Managing Controlling and Improving Quality 1st edition Douglas C. Montgomery, Cheryl L. Jennings, Michele E. Pfund - Solutions

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Assume X is normally distributed with a mean of 10 and a standard deviation of 2. Determine the following:(a) P(X < 13) (b) P(X > 9)(c) P(6 < X < 14) (d) P(2 < X < 4)(e) P(-2 < X < 8)
Assume X is normally distributed with a mean of 10 and a standard deviation of 2. Determine the value for x that solves each of the following:(a) P(X > x) = 0.5(b) P(X > x) = 0.95(c) P(x < X < 10) = 0.2(d) P(-x < X – 10 < x) = 0.95(e) P(-x < X – 10 < x) = 0.99
Assume X is normally distributed with a mean of 5 and a standard deviation of 4. Determine the following: (a) P(X < 11) (b) P(X > 0)(c) P(3 < X < 7) (d) P(-2 < X < 9)(e) P(-2 < X < 8)
Assume X is normally distributed with a mean of 5 and a standard deviation of 4. Determine the value for x that solves each of the following:(a) P(X > x) = 0.5 (b) P(X > x) = 0.95(c) P(x < X < 9) = 0.2 (d) P(3 < X < x) = 0.95(e) P(-x < X < x) = 0.99
The compressive strength of samples of cement can be modeled by a normal distribution with a mean of 6000 kilograms per square centimeter and a standard deviation of 100 kilograms per square centimeter.(a) What is the probability that a sample’s strength is less than 6250 Kg/cm2?(b) What is the
The tensile strength of paper is modeled by a normal distribution with a mean of 35 pounds per square inch and a standard deviation of 2 pounds per square inch.(a) What is the probability that the strength of a sample is less than 40 lb/in2?(b) If the specifications require the tensile strength to
The line width of for semiconductor manufacturing is assumed to be normally distributed with a mean of 0.5 micrometer and a standard deviation of 0.05 micrometer.(a) What is the probability that a line width is greater than 0.62 micrometer?(b) What is the probability that a line width is between
The fill volume of an automated filling machine used for filling cans of carbonated beverage is normally distributed with a mean of 12.4 fluid ounces and a standard deviation of 0.1 fluid ounce.(a) What is the probability a fill volume is less than 12 fluid ounces?(b) If all cans less than 12.1 or
The time it takes a cell to divide (called mitosis) is normally distributed with an average time of one hour and a standard deviation of 5 minutes.(a) What is the probability that a cell divides in less than 45 minutes?(b) What is the probability that it takes a cell more than 65 minutes to
In the previous exercise, suppose that the mean of the filling operation can be adjusted easily, but the standard deviation remains at 0.1 ounce.(a) At what value should the mean be set so that 99.9% of all cans exceed 12 ounces?(b) At what value should the mean be set so that 99.9% of all cans
The reaction time of a driver to visual stimulus is normally distributed with a mean of 0.4 seconds and a standard deviation of 0.05 seconds.(a) What is the probability that a reaction requires more than 0.5 seconds?(b) What is the probability that a reaction requires between 0.4 and 0.5
The speed of a file transfer from a server on campus to a personal computer at a student’s home on a weekday evening is normally distributed with a mean of 60 kilobits per second and a standard deviation of 4 kilobits per second.(a) What is the probability that the file will transfer at a speed
The length of an injection-molded plastic case that holds magnetic tape is normally distributed with a length of 90.2 millimeters and a standard deviation of 0.1 millimeter.(a) What is the probability that a part is longer than 90.3 millimeters or shorter than 89.7 millimeters?(b) What should the
In the previous exercise assume that the process is centered so that the mean is 90 millimeters and the standard deviation is 0.1 millimeter. Suppose that 10 cases are measured, and they are assumed to be independent.(a) What is the probability that all 10 cases are between 89.7 and 90.3
The sick-leave time of employees in a firm in a month is normally distributed with a mean of 100 hours and a standard deviation of 20 hours.(a) What is the probability that the sick-leave time for next month will be between 50 and 80 hours?(b) How much time should be budgeted for sick leave if the
The life of a semiconductor laser at a constant power is normally distributed with a mean of 7000 hours and a standard deviation of 600 hours.(a) What is the probability that a laser fails before 5000 hours?(b) What is the life in hours that 95% of the lasers exceed?(c) If three lasers are used in
The diameter of the dot produced by a printer is normally distributed with a mean diameter of 0.002 inch and a standard deviation of 0.0004 inch.(a) What is the probability that the diameter of a dot exceeds 0.0026 inch?(b) What is the probability that a diameter is between 0.0014 and 0.0026
The weight of a sophisticated running shoe is normally distributed with a mean of 12 ounces and a standard deviation of 0.5 ounce.(a) What is the probability that a shoe weighs more than 13 ounces?(b) What must the standard deviation of weight be in order for the company to state that 99.9% of its
Suppose that X is a binomial random variable with n = 200 and p = 0.4.(a) Approximate the probability that X is less than or equal to 70.(b) Approximate the probability that X is greater than 70 and less than 90.
