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applied statistics and probability for engineers
Applied Statistics For Engineers And Scientists 3rd Edition Jay L. Devore, Nicholas R. Farnum, Jimmy A. Doi - Solutions
=+a. If cork diameter is a normally distributed variable with mean value 3.04 cm and standard deviation .02 cm, what is the probability that 6.2 How Control Charts Work
=+8. A cork intended for use in a wine bottle is considered acceptable if its diameter is between 2.9 cm and 3.1 cm (so the lower specification limit is LSL 5 2.9 cm, and the upper specification limit is USL 5 3.1 cm).
=+what is the effect on the estimated proportions of conforming and nonconforming products?
=+7. If the measuring instrument used in Exercise 6 is out of calibration and is giving readings that are, say, .02 in. higher than the true length of an object,
=+e. Explain the reason for the difference in your answers to parts (c) and (d).
=+d. Assuming that the process from which the data was taken can be described by a normal density function, what percentage of the process data is expected to fall above the USL[use your estimates from part (b)]? What percentage of the process data is expected to fall below the LSL?
=+c. What percentage of these measurements falls above the USL? What percentage of the measurements falls below the LSL?
=+b. Estimate the mean and standard deviation of the process from which this data was taken.
=+6. The following are measurements (in inches) of a quality characteristic with specification limits of 2.506.05 in.:2.54 2.52 2.50 2.52 2.50 2.50 2.47 2.48 2.51 2.53 2.53 2.51 2.50 2.47 2.49 2.50 2.50 2.50 2.46 2.48 2.48 2.50 2.51 2.53 2.51 2.53 2.53 2.52 2.47 2.51a. Create a histogram of the
=+g. The number of errors in 1000 lines of computer code h. The time between breakdowns of a certain machine i. The breaking strength of a molded plastic part
=+f. The torque applied to an airplane wing fastener(bolts and nuts used in aerospace are called fasteners)
=+d. The number of bolts in a batch that have oversize thread diameterse. The proportion of bolts in a batch that have oversize thread diameters
=+vb. The concentration of a chemical solution used in an electroplating processc. The thread diameter of a bolt
=+a. The number of flaws per square foot in a large sheet of metal
=+5. Measurements are to be taken on each of the following characteristics. In each case, indicate whether the resulting measurements would be classified as variables or attributes data.
=+4. Citrus products must have a certain sugar content, measured in degrees Brix, to be judged satisfactory to sell to grocery stores. Suppose that a certain batch of oranges fails to meet the specified Brix level.Which classification would you apply to these oranges, defective or nonconforming?
=+b. What are the penalties for exceeding the upper specification?
=+a. What specification limit does the envelope size place on the page-folding process?
=+3. A standard legal envelope is 4 inches wide by 9.5 inches long. Normally, 8.5-inch by 11-inch pages are folded in thirds before they are inserted into such envelopes. Viewing page folding as a process whose measurable output is the width of the folded page, answer the following questions:
=+to perform such tests. One of the measures of the quality of the services provided by such labs is the waiting time before test results are available. Does the characteristic waiting time have a one- or a twosided tolerance?
=+2. Determining whether structural materials conform to specifications often requires special test equipment (which can be expensive) and test procedures(which require specialized training). Consequently, independent testing and evaluation labs have arisen
=+1. General-purpose resistors are color-coded with a sequence of four rings that identify the nominal value of the resistance (in ohms) and the plus and minus tolerance (expressed as a percentage of the nominal)to be expected in the actual resistance. For example, bands (in order) of green, blue,
=+what is the probability that a 1 was sent? Hint:Use a tree diagram.
=+c. Suppose that 70% of all bits sent from the transmitter are 1s. If a 1 is received by the receiver,
=+b. If a 1 is sent from the transmitter, what is the probability that a 1 is received by the receiver?Hint: Use a tree diagram.
=+a. If a 1 is sent from the transmitter, what is the probability that a 1 is sent by all three relays?
=+78. A message is transmitted using a binary code of 0s and 1s. Each transmitted bit (0 or 1) must pass through three relays before reaching a receiver. At each relay, the probability is .20 that the bit sent is different from the bit received (a reversal). Assume that relays operate independently
=+schedule. If A and B are independent events with P(A) . P(B) and P(A or B) 5 .626, P(A and B) 5.144, determine the values of P(A) and P(B).
=+77. One satellite is scheduled to be launched from Cape Canaveral in Florida, and another launching is scheduled for Vandenberg Air Force Base in California. Let A denote the event that the Vandenberg launch goes off on schedule, and let B represent the event that the Cape Canaveral launch goes
=+Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially
=+what is the probability that it came from line 1?
=+c. Given that the selected can had a surface defect,
=+b. If the selected can came from line 1, what is the probability that it had a blemish?
=+a. What is the probability that the can was produced by line 1? That the reason for nonconformance is a crack?
=+ During this period, line 1 produced 500 nonconforming cans, line 2 produced 400 such cans, and line 3 was responsible for 600 nonconforming cans.Suppose that one of these 1500 cans is randomly selected.
=+76. A factory uses three production lines to manufacture cans of a certain type. The accompanying table gives percentages of nonconforming cans, categorized by type of nonconformance, for each of the three lines during a particular time period:Line 1 Line 2 Line 3 Blemish 15 12 20 Crack 50 44 40
=+the two destinations, what are the posterior probabilities of having flown on airlines #1, #2, and #3? Hint:From the tip of each first-generation branch on a tree diagram, draw three second-generation branches labeled, respectively, 0 late, 1 late, and 2 late.
