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Applied Statistics And Probability For Engineers 6th Edition Douglas C. Montgomery, George C. Runger - Solutions
A manufacturing operations consists of 10 operations. However, five machining operations must be completed before any of the remaining five assembly operations can begin. Within each set of five, operations can be completed in any order. How many different production sequences are possible?
In a sheet metal operation, three notches and four bends are required. If the operations can be done in any order, how many different ways of completing the manufacturing are possible?
A lot of 140 semiconductor chips is inspected by choosing a sample of five chips. Assume 10 of the chips do not conform to customer requirements. (a) How many different samples are possible? (b) How many samples of five contain exactly one nonconforming chip? (c) How many samples of five contain
In the layout of a printed circuit board for an electronic product, there are 12 different locations that can accommodate chips. (a) If five different types of chips are to be placed on the board, how many different layouts are possible? (b) If the five chips that are placed on the board are of
In the laboratory analysis of samples from a chemical process, five samples from the process are analyzed daily. In addition, a control sample is analyzed two times each day to check the calibration of the laboratory instruments. (a) How many different sequences of process and control samples are
In the design of an electromechanical product, seven different components are to be stacked into a cylindrical casing that holds 12 components in a manner that minimizes the impact of shocks. One end of the casing is designated as the bottom and the other end is the top. (a) How many different
The design of a communication system considered the following questions: (a) How many three-digit phone prefixes that are used to represent a particular geographic area (such as an area code) can be created from the digits 0 through 9? (b) As in part (a), how many three-digit phone prefixes are
A byte is a sequence of eight bits and each bit is either 0 or 1. (a) How many different bytes are possible? (b) If the first bit of a byte is a parity check, that is, the first byte is determined from the other seven bits, how many different bytes are possible?
In a chemical plant, 24 holding tanks are used for final product storage. Four tanks are selected at random and without replacement. Suppose that six of the tanks contain material in which the viscosity exceeds the customer requirements. (a) What is the probability that exactly one tank in the
Plastic parts produced by an injection-molding operation are checked for conformance to specifications. Each tool contains 12 cavities in which parts are produced, and these parts fall into a conveyor when the press opens. An inspector chooses 3 parts from among the 12 at random. Two cavities are
A bin of 50 parts contains five that are defective. A sample of two is selected at random, without replacement. (a) Determine the probability that both parts in the sample are defective by computing a conditional probability. (b) Determine the answer to part (a) by using the subset approach
The random variable is the number of nonconforming solder connections on a printed circuit board with 1000 connections.
In a voice communication system with 50 lines, the random variable is the number of lines in use at a particular time.
An electronic scale that displays weights to the nearest pound is used to weigh packages. The display shows only five digits. Any weight greater than the display can indicate is shown as 99999. The random variable is the displayed weight.
A batch of 500 machined parts contains 10 that do not conform to customer requirements. The random variable is the number of parts in a sample of 5 parts that do not conform to customer requirements.
A batch of 500 machined parts contains 10 that do not conform to customer requirements. Parts are selected successively, without replacement, until a nonconforming part is obtained. The random variable is the number of parts selected.
The random variable is the moisture content of a lot of raw material, measured to the nearest percentage point.
The random variable is the number of surface flaws in a large coil of galvanized steel.
The random variable is the number of computer clock cycles required to complete a selected arithmetic calculation.
An order for an automobile can select the base model or add any number of 15 options. The random variable is the number of options selected in an order.
A group of 10,000 people are tested for a gene called Ifi202 that has been found to increase the risk for lupus. The random variable is the number of people who carry the gene.
Use the probability mass function in Exercise 3-11 to determine the following probabilities: (a) P (X < 2) (b) P (0.5 < X < 2.7) (c) P (X > 3) (d) P (0 < X < 2) (e) P (X = 0 or X = 2) Verify that the following functions are probability mass functions, and determine the requested probabilities.
f(x) = (3/4) (1/4)x , x = 0,1,2,... (a) P (X = 2) (b) P (X < 2) (c) P (X >2) (d) P (X > 1)
Marketing estimates that a new instrument for the analysis of soil samples will be very successful, moderately successful, or unsuccessful, with probabilities 0.3, 0.6, and 0.1, respectively. The yearly revenue associated with a very successful, moderately successful, or unsuccessful product is $10
A disk drive manufacturer estimates that in five years a storage device with 1 terabyte of capacity will sell with probability 0.5, a storage device with 500 gigabytes capacity will sell with a probability 0.3, and a storage device with 100 gigabytes capacity will sell with probability 0.2. The
An optical inspection system is to distinguish among different part types. The probability of a correct classification of any part is 0.98. Suppose that three parts are inspected and that the classifications are independent. Let the random variable X denote the number of parts that are correctly
In a semiconductor manufacturing process, three wafers from a lot are tested. Each wafer is classified as pass or fail. Assume that the probability that a wafer passes the test is 0.8 and that wafers are independent. Determine the probability mass function of the number of wafers from a lot that
The distributor of a machine for cytogenics has developed a new model. The company estimates that when it is introduced into the market, it will be very successful with a probability 0.6, moderately successful with a probability 0.3, and not successful with probability 0.1. The estimated yearly
An assembly consists of two mechanical components. Suppose that the probabilities that the first and second components meet specifications are 0.95 and 0.98. Assume that the components are independent. Determine the probability mass function of the number of components in the assembly that meet
An assembly consists of three mechanical components. Suppose that the probabilities that the first, second, and third components meet specifications are 0.95, 0.98, and 0.99. Assume that the components are independent. Determine the probability mass function of the number of components in the
Determine the cumulative distribution function of the random variable in Exercise 3-13.