Suppose that X is a binomial random variable with n = 100 and p = 0.1.(a) Compute the exact probability that X is less than 4.(b) Approximate the probability that X is less than 4 and compare to the result in part (a).(c) Approximate the probability that 8 < X < 12.
The manufacturing of semiconductor chips produces 2% defective chips. Assume the chips are independent and that a lot contains 1000 chips.(a) Approximate the probability that more than 25 chips are defective.(b) Approximate the probability that between 20 and 30 chips are defective.
A supplier ships a lot of 1000 electrical connectors. A sample of 25 is selected at random, without replacement. Assume the lot contains 100 defective connectors.(a) Using a binomial approximation, what is the probability that there are no defective connectors in the sample?(b) Use the normal
An electronic office product contains 5000 electronic components. Assume that the probability that each component operates without failure during the useful life of the product is 0.999, and assume that the components fail independently. Approximate the probability that 10 or more of the original
Suppose that the number of asbestos particles in a sample of 1 squared centimeter of dust is a Poisson random variable with a mean of 1000. What is the probability that 10 squared centimeters of dust contains more than 10,000 particles?
A corporate Web site contains errors on 50 of 1000 pages. If 100 pages are sampled randomly, without replacement, approximate the probability that at least 1 of the pages in error are in the sample
Hits to a high-volume Web site are assumed to follow Poisson distribution with a mean of 10,000 per day. Approximate each of the following:(a) The probability of more than 20,000 hits in a day(b) The probability of less than 9900 hits in a day(c) The value such that the probability that the number
Continuation of Exercise 4-68.(a) Approximate the expected number of days in a year (365 days) that exceed 10,200 hits.(b) Approximate the probability that over a year (365 days) more than 15 days each have more than 10,200 hits.
The percentage of people exposed to a bacteria who become ill is 20%. Assume that people are independent. Assume that 1000 people are exposed to the bacteria. Approximate each of the following:(a) The probability that more than 225 become ill(b) The probability that between 175 and 225 become
A high-volume printer produces minor print-quality errors on a test pattern of 1000 pages of text according to a Poisson distribution with a mean of 0.4 per page.(a) Why is the number of errors on each page independent random variables?(b) What is the mean number of pages with errors (one or
Suppose X has an exponential distribution with λ = 2. Determine the following: (a) P(x < 0) (b) P(X > 2) (c) P(X < 1) (d) P(1) < X < 2) (e) Find the value of x such that P(X < x) = 0.05.
Suppose X has an exponential distribution with mean equal to 10.Determine following:(a) P(X > 10)(b) P(X > 20)(c) P(X > 30)
Suppose the counts recorded by a Geiger counter follow a Poisson process with an average of two counts per minute. (a) What is the probability that there are no counts in a 30- second interval?(b) What is the probability that the first count occurs in less than 10 seconds?(c) What is the
Suppose that the logons to a computer network follow a Poisson process with an average of 3 counts per minute.(a) What is the mean time between counts?(b) What is the standard deviation of the time between counts?(c) Determine x such that the probability that at least one count occurs before time x
The time to failure (in hours) for a laser in a cytometry machine is modeled by an exponential distribution with λ = 0.00004. (a) What is the probability that the laser will last at least 20,000 hours? (b) What is the probability that the laser will last at most 30,000 hours? (c) What is
The time between calls to a plumbing supply business is exponentially distributed with a mean time between calls of 15 minutes.(a) What is the probability that there are no calls within a 30- minute interval? (b) What is the probability that at least one call arrives within a 10-minute interval?(c)
The life of automobile voltage regulators has an exponential distribution with a mean life of six years. You purchase an automobile that is six years old, with a working voltage regulator, and plan to own it for six years.(a) What is the probability that the voltage regulator fails during your
The time to failure (in hours) of fans in a personal computer can be modeled by an exponential distribution with(a) What proportion of the fans will last at least 10,000 hours?(b) What proportion of the fans will last at most 7000 hours?
