(8), pp. 966€“974)] considered a new probabilistic model to balance power demand with large amounts
Derive the convergence results used to obtain a Poisson distribution as the limit of a binomial distribution.
Suppose that 50 sites on a patient might contain lesions. A biopsy selects 8 sites randomly (without replacement). What is the minimum number of sites with lesions so that the probability of at least one selected site contains lesions is greater than or equal to 0.95? Rework for greater than or
Saguaro cacti are large cacti indigenous to the southwestern United States and Mexico. Assume that the number of saguaro cacti in a region follows a Poisson distribution with a mean of 280 per square kilometer. Determine the following:(a) Mean number of cacti per 10,000 square meters.(b)
Messages that arrive at a service center for an information systems manufacturer have been classified on the basis of the number of keywords (used to help route messages) and the type of message, either e-mail or voice. Also, 70% of the messages arrive via e-mail and the rest are voice.Determine
Traffic flow is traditionally modeled as a Poisson distribution. A traffic engineer monitors the traffic flowing through an intersection with an average of six cars per minute. To set the timing of a traffic signal, the following probabilities are used.(a) What is the probability that no cars pass
An automated egg carton loader has a 1% probability of cracking an egg, and a customer will complain if more than one egg per dozen is cracked. Assume that each egg load is an independent event.(a) What is the distribution of cracked eggs per dozen? Include parameter values.(b) What is the
Inclusions are defects in poured metal caused by contaminants. The number of (large) inclusions in cast iron follows a Poisson distribution with a mean of 2.5 per cubic millimeter. Determine the following:(a) Probability of at least one inclusion in a cubic millimeter.(b) Probability of at least
The article “An Association Between Fine Particles and Asthma Emergency Department Visits for Children in Seattle” [Environmental Health Perspectives June, 1999 107(6)] used Poisson models for the number of asthma emergency department (ED) visits per day. For the zip codes studied, the mean ED
Cabs pass your workplace according to a Poisson process with a mean of five cabs per hour. Suppose that you exit the workplace at 6:00 p.m. Determine the following:(a) Probability that you wait more than 10 minutes for a cab.(b) Probability that you wait fewer than 20 minutes for a cab.(c) Mean
The number of views of a page on a Web site follows a Poisson distribution with a mean of 1.5 per minute.(a) What is the probability of no views in a minute?(b) What is the probability of two or fewer views in 10 minutes?(c) Does the answer to the previous part depend on whether the 10-minute
Astronomers treat the number of stars in a given volume of space as a Poisson random variable. The density in the Milky Way Galaxy in the vicinity of our solar system is one star per 16 cubic light-years.(a) What is the probability of two or more stars in 16 cubic light-years?(b) How many cubic
Verify that the following functions are probability mass functions, and determine the requested probabilities.1.(a) P(X ‰¤ 2)(b) P(X > ˆ’2)(c) P(ˆ’1‰¤ X ‰¤ 1)(d) P(X ‰¤ ˆ’1 or X = 2)2. f(x) = (8 / 7) (1 / 2)x, x =
A Web site contains 100 interconnected pages. The random variable is the number of unique pages viewed by a visitor to the Web site.
A healthcare provider schedules 30 minutes for each patient’s visit, but some visits require extra time. The random variable is the number of patients treated in an eight-hour day.