Determine the cumulative distribution function for the random variable in Exercise 3-15; also determine the following probabilities: (a) P(X < 1.25) (b) P (X < 2.2) (c) P (-1.1 < X < 1) (d) P (X > 0)
Determine the cumulative distribution function for the random variable in Exercise 3-17; also determine the following probabilities: (a) P(X < 1.5) (b) P (X < 3) (c) P (X > 2) (d) P (1 < X < 2)
Determine the cumulative distribution function for the random variable in Exercise 3-19.
Determine the cumulative distribution function for the random variable in Exercise 3-20.
Determine the cumulative distribution function for the random variable in Exercise 3-22.
Determine the cumulative distribution function for the variable in Exercise 3-23. Verify that the following functions are cumulative distribution functions, and determine the probability mass function and the requested probabilities.
Errors in an experimental transmission channel are found when the transmission is checked by a certifier that detects missing pulses. The number of errors found in an eight bit byte is a random variable with the following distribution:
The thickness of wood paneling (in inches) that a customer orders is a random variable with the following cumulative distribution function:
If the range of X is the set {0, 1, 2, 3, 4} and P(X =x) =0.2 determine the mean and variance of the random variable.
Determine the mean and variance of the random variable in Exercise 3-13.
Determine the mean and variance of the random variable in Exercise 3-15.
Determine the mean and variance of the random variable in Exercise 3-17.
Determine the mean and variance of the random variable in Exercise 3-19.
Determine the mean and variance of the random variable in Exercise 3-20
Determine the mean and variance of the random variable in Exercise 3-22.
Determine the mean and variance of the random variable in Exercise 3-23.
The range of the random variable X is [0, 1, 2, 3, x], where x is unknown. If each value is equally likely and the mean of X is 6, determine x.
Let the random variable X have a discrete uniform distribution on the integers. Calculate the mean and variance of X.
Let the random variable X have a discrete uniform distribution on the integers. Determine the mean and variance of X.
Let the random variable X be equally likely to assume any of the values 1/8, 1/4, or 3/8. Determine the mean and variance of X.
Thickness measurements of a coating process are made to the nearest hundredth of a millimeter. The thickness measurements are uniformly distributed with values 0.15, 0.16, 0.17, 0.18, and 0.19. Determine the mean and variance of the coating thickness for this process.
Product codes of 2, 3, or 4 letters are equally likely. What is the mean and standard deviation of the number of letters in 100 codes?
The lengths of plate glass parts are measured to the nearest tenth of a millimeter. The lengths are uniformly distributed, with values at every tenth of a millimeter starting at 590.0 and continuing through 590.9. Determine the mean and variance of lengths.
Suppose that X has a discrete uniform distribution on the integers 0 through 9. Determine the mean, variance, and standard deviation of the random variable Y _ 5X and compare to the corresponding results for X.
Show that for a discrete uniform random variable X, if each of the values in the range of X is multiplied by the constant c, the effect is to multiply the mean of X by c and the variance of X by c2. That is, show that (cX) = cE (X) and V(cX) = c2V(X).
The probability of an operator entering alphanumeric data incorrectly into a field in a database is equally likely. The random variable X is the number of fields on a data entry form with an error. The data entry form has 28 fields. Is X a discrete uniform random variable? Why or why not.
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
The random variable X has a binomial distribution with n = 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 with n = 10 and p = 0.5. Determine the following probabilities: (a) P(X =5) (b) P(X < 2) (c) P(X > 9) (d) P(3 < X
Sketch the probability mass function of a binomial distribution with n = 10 and p = 0.01 and comment on the shape of the distribution. (a) What value of X is most likely? (b) What value of X is least likely?
The random variable X has a binomial distribution with n = 10 and p = 0.01. Determine the following probabilities. (a) P(X =5) (b) P(X < 2) (c) P(X > 9) (d) P(3 < X
Determine the cumulative distribution function of a binomial random variable with n = 3 and p = ½.
Determine the cumulative distribution function of a binomial random variable with n = 3 and p = ¼
An electronic product contains 40 integrated circuits. The probability that any integrated circuit is defective is 0.01, and the integrated circuits are independent. The product operates only if there are no defective integrated circuits. What is the probability that the product operates?