The time between the arrival of electronic messages at your computer is exponentially distributed with a mean of two hours.(a) What is the probability that you do not receive a message during a two-hour period?(b) If you have not had a message in the last four hours, what is the probability that
The time between arrivals of taxis at a busy intersection is exponentially distributed with a mean of 10 minutes.(a) What is the probability that you wait longer than one hour for a taxi?(b) Suppose you have already been waiting for one hour for a taxi, what is the probability that one arrives
Continuation of Exercise 4-81. (a) Determine x such that the probability that you wait more than x minutes is 0.10.(b) Determine x such that the probability that you wait less than x minutes is 0.90.(c) Determine x such that the probability that you wait less than x minutes is 0.50.
The distance between major cracks in a highway follows an exponential distribution with a mean of 5 miles.(a) What is the probability that there are no major cracks in a 10-mile stretch of the highway?(b) What is the probability that there are two major cracks in a 10-mile stretch of the
Continuation of Exercise 4-83.(a) What is the probability that the first major crack occurs between 12 and 15 miles of the start of inspection?(b) What is the probability that there are no major cracks in two separate 5-mile stretches of the highway?(c) Given that there are no cracks in the first 5
The lifetime of a mechanical assembly in a vibration test is exponentially distributed with a mean of 400 hours.(a) What is the probability that an assembly on test fails in less than 100 hours?(b) What is the probability that an assembly operates for more than 500 hours before failure?(c) If an
Continuation of Exercise 4-85.(a) If 10 assemblies are tested, what is the probability that at least one fails in less than 100 hours? Assume that the assemblies fail independently.(b) If 10 assemblies are tested, what is the probability that all have failed by 800 hours? Assume the assemblies fail
When a bus service reduces fares, a particular trip from New York City to Albany, New York, is very popular. A small bus can carry four passengers. The time between calls for tickets is exponentially distributed with a mean of 30 minutes. Assume that each call orders one ticket. What is the
The time between arrivals of small aircraft at a county airport is exponentially distributed with a mean of one hour. What is the probability that more than three aircraft arrive within an hour?
Continuation of Exercise 4-88.(a) If 30 separate one-hour intervals are chosen, what is the probability that no interval contains more than three arrivals? (b) Determine the length of an interval of time (in hours) such that the probability that no arrivals occur during the interval is 0.10.
The time between calls to a corporate office is exponentially distributed with a mean of 10 minutes.(a) What is the probability that there are more than three calls in one-half hour?(b) What is the probability that there are no calls within one-half hour?(c) Determine x such that the probability
Continuation of Exercise 4-90.(a) What is the probability that there are no calls within a two-hour interval?(b) If four non-overlapping one-half hour intervals are selected, what is the probability that none of these intervals contains any call?(c) Explain the relationship between the results in
If the random variable X has and exponential distribution with mean θ determine the following: (a) P(X > θ) (b) P(X > 2θ) (c) P(X > 3θ) (d) How do the results depend on θ?
Assume that the flaws along a magnetic tape follow a Poisson distribution with a mean of 0.2 flaws per meter. Let X denotes the distance between two successive flaws.(a) What is the mean of X?(b) What is the probability that there are no flaws in 10 consecutive meters of tape?(c) Does your answer
Continuation of Exercise 4-93. (More difficult questions.)(a) What is the probability that the first time the distance between two flaws exceeds 8 meters is at the fifth flaw?(b) What is the mean number of flaws before a distance between two flaws exceeds 8 meters?
Derive the formula for the mean and variance of an exponential random variable.
Calls to a telephone system follow a Poisson distribution with a mean of five calls per minute.(a) What is the name applied to the distribution and parameter values of the time until the tenth call?(b) What is the mean time until the tenth call?(c) What is the mean time between the ninth and tenth
Continuation of Exercise 4-96.(a) What is the probability that exactly four calls occur within one minute?(b) If 10 separate one-minute intervals are chosen, what is the probability that all intervals contain more than two calls?
