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modern mathematical statistics with applications
Engineering Statistics 2nd Edition Douglas C. Montgomery, George C. Runger, Norma F. Hubele - Solutions
Assume a sample of 40 observations is drawn from a population with mean 20 and variance 2. Compute the following. (a) Mean and variance of X (c) P(X 19) (d) P(X > 22) (e) P(19 X 21.5)
Given that X is normally distributed with mean 50 and standard deviation 4, compute the following for n 25.(a) Mean and variance of(b)(c)(d) P(49 X 51.5)P(X 7 52)P(X 49)X P(96 X 102)P(X 7 103)P(X 98)X
Given that X is normally distributed with mean 100 and standard deviation 9, compute the following for n 16.(a) Mean and variance of(b)(c)(d)
Consider the random variables in Exercise 3-172.Determine the following:(a) E (2X Y ) (b) Cov(X,Y )(c) V(X 3Y ) (d) XY
Consider the random variables in Exercise 3-171.Determine the following:(a) E (2X Y ) (b) Cov(X,Y )(c) V(X 3Y ) (d) XY
The volume V of a cube is defined as the product of the length, L, the width, W, and the height, H. Assume that each of these dimensions is a random variable with mean 2 inches and standard deviation 0.1 inch. Assume independence and compute the mean and variance of V.
Consider X1, X2, and X3 given in Exercise 3-174.Define Y X1X2X3. Compute the mean and variance of Y.
Consider X1 and X2 given in Exercise 3-173. Define Y X1X2. Compute the mean and variance of Y.
Consider the equation for the acceleration due to gravity, G, given in Section 3-12.3. Suppose that E(T ) 5.2 seconds and V(T ) 0.0004 square second. Compute the mean and variance of G.
Consider the equation for the period T of a pendulum given in Section 3-12.3. Suppose that the length L is a random variable with mean 30 feet and standard deviation 0.02 feet.Compute the mean and variance of T.
Consider Example 3-44. Let the current have mean of 40 amperes and a standard deviation of 0.5 ampere. If the electrical circuit has a resistance of 100 ohms, compute the mean and variance of P.
Let X have mean 100 and variance 25. Define YX2 2X 1. Compute the mean and variance of Y.
Let X have mean 20 and variance 9. Define Y 2X2.Compute the mean and variance of Y.
Consider the random variables defined in Exercise 3-175. Assume that the random variables are not independent and have Cov(X1, X2) 5. Compute the mean and variance of Y.
Consider the random variables in Exercise 3-172, but assume that the variables are dependent with Cov (X1, X3) 1, Cov (X2, X3) 2 and that they are normally distributed. Determine the following.(a) V(Y ) (b) P(Y 7 12)
Consider the random variables in Exercise 3-173, but assume that the variables are dependent with covariance 3 and that they are normally distributed. Determine the following.(a) V(Y ) (b)
A U-shaped assembly is to be formed from the three parts A, B, and C. The picture is shown in Fig. 3-42. The length of A is normally distributed with a mean of 10 millimeters and a standard deviation of 0.1 millimeter. The thickness of part B is normally distributed with a mean of 2 millimeters and
The width of a casing for a door is normally distributed with a mean of 24 inches and a standard deviation of inch. The width of a door is normally distributed with a mean of 23 inches and a standard deviation of inch. Assume independence.(a) Determine the mean and standard deviation of the
A plastic casing for a magnetic disk is composed of two halves. The thickness of each half is normally distributed with a mean of 1.5 millimeters and a standard deviation of 0.1 millimeter, and the halves are independent.(a) Determine the mean and standard deviation of the total thickness of the
Consider the variables defined in Exercise 3-174.Assume that X1, X2, and X3 are normal random variables.Compute the following probabilities.(a) P(Y 2.0) (b) P(1.3 Y 8.3)
Consider the variables defined in Exercise 3-173.Assume that X1 and X2 are normal random variables. Compute the following probabilities.(a) P(Y 50)(b) P(25 Y 37)(c) P(14.63 Y 47.37)
If X1, X2, and X3 are independent random variables with 11.2, 20.8, 30.5, 11, 20.25, 32.2, and Y 2.5X1 0.5X2 1.5X3, determine the following.(a) E(Y ) (b) V(Y ) (c) E(3Y ) (d) V(3Y )
If X1 and X2 are independent random variables with16, 21, 12, 24, and Y4X1 2X2, determine the following.(a) E(Y ) (b) V(Y ) (c) E(2Y ) (d) V(2Y )
If X1, X2, and X3 are independent random variables with E(X1) 4, E(X2) 3, E(X3) 2, V(X1) 1, V(X2) 5, V(X3) 2, and Y 2X1 X2 3X3, determine the following.(a) E(Y ) (b) V(Y )
If X1 and X2 are independent random variables with E(X1) 2, E(X2) 5, V(X1) 2, V(X2) 10, and Y 3X1 5X2, determine the following.(a) E(Y ) (b) V(Y )
Suppose that the joint distribution of X and Y has probability density function f (x, y) 0.25xy for 0 x2 and 0 y 2). Determine the following:(a) P(X 6 1, Y 6 1)(b)P(X 6 1, Y 7 1)(c) P(X 7 1, Y 7 1) (d) (e) Whether or not X and Y are independent (d) P(X < 1)
The following circuit operates if and only if there is a path of functional devices from left to right. The probability that each device functions is as shown. Assume that the probability that a device functions does not depend on whether or not other devices are functional. What is the probability
The following circuit operates if and only if there is a path of functional devices from left to right. The probability that each device functions is as shown. Assume that the probability that a device is functional does not depend on whether or not other devices are functional. What is the
Suppose a parallel system has three components C1, C2, and C3, in parallel, with the probability that each component functions equal to 0.90, 0.99, and 0.95, respectively. What
Consider the parallel system described in Example 3-40. Suppose the probability that component C1 functions is 0.85 and the probability that component C2 functions is 0.92.What is the probability that the system operates?