Suppose that a table of part counts is generalized as follows:where a, b, and k are positive integers. Let A denote the event that a part is from supplier 1, and let B denote the event that a part conforms to specifications. Show that A and B are independent events. This exercise illustrates the
Suppose that a lot of washers is large enough that it can be assumed that the sampling is done with replacement. Assume that 60% of the washers exceed the target thickness.(a) What is the minimum number of washers that need to be selected so that the probability that none is thicker than the target
Suppose documents in a lending organization are selected randomly (without replacement) for review. In a set of 50 documents, suppose that 2 actually contain errors.(a) What is the minimum sample size such that the probability exceeds 0.90 that at least 1 document in error is selected?(b) Comment
Consider the hospital emergency room data given below. Let A denote the event that a visit is to hospital 4, and let B denote the event that a visit results in LWBS (at any hospital). Are these events independent? Hospital Total 1 22,252 Total 5292 6991 5640 4329 270 LWBS 195
Samples of skin experiencing desquamation are analyzed for both moisture and melanin content. The results from 100 skin samples are as follows:Let A denote the event that a sample has low melanin content, and let B denote the event that a sample has high moisture content. Determine the
An article in the British Medical Journal [“Comparison of treatment of renal calculi by operative surgery, percutaneous nephrolithotomy, and extracorporeal shock wave lithotripsy” (1986, Vol. 82, pp. 879–892)] provided the following discussion of success rates in kidney stone removals. Open
Heart failures are due to either natural occurrences (87%) or outside factors (13%). Outside factors are related to induced substances (73%) or foreign objects (27%). Natural occurrences are caused by arterial blockage (56%), disease (27%), and infection (e.g., staph infection) (17%).(a) Determine
In the 2012 presidential election, exit polls from the critical state of Ohio provided the following results:What is the probability a randomly selected respondent voted for Obama? Romney 45% Obama Total No college degree (60%) College graduate (40%) 52% 47% 51%
A computer system uses passwords that contain exactly eight characters, and each character is one of the 26 lowercase letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9). Let Ω denote the set of all possible passwords. Suppose that all passwords in Ω are
The article [“Clinical and Radiographic Outcomes of Four Different Treatment Strategies in Patients with Early Rheumatoid Arthritis,” Arthritis & Rheumatism (2005, Vol. 52, pp. 3381– 3390)] considered four treatment groups. The groups consisted of patients with different drug
The article “Term Efficacy of Ribavirin Plus Interferon Alfa in the Treatment of Chronic Hepatitis C,” [Gastroenterology (1996, Vol. 111, no. 5, pp. 1307–1312)], considered the effect of two treatments and a control for treatment of hepatitis C. The following table provides the total patients
Consider the bar code 39 is a common bar code system that consists of narrow and wide bars (black) separated by either wide or narrow spaces (white). Each character contains nine elements (five bars and four spaces). The code for a character starts and ends with a bar (either narrow or wide)
Consider the well failure data given below.(a) What is the probability of a failure given there are more than 1,000 wells in a geological formation?(b) What is the probability of a failure given there are fewer than 500 wells in a geological formation? Wells Geological Formation Group Failed Total
Consider the hospital emergency room data given below. Let A denote the event that a visit is to hospital 4, and let B denote the event that a visit results in LWBS (at any hospital). Determine the following probabilities.(a) P(A | B)(b) P(A€² | B)(c) P(A |
The number of content changes to a Web site follows a Poisson distribution with a mean of 0.25 per day.(a) What is the probability of two or more changes in a day?(b) What is the probability of no content changes in five days?(c) What is the probability of two or fewer changes in five days?
In 1898, L. J. Bortkiewicz published a book entitled The Law of Small Numbers. He used data collected over 20 years to show that the number of soldiers killed by horse kicks each year in each corps in the Prussian cavalry followed a Poisson distribution with a mean of 0.61.(a) What is the
Data from www.centralhudsonlabs.com determined the mean number of insect fragments in 225-gram chocolate bars was 14.4, but three brands had insect contamination more than twice the average. See the U.S. Food and Drug Administration–Center for Food Safety and Applied Nutrition for Defect Action
Suppose that lesions are present at 5 sites among 50 in a patient. A biopsy selects 8 sites randomly (without replacement).(a) What is the probability that lesions are present in at least one selected site?(b) What is the probability that lesions are present in two or more selected sites?(c)
Suppose that a healthcare provider selects 20 patients randomly (without replacement) from among 500 to evaluate adherence to a medication schedule. Suppose that 10% of the 500 patients fail to adhere with the schedule. Determine the following:(a) Probability that exactly 10% of the patients in the
Consider the semiconductor wafer data in Table 2-1. Suppose that 10 wafers are selected randomly (without replacement) for an electrical test. Determine the following:(a) Probability that exactly 4 wafers have high contamination.(b) Probability that at least 1 is from the center of the sputtering