Raw materials are studied for contamination. Suppose that the number of particles of contamination per pound of material is a Poisson random variable with a mean of 0.01 particle per pound.(a) What is the expected number of pounds of material required to obtain 15 particles of contamination?(b)
The time between failures of a laser in a cytogenesis machine is exponentially distributed with a mean of 25,000 hours.(a) What is the expected time until the second failure?(b) What is the probability that the time until the third failure exceeds 50,000 hours?
In a data communication system, several messages that arrive at a node are bundled into a packet before they are transmitted over the network. Assume the messages arrive at the node according to a Poisson process with r = 30 messages per minute. Five messages are used to form a packet.(a) What is
Errors caused by contamination on optical disks occur at the rate of one error every 105 bits. Assume the errors follow a Poisson distribution.(a) What is the mean number of bits until five errors occur?(b) What is the standard deviation of the number of bits until five errors occur?(c) The
Calls to the help line of a large computer distributor follow a Passion distribution with a mean of 20 calls per minute. (a) What is the mean time until the one-hundredth calls?(b) What is the mean time between call numbers 50 and 80?(c) What is the probability that three or more calls occur within
The time between arrivals of customers at an automatic teller machine is an exponential random variable with a mean of 5 minutes.(a) What is the probability that more than three customers arrive in 10 minutes?(b) What is the probability that the time until the fifth customer arrives is less than 15
The time between process problems in a manufacturing line is exponentially distributed with a mean of 30 days. (a) What is the expected time until the fourth problem?(b) What is the probability that the time until the fourth problem exceeds 120 days?
Use the properties of the gamma function to evaluate the following: (a) Г(6) (b) Г(5/2) (c) Г(9/2)
Use integration by parts to show that Г(r) = (r – 1) Г(r – 1).
Show that the gamma density function integrates to f(x, λ, r) integrates to 1.
Use the result for the gamma distribution to determine the mean and variance of a chi-square distribution with r = 7/2.
Suppose that X has a Weibull distribution with β = 0.2 and δ = 100 hours. Determine the mean and variance of X.
Suppose that X has a Weibull distribution β = 0.2 and δ = 100 hours. Determine that following: (a) P(X < 10,000) (b) P(X > 5000)
Assume that the life of a roller bearing follows a Weibull distribution with parameters β = 2 and δ = 10,000 hours. (a) Determine the probability that a bearing lasts at least 8000 hours. (b) Determine the mean time until failure of a bearing. (c) If 10 bearings are in use and failures
The life (in hours) of a computer processing unit (CPU) is modeled by a Weibull distribution with parameters β = 3 and δ = 900 hours. (a) Determine the mean life of the CPU. (b) Determine the variance of the life of the CPU. (c) What is the probability that the CPU fails before 500
Assume the life of a packaged magnetic disk exposed to corrosive gases has a Weibull distribution with β = 0.5 and the mean life is 600 hours. (a) Determine the probability that a packaged disk lasts at least 500 hours. (b) Determine the probability that a packaged disk fails before 400
The life of a recirculating pump follows a Weibull distribution with parameters β = 2, and δ = 700 hours. (a) Determine the mean life of a pump. (b) Determine the variance of the life of a pump. (c) What is the probability that a pump will last longer than its mean?
The life (in hours) of a magnetic resonance imagining machine (MRI) is modeled by a Weibull distribution with parameters β = 2 and δ = 500 hours. (a) Determine the mean life of the MRI. (b) Determine the variance of the life of the MRI. (c) What is the probability that the MRI fails
If X is a Weibull random variable with β = 1, and δ = 1000, what is another name for the distribution of X and what is the mean of X?
Suppose that X has a lognormal distribution with parameters θ = 5 and w2 = 9. Determine the following: (a) P(X < 13, 300) (b) The value for x such that (c) The mean and variance of X
Suppose that X has a lognormal distribution with parameters θ = - 2 and w2. Determine the following: (a) P(500 < X < 1000) (b) The value for x such that (c) The mean and variance of X
Suppose that X has a lognormal distribution with parameters θ = 2 and w2. Determine the following: (a) P(X < 500) (b) The conditional probability that given that (c) What does the difference between the probabilities in parts (a) and (b) imply bout lifetimes of lognormal random variables?
The length of time (in seconds) that a user views a page on a Web site before moving to another page is a lognormal random variable with parameters θ = 2.5 and w2 = 1. (a) What is the probability that a page is viewed for more than 10 seconds? (b) What is the length of time that 50% of
Suppose that X has a lognormal distribution and that the mean and variance of X are 100 and 85,000, respectively. Determine the parameters and of the lognormal distribution. (Hint: define x = exp (θ) and y = exp (w2) and write two equations in terms of x and y.)
