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modern mathematical statistics with applications
Engineering Statistics 2nd Edition Douglas C. Montgomery, George C. Runger, Norma F. Hubele - Solutions
Marketing estimates that a new instrument for the analysis of soil samples will be very successful, moderately successful, or unsuccessful, with probabilities 0.4, 0.5, and 0.1, respectively. The yearly revenue associated with a very successful, moderately successful, or unsuccessful product is$10
Customers purchase a particular make of automobile with a variety of options. The probability mass function of the number of options selected is x 7 8 9 10 f (x) 0.040 0.130 0.190 0.240 x 11 12 13 f (x) 0.300 0.050 0.050(a) What is the probability that a customer will choose fewer than 9
f (x) (12)(x5), x 1, 2, 3, 4(a) P(X 2) (b) P(X 3)(c) P(X 2.5) (d) P(X 1)(e) Mean and variance (f) Graph F(x).
f (x) (87)(12)x, x 1, 2, 3(a) P(X 1) (b) P(X 1)(c) Mean and variance (d) Graph F(x).
x 0 1 2 3 f (x) 0.025 0.041 0.049 0.074 x 4 5 6 7 f (x) 0.098 0.205 0.262 0.123 x 8 9 f (x) 0.074 0.049(a) P(X 1) (b) P(2 X 7.2)(c) P(X 6) (d) Mean and variance(e) Graph F(x).
x 1 2 3 4 f (x) 0.326 0.088 0.019 0.251 x 5 6 7 f (x) 0.158 0.140 0.018(a) P(X 3) (b) P(3 X 5.1)(c) P(X 4.5) (d) Mean and variance(e) Graph F(x).
The following data are the temperatures of effluent at discharge from a sewage treatment facility on consecutive days:Determine which of the probability models studied appears to provide the most suitable fit to the data. 43 47 51 48 52 45 52 46 51 44 49 49 45 44 50 48 5550 44 46 49 46 51 49 50
The following data are direct solar intensity measurements on different days at a location in southern Spain that was analyzed in Chapter 2: 562, 869, 708, 775, 775, 704, 809, 856, 655, 806, 878, 909, 918, 558, 768, 870, 918, 940, 946, 661, 820, 898, 935, 952, 957, 693, 835, 905, 939, 955, 960,
Thirty measurements of the time-to-failure of a critical component in an electronics assembly are recorded.Determine which probability density model—normal, lognormal, or Weibull—appears to provide the most suitable fit to the data. 1.9 20.7 3.0 11.9 6.3 0.4 8.3 2.3 1.6 5.3 4.6 1.9 5.1 10.9 7
The duration of an inspection task is recorded in minutes.Determine which probability density model—normal, lognormal, or Weibull—appears to provide the most suitable fit to the data. 5.15 0.30 6.66 3.76 4.29 9.54 4.38 0.60 7.06 4.34 0.80 5.12 3.69 5.94 3.18 4.47 4.65 8.93 4.70 1.04
Twenty-five measurements of the time a client waits for a server is recorded in seconds. Which probability density model—normal, lognormal, or Weibull—appears to provide the most suitable fit to the data? 1.21 4.19 1.95 6.88 3.97 9.09 6.91 1.90 10.60 0.51 2.23 13.99 8.22 8.08 4.70 4.67 0.50
A quality control inspector is interested in maintaining a flatness specification for the surface of metal disks.Thirty flatness measurements in (0.001 inch) were collected.Which probability density model—normal, lognormal, or Weibull—appears to provide the most suitable fit to the data? 2.49
In studying the uniformity of polysilicon thickness on a wafer in semiconductor manufacturing, Lu, Davis, and Gyurcsik (Journal of the American Statistical Association, Vol. 93, 1998) collected data from 22 independent wafers:494, 853, 1090, 1058, 517, 882, 732, 1143, 608, 590, 940, 920, 917, 581,
After examining the data from the two machines in Exercise 3-82, the process engineer concludes that machine 2 has higher part-to-part variability. She makes some adjustments to the machine that should reduce the variability, and she obtains another sample of 20 parts. The measurements on those
Samples of 20 parts are selected from two machines, and a critical dimension is measured on each part. The data are shown next. Plot the data on normal probability paper.Does this dimension seem to have a normal distribution?What tentative conclusions can you draw about the two machines? Machine 1
A soft-drink bottler is studying the internal pressure strength of 1-liter glass bottles. A random sample of 16 bottles is tested, and the pressure strengths are obtained. The data are shown next. Plot these data on normal probability paper. Does it seem reasonable to conclude that pressure
The length of stay at a specific emergency department in Phoenix, Arizona, in 2009 had a mean of 4.6 hours with a standard deviation of 2.9. Assume that the length of stay is normally distributed.(a) What is the probability of a length of stay greater than 10 hours?(b) What length of stay is
An article under review for Air Quality, Atmosphere &Health titled “Linking Particulate Matter (PM10) and Childhood Asthma in Central Phoenix” used PM10 (particulate matter 10 m in diameter) air quality data measured hourly form sensors in Phoenix, Arizona. The 24-hour (daily) mean PM10 for a
Suppose X has a beta distribution with parameters 2.5 and 2.5. Sketch an approximate graph of the probability density function. Is the density symmetric?