Consider the nonfailed wells in Exercises 3-35. Assume that four wells are selected randomly (without replacement) for inspection.(a) What is the probability that exactly two are selected from the Loch Raven Schist?(b) What is the probability that one or more is selected from the Loch Raven
Consider the visits that result in leave without being seen (LWBS) at an emergency department in Example 2-8. Assume that four visits that result in LWBS are to be randomly selected (without replacement) for a follow-up interview.(a) What is the probability that all selected visits are from
The analysis of results from a leaf transmutation experiment (turning a leaf into a petal) is summarized by the type of transformation completed:A naturalist randomly selects three leaves from this set without replacement. Determine the following probabilities.(a) Exactly one has undergone both
Customers visit a Web site, and the probability of an order if a customer views five or fewer pages is 0.01. However, if a customer views more than five pages, the probability of an order is 0.1. The probability a customer views five or more pages is 0.25. The customers behave independently.(a) Is
Consider the patient data in Example 2-8. Suppose that patients are randomly selected with replacement, from the total for hospital 4. Determine the following:(a) Probability that the first patient admitted is the first one selected.(b) Probability that four or fewer patients are selected to admit
An array of 30 LED bulbs is used in an automotive light. The probability that a bulb is defective is 0.001 and defective bulbs occur independently. Determine the following:(a) Probability that an automotive light has two or more defective bulbs.(b) Expected number of automotive lights to check to
Consider the visits that result in leave without being seen (LWBS) at an emergency department in Example 2-8. Assume that people independently arrive for service at hospital l.(a) What is the probability that the fifth visit is the first one to LWBS?(b) What is the probability that either the fifth
A Web site randomly selects among 10 products to discount each day. The color printer of interest to you is discounted today.(a) What is the expected number of days until this product is again discounted?(b) What is the probability that this product is first discounted again exactly 10 days from
In the process of meiosis, a single parent diploid cell goes through eight different phases. However, only 60% of the processes pass the first six phases and only 40% pass all eight. Assume that the results from each phase are independent.(a) If the probability of a successful pass of each one of
A computer system uses passwords constructed from the 26 letters (a–z) or 10 integers (0–9). Suppose that 10,000 users of the system have unique passwords. A hacker randomly selects (with replacement) passwords from the potential set.(a) Suppose that 9900 users have unique six-character
Consider the patient data in Example 2-8. Suppose that patients are randomly selected with replacement from the total for hospital 4. What is the smallest sample size needed so that the probability is at least 90% that at least one patient is LWBS?
Consider the time to recharge the flash in cell-phone cameras as in Example 3-2. Assume that the probability that a camera passes the test is 0.8 and the cameras perform independently. What is the smallest sample size needed so that the probability of at least one camera failing is at least 95%?
Consider the circuit in Example 2-34. Assume that devices fail independently. What is the probability mass function of the number of device failures? Explain why a binomial distribution does not apply to the number of device failures in Example 2-32.
The probability that a visitor to a Web site provides contact data for additional information is 0.01. Assume that 1000 visitors to the site behave independently. Determine the following probabilities:(a) No visitor provides contact data.(b) Exactly 10 visitors provide contact data.(c) More than 3
Consider the endothermic reactions in Exercise 3-32. A total of 20 independent reactions are to be conducted.(a) What is the probability that exactly 12 reactions result in a final temperature less than 272 K?(b) What is the probability that at least 19 reactions result in a final temperature less
Assume that a Web site changes its content according to the distribution in Exercise 3-34. Assume that 10 changes are made independently.(a) What is the probability that the change is made in less than 4 days in 7 of the 10 updates?(b) What is the probability that the change is made in less than 4
Consider the patient data in Example 2-8. Suppose that five patients are randomly selected with replacement from the total for hospital 4. Determine the following probabilities:(a) Exactly one is LWBS.(b) Two or more are LWBS.(c) At least one is LWBS.
Consider the lengths of stay at a hospital’s emergency department in Exercise 3-33. Assume that five persons independently arrive for service.(a) What is the probability that the length of stay of exactly one person is less than or equal to 4 hours?(b) What is the probability that exactly two
An article in Information Security Technical Report [€œMalicious Software€”Past, Present and Future€ (2004, Vol. 9, pp. 6€“18)] provided the following data on the top 10 malicious software instances for 2002. The clear leader in the number of registered
Samples of rejuvenated mitochondria are mutated (defective) in 1% of cases. Suppose that 15 samples are studied and can be considered to be independent for mutation. Determine the following probabilities. The binomial table in Appendix A can help.(a) No samples are mutated.(b) At most one sample is
Consider the hospital data in Example 2-8. Suppose a patient is selected randomly from the collection in the table. Let X denote the hospital number of the selected patient (either 1, 2, 3, or 4). Does X have a discrete uniform distribution? Why or why not?