Suppose a series system has three components C1, C2, and C3 with the probability that each component functions equal to 0.90, 0.99, and 0.95, respectively. What is the probability that the system operates?
Consider the series system described in Example 3-39.Suppose that the probability that component C1 functions is 0.95 and that the probability that component C2 functions is 0.92.What is the probability that the system operates?
The yield in pounds from a day’s production is normally distributed with a mean of 1500 pounds and a variance of 10,000 pounds squared. Assume that the yields on different days are independent random variables.(a) What is the probability that the production yield exceeds 1400 pounds on each of 5
The inside thread diameter of plastic caps made using an injection molding process is an important quality characteristic. The mold has four cavities. A cap made using cavity i is considered independent from any other cap and can have one of three quality levels: first, second, or third (worst).The
The time between surface finish problems in a galvanizing process is exponentially distributed with a mean of 40 hours. A single plant operates three galvanizing lines that are assumed to operate independently.(a) What is the probability that none of the lines experiences a surface finish problem
Two independent vendors supply cement to a highway contractor. Through previous experience it is known that the compressive strength of samples of cement can be modeled by a normal distribution, with 1 6000 kilograms per square centimeter and 1 100 kilograms per square centimeter for vendor
Let X be an exponential random variable with mean equal to 5 and Y be an exponential random variable with mean equal to 8. Assume X and Y are independent. Find the following probabilities.(a) P(X 5, Y 8)(b) P(X 5, Y 6)(c) P(3 X 7, Y 7)(d) P(X 7, 5 Y 7)
Let X be a Poisson random variable with 2 and Y be a Poisson random variable with 4. Assume X and Y are independent. Find the following probabilities.(a) P(X 4, Y 4)(b) P(X 2, Y 4)(c) P(2 X 4, Y 3)(d) P(X 5, 1 Y 4)
Let X be a normal random variable with 15.0 and 3 and Y be a normal random variable with 20 and 1. Assume X and Y are independent. Find the following probabilities.(a) P(X 12, Y 19)(b) P(X 16, Y 18)(c) P(14 X 16, Y 22)(d) P(11 X 20, 17.5 Y 21)
Let X be a normal random variable with 10 and 1.5 and Y be a normal random variable with 2 and 0.25. Assume X and Y are independent. Find the following probabilities.(a) P(X 9, Y 2.5)(b) P(X 8, Y 2.25)(c) P(8.5 X 11.5, Y 1.75)(d) P(X 13, 1.5 Y 1.8)
The number of visits to the home page of a Web site in a day is modeled with a Poisson distribution with a mean of 200. Approximate the probabilities for the following events:(a) More than 225 visitors arrive in a day(b) Fewer than 175 visitors arrive in a day(c) The number of visitors is greater
The probability a visitor to the home page of a Web site views another page on the site is 0.2. Assume that 200 visitors arrive at the home page and that they behave independently.Approximate the probabilities for the following events:(a) More than 40 visitors view another page(b) At least 30
The number of calls to a health-care provider follows a Poisson distribution with a mean of 36 per hour.Approximate the following probabilities.(a) More than 42 calls in an hour(b) Less than 30 calls in an hour(c) More than 300 calls in an 8-hour day
The number of spam e-mails received each day follows a Poisson distribution with a mean of 50. Approximate the following probabilities.(a) More than 40 and less than 60 spam e-mails in a day(b) At least 40 spam e-mails in a day(c) Less than 40 spam e-mails in a day(d) Approximate the probability
Suppose that the number of asbestos particles in a sample of 1 square centimeter of dust is a Poisson random variable with a mean of 1000. Approximate the probability that 10 square centimeters of dust contain more than 10,000 particles.