The maximum time to complete a task in a project is 2.5 days. Suppose that the completion time as a proportion of this maximum is a beta random variable with 2 and 3.What is the probability that the task requires more than 2 days to complete?
The length of stay at an emergency department is the sum of the waiting and service times. Let X denote the proportion of time spent waiting and assume a beta distribution with 10 and 1. Determine the following:(a) P(X 0.9)(b) P(X 0.5)(c) Mean and variance
A European standard value for a low-emission window glazing uses 0.59 as the proportion of solar energy that enters a room. Suppose that the distribution of the proportion of solar energy that enters a room is a beta random variable.(a) Calculate the mode, mean, and variance of the distribution for
Suppose X has a beta distribution with parameters 1 and 4.2. Determine the following:(a) P(X 0.25)(b) P(0.5 X )(c) Mean and variance
Suppose X has a beta distribution with parameters 2.5 and 1. Determine the following:(a) P(X 0.25)(b) P(0.25 X 0.75)(c) Mean and variance
Suppose that X represents time measurements from a gamma distribution with a mean of 4 minutes and a variance of 2 minutes2. Find the parameters and r.
Suppose that X represents length measurements from a gamma distribution with a mean of 4.5 inches and a variance of 6.25 inches2. Find the parameters and r.
Suppose that X represents diameter measurements from a gamma distribution with a mean of 3 millimeters and a variance of 1.5 millimeters 2. Find the parameters and r.
Suppose that X has a gamma distribution with 2.5 and r 3.2. Determine the mean and variance of X.
Suppose that X has a gamma distribution with 3 and r 6. Determine the mean and variance of X.
Use the properties of the gamma function to evaluate the following:(a)(6) (b)(52) (c)(92)
An article in the Journal of the Indian Geophysical Union, titled “Weibull and Gamma Distributions for Wave Parameter Predictions” (Vol. 9, 2005, 55–64), used the Weibull distribution to model ocean wave heights. Assume that the mean wave height at the observation station is 2.5 meters and
The life (in hours) of a computer processing unit(CPU) is modeled by a Weibull distribution with parameters 3 and 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 hours?
Assume that the life of a roller bearing follows a Weibull distribution with parameters 2 and 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 occur
Suppose that X has a Weibull distribution with 0.2 and 100 hours. Determine the following:(a) P(X 10,000) (b) P(X 5000)
Suppose that X has a Weibull distribution with 0.2 and 100 hours. Determine the mean and variance of X.
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 hours, respectively.(a) Calculate the parameters of the lognormal distribution.(b) Determine the probability that a lifetime exceeds 10,000
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 0.5 and 2 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 users view the
Suppose that X has a lognormal distribution with parameters 2 and 2 4. Determine the following:(a) P(X 500)(b) P(500 X 1000)(c) P(1500 X 2000)(d) What does the difference of the probabilities in parts (a),(b), and (c) imply about the probabilities of lognormal random variables?