Each multiple-choice question on an exam has four choices. Suppose that there are 10 questions and the choice is selected randomly and independently for each question. Let X denote the number of questions answered correctly. Does X have a discrete uniform distribution? Why or why not?
The probability that data are entered incorrectly into a field in a database is 0.005. A data entry form has 28 fields, and errors occur independently for each field. The random variable X is the number of fields on the form with an error. Does X have a discrete uniform distribution? Why or why not?
Suppose that 1000 seven-digit telephone numbers within your area code are dialed randomly. What is the probability that your number is called?
Suppose that nine-digit Social Security numbers are assigned at random. If you randomly select a number, what is the probability that it belongs to one of the 300 million people in the United States?
The number of pages in a PDF document you create has a discrete uniform distribution from five to nine pages (including the end points). What are the mean and standard deviation of the number of pages in the document?
Assume that the wavelengths of photosynthetically active radiations (PAR) are uniformly distributed at integer nanometers in the red spectrum from 675 to 700 nm.(a) What are the mean and variance of the wavelength distribution for this radiation?(b) If the wavelengths are uniformly distributed at
Let the random variable X have a discrete uniform distribution on the integers 1 ≤ x ≤ 3. Determine the mean and variance of X.
Let the random variable X have a discrete uniform distribution on the integers 0 ≤ x ≤ 99. Determine the mean and variance of X.
An article in the Journal of Database Management [“Experimental Study of a Self-Tuning Algorithm for DBMS Buffer Pools” (2005, Vol. 16, pp. 1–20)] provided the workload used in the Transaction Processing Performance Council’s Version C On-Line Transaction Processing (TPC-C OLTP) benchmark,
In a NiCd battery, a fully charged cell is composed of nickelic hydroxide. Nickel is an element that has multiple oxidation states. Assume the following proportions of the states:Nickel Charge Proportions
Consider the circuit in Example 2-32. Assume that devices fail independently. What is the probability mass function of the number of failed devices?
Consider the wafers with contamination particles in Example 2-17. Assume that wafers are independent with respect to contamination particles. Wafers are selected until one with five or more contamination particles occurs. What is the probability mass function of the number of wafers selected?
The following table shows the typical depth (rounded to the nearest foot) for nonfailed wells in geological formations in Baltimore County (The Journal of Data Science, 2009, Vol. 7, pp. 111€“127).Calculate the probability mass function of depth for nonfailed wells from the table.
The distribution of the time until a Web site changes is important to Web crawlers that search engines use to maintain current information about Web sites. The distribution of the time until change (in days) of a Web site is approximated in the following table.Days until Changes
Actual lengths of stay at a hospital€™s emergency department in 2009 are shown in the following table (rounded to the nearest hour). Length of stay is the total of wait and service times. Some longer stays are also approximated as 15 hours in this table.Calculate the probability mass
The data from 200 endothermic reactions involving sodium bicarbonate are summarized as follows:Final Temperature Conditions Number of Reactions266 K……………………………………………………………..48271
The space shuttle flight control system called Primary Avionics Software Set (PASS) uses four independent computers working in parallel. At each critical step, the computers “vote” to determine the appropriate step. The probability that a computer will ask for a roll to the left when a roll to
An article in Knee Surgery, Sports Traumatology, Arthroscopy [“Arthroscopic Meniscal Repair with an Absorbable Screw: Results and Surgical Technique” (2005, Vol. 13, pp. 273–279)] cites a success rate of more than 90% for meniscal tears with a rim width under 3 mm, but only a 67% success rate
Two suppliers each supplied 2000 parts that were evaluated for conformance to specifications. One part type was more complex than the other. The proportion of nonconforming parts of each type are shown in the table.One part is selected at random from each supplier. For each supplier, separately
Natural red hair consists of two genes. People with red hair have two dominant genes, two regressive genes, or one dominant and one regressive gene. A group of 1000 people was categorized as follows:Let A denote the event that a person has a dominant red hair gene, and

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Applied Statistics And Probability For Engineers 6th Edition Douglas C. Montgomery, George C. Runger - Solutions

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