Phoenix water is provided to approximately 1.4 million people, who are served through more than 362,000 accounts (http://phoenix.gov/WATER/wtrfacts.html). All accounts are metered and billed monthly. The probability that an account has an error in a month is 0.001; accounts can be assumed to be
There were 49.7 million people with some type of long-lasting condition or disability living in the United States in 2000. This represented 19.3% of the majority of civilians age 5 and over (http://factfinder.census.gov). A sample of 1000 persons is selected; it can be assumed the disability
The manufacturing of semiconductor chips produces 2% defective chips. Assume that 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 large electronic office product contains 2000 electronic components. Assume that the probability that each component operates without failure during the useful life of the product is 0.995, and assume that the components fail independently.Approximate the probability that 5 or more of the
A particular vendor produces parts with a defect rate of 8%. Incoming inspection to a manufacturing plant samples 100 delivered parts from this vendor and rejects the delivery if 8 defective parts are discovered.(a) Compute the exact probability that the inspector accepts delivery.(b) Approximate
Suppose that X has a Poisson distribution with mean of 50. Approximate the following probabilities (a) P(X 55)
Suppose that X has a binomial distribution with n 300 and Approximate the following probabilities. (a) P(X 100) (c) P(X >130) (b) P(80X100)
The time a visitor to a Web site views the home page is modeled with an exponential distribution with a mean of 20 seconds.(a) Determine the probability that the home page is viewed for more than 30 seconds.(b) Determine the probability that the home page is viewed for a time greater than the
The number of serious infections at a hospital is modeled with a Poisson distribution with a mean of 3.5 per month. Determine the following:(a) Probability of exactly three infections in a month(b) Probability of no infections in a month(c) Probability of at least three infections in a month(d)
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 hour? (b) What is the probability that there are no calls within hour?(c) Determine x such that the probability that there are no
The time between the arrival of e-mail messages at your computer is exponentially distributed with a mean of 2 hours.(a) What is the probability that you do not receive a message during a 2-hour period?(b) If you have not had a message in the last 4 hours, what is the probability that you do not
The time to failure of a certain type of electrical component is assumed to follow an exponential distribution with a mean of 4 years. The manufacturer replaces free all components that fail while under guarantee.(a) What percentage of the components will fail in 1 year?(b) What is the probability
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
A remotely operated vehicle (ROV ) detects debris from a sunken craft at a rate of 50 pieces per hour. The time to detect debris can be modeled using an exponential distribution.(a) What is the probability that the time to detect the next piece of debris is less than 2 minutes?(b) What is the
The time between calls to a health-care provider is exponentially distributed with a mean time between calls of 12 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) What
Suppose the counts recorded by a Geiger counter follow a Poisson process with an average of three 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 X has an exponential distribution with mean equal to 5. Determine the following.(a) P(X 5) (b) P(X 15) (c) P(X 20)(d) Find the value of x such that P(X x) 0.95.
Suppose X has an exponential distribution with 3. Determine the following.(a) P(X 0) (b) P(X 3)(c) P(X 2) (d) P(2 X 3)(e) Find the value of x such that P(X x) 0.05.
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.centralhudsonlab.com determined that 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
Messages arrive to a computer server according to a Poisson distribution with a mean rate of 10 per hour.(a) What is the probability that three messages will arrive in 1 hour?(b) What is the probability that six messages arrive in 30 minutes?
Flaws occur in Mylar material according to a Poisson distribution with a mean of 0.01 flaw per square yard.(a) If 25 square yards are inspected, what is the probability that there are no flaws?(b) What is the probability that a randomly selected square yard has no flaws?(c) Suppose that the Mylar
A telecommunication station is designed to receive a maximum of 10 calls per second. If the number of calls to the station is modeled as a Poisson random variable with a mean of 9 calls per second, what is the probability that the number of calls will exceed the maximum design constraint of the
When network cards are communicating, bits can occasionally be corrupted in transmission. Engineers have determined that the number of bits in error follows a Poisson distribution with mean of 3.2 bits/kb (per kilobyte).(a) What is the probability of 5 bits being in error during the transmission of
The number of failures of a testing instrument from contamination particles on the product is a Poisson random variable with a mean of 0.04 failure per hour.(a) What is the probability that the instrument does not fail in an 8-hour shift?(b) What is the probability of at least three failures in a
The number of surface flaws in a plastic roll used in the interior of automobiles has a Poisson distribution with a mean of 0.05 flaw per square foot of plastic roll. Assume an automobile interior contains 10 square feet of plastic roll.(a) What is the probability that there are no surface flaws in
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 earthquake tremors in a 12-month period appears to be distributed as a Poisson random variable with a mean of 6. Assume the number of tremors from one 12-month period is independent of the number in the next 12-month period.(a) What is the probability that there are 10 tremors in 1
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 20 calls per hour.(a) What is the probability that there are exactly 18 calls in 1 hour?(b) What is the probability that there are 3 or fewer calls in 30
Suppose that the number of customers who enter a bank in an hour is a Poisson random variable and that P(X0) 0.04.Determine the mean and variance of X.