Suppose that X has a lognormal distribution with parameters 5 and 2 9. Determine the following:(a) P(X 13,300)(b) The value for x such that P(X x) 0.95(c) The mean and variance of X
The weight of a human joint replacement part is normally distributed with a mean of 2 ounces and a standard deviation of 0.05 ounce.(a) What is the probability that a part weighs more than 2.10 ounces?(b) What must the standard deviation of weight be for the company to state that 99.9% of its parts
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 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?
A device that monitors the levels of pollutants has sensors that detect the amount of CO in the air. Placed in a particular location, it is known that the amount of CO is normally distributed with a mean of 6.23 ppm and a variance of 4.26 ppm2.(a) What is the probability that the CO level exceeds 9
Operators of a medical linear accelerator are interested in estimating the number of hours until the first software failure. Prior experience has shown that the time until failure is normally distributed with mean 1000 hours and standard deviation 60 hours.(a) Find the probability that the software
The length of an injected-molded plastic case that holds tape is normally distributed with a mean 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
The reaction time of a driver to visual stimulus is normally distributed with a mean of 0.4 second and a standard deviation of 0.05 second.(a) What is the probability that a reaction requires more than 0.5 second?(b) What is the probability that a reaction requires between 0.4 and 0.5 second?(c)
Consider the filling machine in Exercise 3-48.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
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 that a fill volume is less than 12 fluid ounces?(b) If all cans less than
The line width of a tool used 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
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 39 lb/in.2?(b) If the specifications require the tensile strength to
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
Assume that X is normally distributed with a mean of 6 and a standard deviation of 3. 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) 0.8
Assume that X is normally distributed with a mean of 37 and a standard deviation of 2. Determine the following.(a) P(X 31) (b) P(X 30)(c) P(33 X 37) (d) P(32 X 39)(e) P(30 X 38)
Assume that X is normally distributed with a mean of 20 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 20) 0.2
Assume that X is normally distributed with a mean of 20 and a standard deviation of 2. Determine the following.(a) P(X 24) (b) P(X 18)(c) P(18 X 22) (d) P(14 X 26)(e) P(16 X 20) (f ) P(20 X 26)
Assume that Z has a standard normal distribution. Use Appendix A Table I to determine the value for z that solves each of the following.(a) P(z Z z) 0.95 (b) P(z Z z) 0.99(c) P(z Z z) 0.68 (d) P(z Z z) 0.9973
Assume that Z has a standard normal distribution. Use Appendix A Table I to determine the value for z that solves each of the following.(a) P(Z z) 0.50000 (b) P(Z z) 0.001001(c) P(Z z) 0.881000 (d) P(Z z) 0.866500(e) P(1.3 Z z) 0.863140
Use Appendix A Table I to determine the following probabilities for the standard normal random variable Z.(a) P(1 Z 1) (b) P(2 Z 2)(c) P(3 Z 3) (d) P(Z 3)(e) P(0 Z 3)
The lifetime of a semiconductor laser has a lognormal distribution with 10 and 1.5 hours. What is the probability the lifetime exceeds 10,000 hours?
The diameter of a shaft in a storage drive is normally distributed with mean 0.2508 inch and standard deviation 0.0005 inch. The specifications on the shaft are 0.2500 0.0015 inch. What proportion of shafts conforms to specifications?
The waiting time until service at a hospital emergency department is modeled with the pdf f (x) (19)x for hours and for Determine the following:(a) Probability the wait is less than 4 hours(b) Probability the wait is more than 5 hours(c) Probability the wait is less than or equal to 30
Given the cdf F(x) 0 for for 0 6 x, determine the following: (a) P(X < 1) (b) P(X > 2) (c) P(1 X < 2) (d) x such that P(X < x) = 0.95 (e) pdf
The probability density function of the weight of packages delivered by a post office is for(a) What is the probability a package weighs less than 10 pounds?(b) Determine the mean and variance of package weight.(c) If the shipping cost is $3 per pound, what is the average shipping cost of a package?
