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applied statistics and multivariate
Applied Statistics And Probability For Engineers 5th Edition Douglas C. Montgomery, George C. Runger - Solutions
Suppose the cumulative distribution function of the random variable X is x
The probability density function of the length of a metal rod is f(x) = 2 for 23
The probability density function of the length of a cut- ting blade isf(x) = 1.25 for 74.6 75.2) (c) If the specifications for this process are from 74.7 to 75.3 millimeters, what proportion of blades meets specifications?
The probability density function of the net weight in pounds of a packaged chemical herbicide is f(x) = 2.0 for 49.75 x50.25 pounds. (a) Determine the probability that a package weighs more than 50 pounds. (b) How much chemical is contained in 90% of all packages?
Approximate probabilities for some binomial and Poisson distributions
Use the table for the cumulative distribution function of a standard normal distribution to calculate probabilities
Standardize normal random variables
Calculate probabilities, determine means and variances for some common continuous probability distributions
Select an appropriate continuous probability distribution to calculate probabilities in specific applications
Understand the assumptions for some common continuous probability distributions
Calculate means and variances for continuous random variables
Determine probabilities from cumulative distribution functions and cumulative distribution functions from probability density functions, and the reverse
Determine probabilities from probability density functions
A large bakery can produce rolls in lots of either 0, 1000, 2000, or 3000 per day. The production cost per item is $0.10. The demand varies randomly according to the following distribution: demand for rolls 0 1000 2000 3000 0.2 probability of demand 0.3 0.2 0.3 Every roll for which there is a
A manufacturer stocks components obtained from a supplier. Suppose that 2% of the components are defective and that the defective components occur inde- pendently. How many components must the manufacturer have in stock so that the probability that 100 orders can be completed without reordering
An air flight can carry 120 passengers. A pas- senger with a reserved seat arrives for the flight with probability 0.95 Assume the passengers behave inde- pendently. (Computer software is expected.) (a) What is the minimum number of seats the airline should reserve for the probability of a full
Derive the expression for the variance of a geometric random variable with parameter p.
Derive the formula for the mean and standard deviation of a discrete uniform random variable over the range of integersa, a +1.....b.
From 500 customers, a major appliance manufac- turer will randomly select a sample without replacement. The company estimates that 25% of the customers will provide useful data. If this estimate is correct, what is the probability mass function of the number of customers that will provide useful
Assume the number of errors along a magnetic recording surface is a Poisson random variable with a mean of one error every 10 hits. A sector of data consists of 4096 eight-bit bytes. (a) What is the probability of more than one error in a sector? (b) What is the mean number of sectors until an
Determine the probability mass function for the random variable with the following cumulative distribution function: 70 0.2 x
The random variable X' has the following probability distribution: probability 2 3 5 8 0.2 04 0.3 0.1 (b) P(X >25) Determine the following: (a) P(X 3) (c) P(2.7 <
Messages that arrive at a service center for an in- formation systems manufacturer have been classified on the basis of the number of keywords (used to help route mes- sages) and the type of message, either e-mail or voice. Also, 70% of the messages arrive via e-mail and the rest are voice. number
Determine the constant c so that the following function is a probability mass function: f(x) = cx for x = 1,2,3,4.
Continuation of Exercise 3-163.Determine the minimum number of assemblies that need to be checked so that the probability of at least one defective assembly exceeds 0.95.
Patient response to a generic drug to control pain is scored on a 5-point scale, where a 3 indicates complete relief. Historically, the distribution of scores is 1 2 3 4 5 0.05 0.1 0.2 0.25 0.4 Two patients, assumed to be independent, are each scored. (a) What is the probability mass function of
The probability that an individual recovers from an illness in a one-week time period without treatment is 0.1. Suppose that 20 independent individuals suffering from this illness are treated with a drug and four recover in a one-week time period. If the drug has no effect, what is the probability
The number of errors in a textbook follows a Poisson distribution with a mean of 0.01 error per page. What is the probability that there are three or less errors in 100 pages?
The number of messages sent to a computer bulletin board is a Poisson random variable with a mean of five mes- sages per hour. (a) What is the probability that five messages are received in 1 hour? (b) What is the probability that 10 messages are received in 1.5 hours? (c) What is the probability
Continuation of Exercise 3-155.(a) What is the probability that you must call six times in order for two of your calls to be answered in less than 30 seconds? (b) What is the mean number of calls to obtain two answers in less than 30 seconds?