Suppose that the number of customers who enter a post office in a 30-minute period is a Poisson random variable and that P(X 0) 0.02. Determine the mean and variance of X.
Suppose X has a Poisson distribution with a mean of 5. Determine the following probabilities.(a) P(X 0) (b) P(X 3)(c) P(X 6) (d) P(X 9)
Suppose X has a Poisson distribution with a mean of 0.3. Determine the following probabilities.(a) P(X 0) (b) P(X 3)(c) P(X 6) (d) P(X 2)
The probability a visitor to the home page of a Web site views another page on the site is 0.2. Assume that 20 visitors arrive at the home page and that they behave independently.Determine the following:(a) Probability that exactly one visitor views another page(b) Probability two or more visitors
Consider the visits that result in leave without being seen (LWBS) at an emergency department in Example 3-1.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 people
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” (Vol. 9, 2004, pp. 6–18), provided the following data on the top 10 malicious software instances for 2002. The clear leader in the number of registered incidences for the year 2002 was the
Traffic engineers install 10 streetlights with new bulbs. The probability that a bulb fails within 50,000 hours of operation is 0.25. Assume that each of the bulbs fails independently.(a) What is the probability that fewer than two of the original bulbs fail within 50,000 hours of operation?(b)
The probability of successfully landing a plane using a flight simulator is given as 0.80. Nine randomly and independently chosen student pilots are asked to try to fly the plane using the simulator.(a) What is the probability that all the student pilots successfully land the plane using the
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
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 shows
In 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
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 five. Assume that the number of nonconforming springs in a batch, denoted as X, is a binomial random variable.(a) What
The phone lines to an airline reservation system are occupied 45% of the time. Assume that the events that the lines are occupied on successive calls are independent.Assume that eight calls are placed to the airline.(a) What is the probability that for exactly two calls the lines are occupied?(b)
A hip joint replacement part is being stress-tested in a laboratory. The probability of successfully completing the test is 0.80. Seven randomly and independently chosen parts are tested. What is the probability that exactly two of the seven parts successfully complete the test?
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?
The random variable X has a binomial distribution with n10 and p0.1. Determine the following probabilities.(a) P(X 5) (b) P(X 2)(c) P(X 9) (d) P(3 X 5)
Given that X has a binomial distribution with n 10 and p 0.01(a) Sketch the probability mass function.(b) Sketch the cumulative distribution function.(c) What value of X is most likely?(d) What value of X is least likely?
The random variable X has a binomial distribution with n20 and p0.5. Determine the following probabilities.(a) P(X 15) (b) P(X 12)(c) P(X 19) (d) P(13 X 15)(e) Sketch the cumulative distribution function.
The random variable X has a binomial distribution with n 10 and p 0.5.(a) Sketch the probability mass function of X.(b) Sketch the cumulative distribution.(c) What value of X is most likely?(d) What value(s) of X is (are) least likely?
For each scenario described, 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 sample
Let X denote the waiting time in seconds (rounded to the nearest tenth) for a large database update to completed.The probability mass function for X is x 0.1 0.2 0.3 0.4 0.5 0.6 f (x) 0.1 0.1 0.3 0.2 0.2 0.1 Determine the following:(a) P(X 0.25)(b) P(0.15 X 4.5)(c) F(x)(d) E(X)
Let X denote the time in minutes (rounded to the nearest half minute) for a blood sample to be taken. The probability mass function for X is x 0 0.5 1 1.5 2 2.5 f (x) 0.1 0.2 0.3 0.2 0.1 0.1 Determine the following:(a) P(X 2.25)(b) P(0.75 X 1.5)(c) F(x)(d) E(X )
Let X denote the number of patients who suffer an infection within a floor of a hospital per month with the following probabilities:x 0 1 2 3 0.7 0.15 0.1 0.05 Determine the following:(a) F(x)(b) Mean and variance(c) P(X 1.5)(d) P(X 2.0)
Let X denote the number of bars of service on your cell phone whenever you are at an intersection with the following probabilities:x 0 1 2 3 4 5 0.1 0.15 0.25 0.25 0.15 0.1 Determine the following:(a) F(x)(b) Mean and variance(c) P(X 2)(d) P(X 3.5)
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