The lifetime of a semiconductor laser has a lognormal distribution, and it is known that the mean and standard deviation of lifetime are 10,000 and 20,000, respectively.(a) Calculate the parameters of the lognormal distribution(b) Determine the probability that a lifetime exceeds 10,000 hours(c)
Derive the probability density function of a lognormal random variable from the derivative of the cumulative distribution function.
Suppose that f(x) = 0.5 – 1 for 2 < x < 4. Determine the following:(a) P(X < 2.5)(b) P(X > 3)(c) P(2.5 < X < 3.5)
Continuation of Exercise 4-124. Determine the cumulative distribution function of the random variable.
Continuation of Exercise 4-124. Determine the mean and variance of the random variable.
The time between calls is exponentially distributed with a mean time between calls of 10 minutes.(a) What is the probability that the time until the first call is less than 5 minutes?(b) What is the probability that the time until the first call is between 5 and 15 minutes?(c) Determine the length
Continuation of Exercise 4-127.(a) If there has not been a call in 10 minutes, what is the probability that the time until the next call is less than 5 minutes?(b) What is the probability that there are no calls in the intervals from 10:00 to 10:05, from 11:30 to 11:35, and from 2:00 to 2:05?
Continuation of Exercise 4-127.(a) What is the probability that the time until the third call is greater than 30 minutes?(b) What is the mean time until the fifth call?
The CPU of a personal computer has a lifetime that is exponentially distributed with a mean lifetime of six years. You have owned this CPU for three years. What is the probability that the CPU fails in the next three years?
Continuation of Exercise 4-130. Assume that your corporation has owned 10 CPUs for three years, and assume that the CPUs fail independently. What is the probability that at least one fails within the next three years?
Suppose that X has a lognormal distribution with parameters θ = 0 and w2 = 4. Determine the following: (a) P(10 < X < 50) (b) The value for x such that (c) The mean and variance of X
Suppose that X has a lognormal distribution and that the mean and variance of X are 50 and 4000, respectively. Determine the following:(a) The parameters and of the lognormal distribution(b) The probability that X is less than 150
Asbestos fibers in a dust sample are identified by an electron microscope after sample preparation. Suppose that the number of fibers is a Poisson random variable and the mean number of fibers per squared centimeter of surface dust is 100. A sample of 800 square centimeters of dust is analyzed.
Without an automated irrigation system, the height of plants two weeks after germination is normally distributed with a mean of 2.5 centimeters and a standard deviation of 0.5 centimeters.(a) What is the probability that a plant’s height is greater than 2.25 centimeters?(b) What is the
Continuation of Exercise 4-135. With an automated irrigation system, a plant grows to a height of 3.5 centimeters two weeks after germination.(a) What is the probability of obtaining a plant of this height or greater from the distribution of heights in Exercise 4-135.(b) Do you think the automated
The thickness of a laminated covering for a wood surface is normally distributed with a mean of 5 millimeters and a standard deviation of 0.2 millimeter.(a) What is the probability that a covering thickness is greater than 5.5 millimeters?(b) If the specifications require the thickness to be
The diameter of the dot produced by a printer is normally distributed with a mean diameter of 0.002 inch. Suppose that the specifications require the dot diameter to be between 0.0014 and 0.0026 inch. If the probability that a dot meets specifications is to be 0.9973, what standard deviation is
Continuation of Exercise 4-138. Assume that the standard deviation of the size of a dot is 0.0004 inch. If the probability that a dot meets specifications is to be 0.9973, what specifications are needed? Assume that the specifications are to be chosen symmetrically around the mean of 0.002.
The life of a semiconductor laser at a constant power is normally distributed with a mean of 7000 hours and a standard deviation of 600 hours.(a) What is the probability that a laser fails before 5,800 hours?(b) What is the life in hours that 90% of the lasers exceed?
Continuation of Exercise 4-140. What should the mean life equal in order for 99% of the lasers to exceed 10,000 hours before failure?
Continuation of Exercise 4-140. A product contains three lasers, and the product fails if any of the lasers fails. Assume the lasers fail independently. What should the mean life equal in order for 99% of the products to exceed 10,000 hours before failure?

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Managing Controlling and Improving Quality 1st edition Douglas C. Montgomery, Cheryl L. Jennings, Michele E. Pfund - Solutions

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