Continuation of Exercise 3-155.(a) What is the probability that you must call four times to obtain the first answer in less than 30 seconds! (b) What is the mean number of calls until you are answered in less than 30 seconds?
The probability that your call to a service line is an- swered in less than 30 seconds is 0.75. Assume that your calls are independent. (a) If you call 10 times, what is the probability that exactly nine of your calls are answered within 30 seconds? (b) If you call 20 times, what is the probability
Traffic flow is traditionally modeled as a Poisson dis- tribution. 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 of no cars
The probability that an eagle kills a jackrabbit in a day of hunting is 10%. Assume that results are independent between days. (a) What is the distribution of the number of days until a sac- cessful jackrabbit hunt? (b) What is the probability that the eagle must wait five days for its first
A particularly long traffic light on your morning commute is green 20% of the time that you approach it. Assume that each morning represents an independent trial. (a) What is the probability that the first morning that the light is green is the fourth morning that you approach it? (b) What is the
A congested computer network has a 1% chance of losing a data packet and packet losses are independent events. An e-mail message requires 100 packets. (a) What is the distribution of data packets that must be re- sent? Include the parameter values. (b) What is the probability that at least one
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 mecha- nism.
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 each egg load is an inde pendent event. (a) What is the distribution of cracked eggs per dozen? Include parameter values. (b) What are the
Batches that consist of 50 coil springs from a produc- tion process are checked for conformance to customer require ments. The mean number of nonconforming coil springs in a batch is five. Assume that the number of nonconforming springs in a batch, denoted as X, is a binomial random variable. (a)
Let X denote the number of bits received in error in a digital communication channel, and assume that X is a binomial random variable with p=0.001. If 1000 bits are transmitted, determine the following: (a) PX-1) (b) P(X1) (c) P(X 2) (d) mean and variance of X
The number of views of a page on a Web site follows a Poisson distribution with a mean of 1.5 per minute. am (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
The number of failures of a testing instrument from contamination particles on the product is a Poisson random variable with a mean of 0.02 failure per hour. (a) What is the probability that the instrument does not fail in an eight-hour shift? (b) What is the probability of at least one failure in
The number of surface flaws in plastic panels used in the interior of automobiles has a Poisson distribution with a mean of 0.05 Blaw per square foot of plastic panel. Assume an automobile interior contains 10 square feet of plastic panel.(a) What is the probability that there are no surface flaws
The number of cracks in a section of interstate highway that are significant enough to require repair is assumed to follow a Poisson distribution with a mean of two cracks per mile. (a) What is the probability that there are no cracks that require repair in 5 miles of highway? (b) What is the
The number of flaws in bolts of cloth in textile man- ufacturing is assumed to be Poisson distributed with a mean of 0.1 flaw per square meter. (a) What is the probability that there are two flaws in 1 square meter of cloth? (b) What is the probability that there is one flaw in 10 square meters of
Data from www.centralhudsonlabs 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 US. Food and Drug Administration-Center for Food Safety and Applied Nutrition for Defect Action Levels
The number of telephone calls that arrive at a phone exchange is often modeled as a Poisson random variable. Assume that on the average there are 10 calls per hour. (a) What is the probability that there are exactly five calls in one hour? (b) What is the probability that there are three or fewer
Suppose that the number of customers who enter a bank in an hour is a Poisson random variable, and suppose that P(X = 0) = 0.05. Determine the mean and variance of X.