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) P(X =5) (b) P(X < 2) (c) P(X > 9) (d) mean and variance of X
The phone lines to an airline reservation system are occupied 40% of the time. Assume that the events that the lines are occupied on successive calls are independent. Assume that 10 calls are placed to the airline. (a) What is the probability that for exactly three calls the lines are
Batches that consist of 50 coil springs from a production process are checked for conformance to customer requirements. The mean number of nonconforming coil springs in a batch is 5. Assume that the number of nonconforming springs in a batch, denoted as X, is a binomial random variable. (a) What
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 than three standard deviations. (a) If the
Because not all airline passengers show up for their reserved seat, an airline sells 125 tickets for a flight that holds only 120 passengers. The probability that a passenger does not show up is 0.10, and the passengers behave independently. (a) What is the probability that every passenger who
This exercise illustrates that poor quality can affect schedules and costs. A manufacturing process has 100 customer orders to fill. Each order requires one component part that is purchased from a supplier. However, typically, 2% of the components are identified as defective, and the components can
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 5 questions correctly?
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) Over five mornings, what is the probability that the light is green on exactly one day? (b) Over 20 mornings, what is the
Suppose the random variable X has a geometric distribution with p _ 0.5. Determine the following probabilities: (a) P(X = 1) (b) P(X =4) (c) P(X =8) (d) P(X < 2) (e) P(X =2)
Suppose the random variable X has a geometric distribution with a mean of 2.5. Determine the following probabilities: (a) P(X = 1) (b) P(X =4) (c) P(X =5) (d) P(X < 3) (e) P(X =3)
The probability of a successful optical alignment in the assembly of an optical data storage product is 0.8. Assume the trials are independent. (a) What is the probability that the first successful alignment requires exactly four trials? (b) What is the probability that the first successful
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 4 or more people will have to be tested before 2 with the gene are detected? (b) How many people are
Assume that each of your calls to a popular radio station has a probability of 0.02 of connecting, that is, of not obtaining a busy signal. Assume that your calls are independent. (a) What is the probability that your first call that connects is your tenth call? (b) What is the probability that
In Exercise 3-70, recall that 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
A trading company has eight computers that it uses to trade on the New York Stock Exchange (NYSE). The probability 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
In Exercise 3-66, recall that 20 parts are checked each hour and that X denotes the number of parts in the sample of 20 that require rework. (a) If the percentage of parts that require rework remains at 1%, what is the probability that hour 10 is the first sample at which X exceeds 1? (b) If the
Consider a sequence of independent Bernoulli trials with p _ 0.2. (a) What is the expected number of trials to obtain the first success? (b) After the eighth success occurs, what is the expected number of trials to obtain the ninth success?
Show that the probability density function of a negative 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 geometric
Suppose that X is a negative binomial random variable with p _ 0.2 and r _ 4. Determine the following: (a) E(X1) (b) P(X =20) (c) P(X =19) (d) P(X =21) (e) The most likely value for X
The probability is 0.6 that a calibration of a transducer in an electronic instrument conforms to specifications for the measurement system. Assume the calibration attempts are independent. What is the probability that at most three calibration attempts are required to meet the specifications for
An electronic scale in an automated filling operation stops the manufacturing line after three underweight packages are detected. Suppose that the probability of an underweight package is 0.001 and each fill is independent. (a) What is the mean number of fills before the line is stopped? (b) What
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 immediately switched online. After the second computer fails, the last computer can be immediately switched online. Assume
Derive the expressions for the mean and variance of a geometric random variable with parameter p. (Formulas for infinite series are required.)
Suppose X has a hypergeometric distribution with N = 100, n = 4, and K = 20. Determine the following: (a) P(X = 1) (b) P(X =6) (c) P(X =4) (d) Determine the mean and variance of X.
Suppose X has a hypergeometric distribution with N = 20, n = 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 = 10, n = 3, and K = 4. Sketch the probability mass function of X.
Determine the cumulative distribution function for X in Exercise 3-88.
A lot of 75 washers contains 5 in which the variability in thickness around the circumference of the washer is unacceptable. A sample of 10 washers is selected at random, without replacement. (a) What is the probability that none of the unacceptable washers is in the sample? (b) What is the
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 1 man has the marker? (b)
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 functional testing. (a) If 20 cards are defective, what is the probability that at least 1 defective card is in the sample?
Magnetic tape is slit into half-inch widths that are wound into cartridges. A slitter assembly contains 48 blades. Five blades are selected at random and evaluated each day for sharpness. If any dull blade is found, the assembly is replaced with a newly sharpened set of blades. (a) If 10 of the
A state runs a lottery in which 6 numbers are randomly selected from 40, without replacement. A player chooses 6 numbers before the state’s sample is selected. (a) What is the probability that the 6 numbers chosen by a player match all 6 numbers in the state’s sample? (b) What is the
Continuation of Exercises 3-86 and 3-87. (a) Calculate the finite population corrections for Exercises 3-86 and 3-87. For which exercise should the binomial approximation to the distribution of X be better? (b) For Exercise 3-86, calculate P(X = 1) and P(X = 4) assuming that X has a binomial
Use the binomial approximation to the hypergeometric distribution to approximate the probabilities in Exercise 3-92. What is the finite population correction in this exercise?
Suppose X has a Poisson distribution with a mean 4. Determine the following probabilities: (a) P(X = 0) (b) P(X < 2) (c) P(X = 4) (d) P(X = 8)
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