The probability density function of the time a customer arrives at a terminal (in minutes after 8:00 A.M.) is for Determine the probability that(a) The customer arrives by 9:00 A.M.(b) The customer arrives between 8:15 A.M. and 8:30 A.M.(c) Determine the time at which the probability of an earlier
A medical linear accelerator is used to accelerate electrons to create high-energy beams that can destroy tumors with minimal impact on surrounding healthy tissue. The beam energy fluctuates between 200 and 210 MeV (million electron volts). The cumulative distribution function isDetermine the
The thickness of a conductive coating in micrometers has a density function of 600x2 for 100 m x120 m and zero for x elsewhere.(a) Determine the mean and variance of the coating thickness.(b) If the coating costs $0.50 per micrometer of thickness on each part, what is the average cost of the
Suppose the cumulative distribution function of the length (in millimeters) of computer cables is(a) Determine P(x 1208).(b) If the length specifications are 1195 x 1205 millimeters, what is the probability that a randomly selected computer cable will meet the specification requirement? F(x) = 0.1x
(Integration by parts is required in this exercise.) The probability density function for the diameter of a drilled hole in millimeters is 10e10(x5) for x 5 mm and zero for x 5 mm. Although the target diameter is 5 millimeters, vibrations, tool wear, and other factors can produce diameters
Suppose that contamination particle size (in micrometers)can be modeled as f (x) 2x3 for 1 x and f (x) 0 for x 1.(a) Confirm that f(x) is a probability density function.(b) Give the cumulative distribution function.(c) Determine the mean.(d) What is the probability that the size of a random
The thickness measurement of a wall of plastic tubing, in millimeters, varies according to a cumulative distribution functionDetermine the following.(a) P(X 2.0080) (b) P(X 2.0055)(c) If the specification for the tubing requires that the thickness measurement be between 2.0090 and 2.0100
The temperature readings from a thermocouple in a furnace fluctuate according to a cumulative distribution functionDetermine the following.(a) P(X 805) (b) P(800 X 805) (c) P(X 808)(d) If the specifications for the process require that the furnace temperature be between 802 and 808C, what is the
The probability density function of the net weight in ounces of a packaged compound is f (x) 2.0 for 19.75 x20.25 ounces and f (x) 0 for x elsewhere.(a) Determine the probability that a package weighs less than 20 ounces.(b) Suppose that the packaging specifications require that the weight be
The pdf of the time to failure of an electronic component in a copier (in hours) is f (x) [exp (x3000)]3000 for x0 and f (x) 0 for x0. Determine the probability that(a) A component lasts more than 1000 hours before failure.(b) A component fails in the interval from 1000 to 2000 hours.(c) A
Suppose that f (x) 1.5x2 for 1 x1 and f (x) 0 otherwise. Determine the following probabilities.(a) P(0 X ) (b) P(0.5 X)(c) P(0.5 X 0.5) (d) P(X2)(e) P(X 0 or X 0.5)(f ) Determine x such that P(x X ) 0.05.
Suppose that f (x) e(x6) for 6 x and f (x) 0 for x 6. Determine the following probabilities.(a) P(X 6) (b) P(6 X 8)(c) P(X 8) (d) P(X 8)(e) Determine x such that P(X x) 0.95.
For each of the density functions in Exercise 3-21, perform the following.(a) Graph the density function and mark the location of the mean on the graph.(b) Find the cumulative distribution function.(c) Graph the cumulative distribution function.