Suppose X has a Poisson distribution with a mean of 0.4. Determine the following probabilities: (a) P(X 0) (c) PX 4) (b) P(X 2) (d) P(X = 8)
Suppose X has a Poisson distribution with a mean of 4.Determine the following probabilities: (a) P(X 0) (b) P(X = 2) (c) PX-4) (d) P(X-8)
Consider the non-failed wells in Exercises 3-31.Assume that four wells are selected randomly (without re- placement) 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
(a) Calculate the finite population corrections for Exercises 3-117 and 3-118.For which exercise should the binomial approximation to the distribution of X be better? (b) For Exercise 3-117, calculate P(X = 1) and P(X = 4) assuming that .X' has a binomial distribution and compare these results to
A state runs a lottery in which six numbers are randomly selected from 40, without replacement. A player chooses six numbers before the state's sample is selected. (a) What is the probability that the six numbers chosen by a player match all six numbers in the state's sample? (b) What is the
The analysis of results from a leaf transmutation experiment (turning a leaf into a petal) is summarized by type al of transformation completed: Total Textural Transformation Yes No Total Color Yes 243 26 Transformation No 13 18 A naturalist randomly selects three leaves from this set, without
Printed circuit cards are placed in a functional test after being populated with semiconductor chips. A lot contains 140 cards, and 20 are selected without replacement for func- tional testing (a) If 20 cards are defective, what is the probability that at least 1 defective card is in the sample?(b)
A company employs 800 men under the age of 55.Suppose that 30% carry a marker on the male chromosome that indicates an increased risk for high blood pressure. (a) If 10 men in the company are tested for the marker in this chromosome, what is the probability that exactly one man has the marker? (b)
A batch contains 36 bacteria cells and 12 of the cells are not capable of cellular replication. Suppose you examine three bacteria cells selected at random, without replacement. (a) What is the probability mass function of the number of cells in the sample that can replicate? (b) What are the mean
Suppose X has a hypergeometric distribution with N = 10, 3, and K = 4.Sketch the probability mass func- tion of X. Determine the cumulative distribution function for X.
Suppose X has a hypergeometric distribution with N=20, 4, and K-4. Determine the following: (a) P(X-1) (b) P(X = 4) (c) P(X 2) (d) Determine the mean and variance of X.
Suppose X has a hypergeometric distribution with N=100, 4, and K 20.Determine the following: (a) P(X = 1) (b) P(X6) (c) P(X=4) (d) Determine the mean and variance of X.
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 1.(a) What is the probability that the fifth visit is the first one to LWBS? (b) What is the probability that either the fifth
Consider the endothermic reactions in Exercise 3-28.Assume independent reactions are conducted. (a) What is the probability that the first reaction to result in a final temperature less than 272 K is the tenth reaction? (b) What is the mean number of reactions until the first final temperature is
Show that the probability density function of a nega tive binomial random variable equals the probability density function of a geometric random variable when r = 1.Show that the formulas for the mean and variance of a negative binomial random variable equal the corresponding results for a
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 the results from each phase are independent. (a) If the probability of a successful pass of each one of the
A fault-tolerant system that processes transactions for a financial services firm uses three separate computers. If the operating computer fails, one of the two spares can be im- mediately switched online. After the second computer fails. the last computer can be immediately switched online. Assume
Assume that 20 parts are checked each hour and that X denotes the number of parts in the sample of 20 that require rework. Parts are assumed to be independent with respect to rework. (a) If the percentage of parts that require rework remains at 1%, what is the probability that hour 10 is the first
A trading company has eight computers that it uses to trade on the New York Stock Exchange (NYSE). The probabil- ity of a computer failing in a day is 0.005, and the computers fail independently. Computers are repaired in the evening and each day is an independent trial. (a) What is the probability
A computer system uses passwords constructed from the 26 letters (a-2) or 10 integers (0-9). Suppose there are 10,000 users of the system with unique passwords. A hacker randomly selects (with replacement) passwords from the potential set (a) Suppose there are 9900 users with unique six-character.
Heart failure is due to either natural occurrences (87%) or outside factors (13%). Outside factors are related to induced substances or foreign objects. Natural occurrences are caused by arterial blockage, disease, and infection. Assume that causes of heart failure between individuals are
A player of a video game is confronted with a series of opponents and has an 80% probability of defeating each one. Success with any opponent is independent of previous encoun ters. The player continues to contest opponents until defeated. (a) What is the probability mass function of the number of
In a clinical study, volunteers are tested for a gene that has been found to increase the risk for a disease. The probability that a person carries the gene is 0.1. (a) What is the probability four or more people will have to be tested before two with the gene are detected? (b) How many people are
Suppose that X' is a negative binomial random vari- able with p = 0.2 and r = 4.Determine the following: (a) E(X) (c) P(X = 19) (b) P(X=20) (d) P(X = 21) (e) The most likely value for X
Suppose the random variable X has a geometric distribu tion with p=0.5. Determine the following probabilities: (a) P(X) (b) P(X = 4) (c) P(X 8) (d) P(X 2) (e) P(X2)
Consider the endothermic reactions in Exercise 3-28.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 a Web site changes its content according to the distribution in Exercise 3-30.Assume 10 changes are made independently. (a) What is the probability that the change is made in less than 4 days in seven of the 10 updates?(b) What is the probability that the change is made in less than 4 days
Consider the visits that result in leave without being seen (LWBS) at an emergency department in Example 2-8.Assume that four persons independently arrive for service at Hospital 1.(a) What is the probability that exactly one person will LWBS? (b) What is the probability, that two or more two
Consider the lengths of stay at a hospital's emergency department in Exercise 3-29.Assume that five persons inde- pendently 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
A statistical process control chart example. Samples of 20 parts from a metal punching process are selected every hour. Typically, 1% of the parts require rework. Let X denote the number of parts in the sample of 20 that require rework. A process problem is suspected if X exceeds its mean by more
A computer system uses passwords that are exactly six characters and each character is one of the 26 letters (a-2) or 10 integers (0-9). Suppose there are 10,000 users of the system with unique passwords. A hacker randomly selects (with replacement) one billion passwords from the potential set, and
Heart failure is due to either natural occurrences (87%) or outside factors (13%). Outside factors are related to induced substances or foreign objects. Natural occurrences are caused by arterial blockage, disease, and infection. Suppose that 20 patients will visit an emergency room with heart
An article in Information Security Technical Report ["Malicious Software Past, Present and Future" (2004, Vol. 9, pp. 6-18)] provided the following data on the top ten mali- cious software instances for 2002. The clear leader in the num- ber of registered incidences for the year 2002 was the
Samples of rejuvenated mitochondria are mutated (defective) in 1% of cases. Suppose 15 samples are studied,and they 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
A particularly long traffic light on your morning com- mute is green 20% of the time that you approach it. Assume that each morning represents an independent trial. (a) Over five mornings, what is the probability that the light is green on exactly one day? (b) Over 20 momings, what is the
A multiple-choice test contains 25 questions, each with four answers. Assume a student just guesses on each question. (a) What is the probability that the student answers more than 20 questions correctly? (b) What is the probability the student answers less than five questions correctly?
Determine the cumulative distribution function of a binomial random variable with = 3 and p = 1/4.
Determine the cumulative distribution function of a binomial random variable with 3 and p = 1/2
Sketch the probability mass function of a binomial distribution with = 10 and p = 0.01 and comment on the shape of the distribution. (a) What value of Xis most likely? (b) What value of X is least likely?
The random variable X has a binomial distribution with = 10 and p = 0.5. Sketch the probability mass function of .X. (a) What value of X' is most likely? (b) What value(s) of X is(are) least likely?
The random variable X has a binomial distribution 10 and p = 0.01. Determine the following proba- with bilities. (a) P(X = 5) (c) P(X = 9) (b) P(X2) (d) P(3X
The random variable X has a binomial distribution with 10 and p=0.5. Determine the following proba- bilities: (a) P(X5) (c) P(X9) (b) P(X2) (d) P(3X
Let X be a binomial random variable with p = 0.1 and n = 10.Calculate the following probabilities from the binomial probability mass function and also from the binomial table in Appendix A and compare results. (a) P(X2) (c) P(X = 4) (b) P(X>8) (d) P(5x7)
Let X be a binomial random variable with p = 0.2 and = 20.Use the binomial table in Appendix A to deter- mine the following probabilities. (a) P(X3) (b) P(X>10) (c) P(X = 6) (d) P(6x11)
For each scenario described below, state whether or not the binomial distribution is a reasonable model for the random variable and why. State any assumptions you make. (a) A production process produces thousands of temperature transducers. Let X denote the number of nonconforming transducers in a
Show that for a discrete uniform random variable .X, if each of the values in the range of X is multiplied by the constantc, the effect is to multiply the mean of X by c and the variance of X by . That is, show that E(X) = E(X) and ex)=x
Assume that the wavelengths of photosynthetically active radiations (PAR) are uniformly distributed at integer nanometers in the red spectrum from 675 to 700 m. (a) What is the mean and variance of the wavelength distribu- tion for this radiation? (b) If the wavelengths are uniformly distributed at
The lengths of plate glass parts are measured to the nearest tenth of a millimeter. The lengths are uniformly dis- tributed, with values at every tenth of a millimeter starting at 590.0 and continuing through 590.9. Determine the mean and variance of the lengths.
Product codes of two, three, four, or five letters are equally likely. What is the mean and standard deviation of the number of letters in the codes?
Let the random variable X have a discrete uniform distribution on the integers 15x3. Determine the mean and variance of .X.
Calculate the mean and variance for the random variable in Exercise 3-31.
Calculate the mean and variance for the random variable in Exercise 3.30.
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