Show that the following functions are probability density functions for some value of k and determine k. Then determine the mean and variance of X.(a) f (x) kx2 for 0 x4(b) f (x) k(1 2x) for 0 x2(c) f (x) kex for 0 x(d) f (x) k where k 0 and 100 x100 k
Consider the hospital emergency room data in Example 3-1. Let A denote the event that a visit is to Hospital 4 and let B denote the event that a visit results in LWBS (at any hospital). Determine the following probabilities. (a) P(AB) (b) P(A') (c) P(AUB) (d) P(A U B') (e) P(A'B')
Let X denote the number of unique visitors to a Web site in a month with the following probabilities:the following probabilities: P(0X9) 0.4, P(0 X 19)=0.7,P(0 X 29)= 0.8, P(0X39)=0.9, P(0 X 49) = 1. Determine
Let X denote the number of patients who suffer an infection within a floor of a hospital per month with the following probabilities:Determine the following probabilities:(a) Less than one infection (b) More than three infections (c) At least one infection (d) No infections x 1 2 3 P(X = x) 0.7 0.15
Let X denote the number of bars of service on your cell phone whenever you are at an intersection with the following probabilities:Determine the following probabilities:(a) Two or three bars (b) Fewer than two bars (c) More than three bars (d) At least one bar x 0 1 2 3 4 5 P(X = x) 0.1 0.15 0.25
Let E1 denote the event that a structural component fails during a test and E2 denote the event that the component shows some strain but does not fail. Given P(E1) 0.15 and P(E2) 0.30,(a) What is the probability that a structural component does not fail during a test?(b) What is the probability
Let X denote the life of a semiconductor laser (in hours) with the following probabilities:and(a) What is the probability that the life is less than or equal to 7000 hours?(b) What is the probability that the life is greater than 5000 hours?(c) What is P(5000 X 7000)?P(X 7 7000) 0.45.P(X
Suppose that an ink cartridge is classified as being overfilled, medium filled, or underfilled with a probability of 0.40, 0.45, and 0.15, respectively.(a) What is the probability that a cartridge is classified as not underfilled?(b) What is the probability that a cartridge is either overfilled or
Let P(X 15) 0.3, P(15 X 24) 0.6, and P(X 20) 0.5.(a) Find P(X 15).(b) Find P(X 24).(c) Find P(15 X 20).(d) If P(18 X 24) 0.4, find P(X 18).
If P(X A) 0.3, P(X B) 0.25, P(X C) 0.60, P(XA B) 0.55, and P(XB C) 0.70, determine the following probabilities.(a) P(XA) (b) P(XB) (c) P(XC)(d) Are A and B mutually exclusive?(e) Are B and C mutually exclusive?
If P(XA) 0.4, and P(XB) 0.6 and the intersection of sets A and B is empty,(a) Are sets A and B mutually exclusive?(b) Find P(X A).(c) Find P(X B).(d) Find P(X A B).
State the complement of each of the following sets:(a) Engineers with less than 36 months of full-time employment.(b) Samples of cement blocks with compressive strength less than 6000 kilograms per square centimeter.(c) Measurements of the diameter of forged pistons that do not conform to
The concentration of organic solids in a water sample. Decide whether a discrete or continuous random variable is the best model for each
The number of molecules in a sample of gas. Decide whether a discrete or continuous random variable is the best model for each
The weight of an injection-molded plastic part. Decide whether a discrete or continuous random variable is the best model for each
The proportion of defective solder joints on a circuit board. Decide whether a discrete or continuous random variable is the best model for each
The number of convenience options selected by an automobile buyer. Decide whether a discrete or continuous random variable is the best model for each
The strength of a concrete specimen. Decide whether a discrete or continuous random variable is the best model for each
The number of times a transistor in a computer memory changes state in a time interval. Decide whether a discrete or continuous random variable is the best model for each
The lifetime of a biomedical device after implant in a patient. Decide whether a discrete or continuous random variable is the best model for each
Figure 2-5 illustrates the stem-and-leaf diagram for 25 observations on batch yields from a chemical process. In Fig. 2-5a we used 6, 7, 8, and 9 as the stems. This results in too few stems, and the stem-andleaf diagram does not provide much information about the data. In Fig. 2-5b we divided each
An article in Quality Engineering (Vol. 4, 1992, pp. 487–495) presents viscosity data from a batch chemical process. A sample of these data is presented next. (Read down the entire column, then left to right.)(a) Draw a time series plot of all the data and comment on any features of the data that
A study was performed on wear of a bearing y and its relationship to viscosity and The following data were obtained.(a) Create two scatter diagrams of the data. What do you anticipate will be the sign of each sample correlation coefficient?(b) Compute and interpret the two sample correlation
Consider the distillation column described in Section 1-2. Suppose that the engineer runs this column for 24 consecutive hours and records the acetone concentration at the end of each hour. Is this a random sample?
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