=+For airline #1, flights are late into DC 30% of the time and late into LA 10% of the time. For airline #2, these percentages are 25% and 20%, whereas for airline #3 the percentages are 40% and 25%. If we learn that on a particular trip she arrived late at exactly one of
=+75. A friend who lives in Los Angeles makes frequent consulting trips to Washington, DC; 50% of the time she travels on airline #1, 30% of the time on airline#2, and the remaining 20% of the time on airline #3.
=+d. Explain how you would use the information in part (c) to calculate the probability of a common birth date.
=+what is the probability that all births occurred on March 11? Hint: The deviation of birth date from due date is normally distributed with mean 0.
=+a normal distribution with mean value 280 days and standard deviation 19.88 days. The due dates for the three Utah sisters were March 15, April 1, and April 4, respectively. Assuming that all three due dates are at the mean of the distribution,
=+c. The author suggested that, based on extensive data, the length of gestation (time between conception and birth) could be modeled as having
=+b. With the assumptions used in part (a), what is the probability that three randomly selected births all occur on the same day?
=+a. Disregarding leap year and assuming that the other 365 days are equally likely, what is the probability that three randomly selected births all occur on March 11? Be sure to indicate what, if any, extra assumptions you are making.
=+74. The article “Three Sisters Give Birth on the Same Day” (Chance, Spring 2001, 23–25) used the fact that three Utah sisters had all given birth on March 11, 1998 as a basis for posing some interesting questions regarding birth coincidences.
=+c. Find the number of resistors, n, for which p1490 # T # 5102 5 .95, where T denotes the total resistance in the circuit.
=+b. What is the probability that the total resistance in the circuit differs from 500 ohms by more than 11 ohms?
=+a. What is the probability that the average resistance in the circuit exceeds 105 ohms?
=+73. Five randomly selected 100-ohm resistors are connected in a series circuit. Suppose that it is known that the population of all such resistors has a mean resistance of exactly 100 ohms with a standard deviation of 1.7 ohms.
=+b. What is the mean of the sampling distribution of the total voltage in four randomly selected 1.5-volt batteries?
=+a. What are the mean and standard error of the sampling distribution of the average voltage in four randomly selected 1.5-volt batteries?
=+72. An electrical appliance uses four 1.5-volt batteries.The batteries are connected in series so that the total voltage supplied to the appliance is the sum of the voltages in the four batteries. Suppose that the actual voltage of all 1.5-volt batteries is known to have a mean of 1.5 volts and
=+from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.244 chapter 5
=+c. Compare the shapes of the histograms in parts (a) and (b), and offer an explanation for any differences that you observe.Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may
=+b. Repeat part (a) by generating 100 samples of size 10 from an exponential distribution with a mean of 5.
=+a. Generate at least 100 samples of size 10 from a uniform distribution on the interval [10, 20].Create a histogram of the 100 sample means, and describe the shape of the histogram.
=+71. Use spreadsheet (e.g., Excel™) or other software to approximate the sampling distribution of the sample mean.
=+c. What is the probability that more than five errors per second will be transmitted?
=+b. Find the probability of transmitting two or more errors per second.
=+a. In any given 1-second period, what is the probability that no errors are transmitted?
=+70. A continuous signal is sent over a communication channel. The number of errors per second, x, at the receiving end of the channel has a normal distribution with a mean and standard deviation of 3 and .8 errors per second, respectively.
=+c. What is the probability of finding two or more errors on a traveler?
=+b. What is the probability that a traveler is free of errors?
=+a. What is the probability that a given traveler will contain at least one incorrect field?
=+69. “Travelers” are documents that accompany a product as it sequences through various production steps. Travelers contain manufacturing instructions pertaining to the particular item or order. Suppose that each of 30 data fields on a particular traveler has a .5% chance of being filled out
=+c. Suppose an inspector inspects two different panels, one with a crack size of c and the other with a crack size of 2c . Again assuming 5 4 and also that the results of the two inspections are independent of one another, what is the probability that exactly one of the two cracks will be
=+b. What is Pd(2c ) when 5 4?
=+a. Verify that Pd(c ) 5 .5.
=+ where c is the crack size that corresponds to a .5 detection probability (and thus is an assessment of the quality of the inspection process).
=+68. According to the article “Optimization of Distribution Parameters for Estimating Probability of Crack Detection” (J. of Aircraft, 2009:2090–2097), the following “Palmberg” equation is commonly used to determine the probability Pd(c) of detecting a crack of size c in an aircraft
=+c. Find the standard deviation of the variable x.
=+b. Find the mean of the variable x.
=+67. A continuous random variable x has a density function of the form f1x2 5 .5x over the interval [0, b].a. Find b.
=+b. Can two or more zip codes have the same value of x?
=+a. List the possible values of the random variable x.
=+66. Let x denote the number of nonzero digits in a randomly selected zip code.
=+A and B are independent, then so are the pairs of events A=and B, A and B=, and A=and B=.)
=+65. Find a formula for the probability that at least one of two independent events occurs. (Hint: If events
=+64. Two pumps that are connected in parallel fail independently of one another on any given day. The probability that only one pump fails is .10, and the probability that neither of the two pumps fails is .05.What is the probability that both pumps fail on a given day? Hint: Use a Venn diagram.
=+63. A battery-operated tool requires that each of its four batteries operate correctly to provide sufficient power to the tool. If each battery operates independently of the others and each has a .10 chance of failing over a 30-hour period of operation, what is the probability that the tool will
=+a. Give a verbal description of the expressions A|B and B|A.b. Does P(A|B) 5 P(B|A)?
=+deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.Supplementary Exercises 243
=+Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has
=+62. A complex assembly contains 20 critical components (labeled C1, C2, . . .), each having a probability of .95 of functioning correctly. Each component must function correctly for the entire assembly to function. Let A denote the event that the assembly fails to function correctly and let B
=+b. What is the probability that a randomly chosen tree does not come from parcel 5?
=+a. What is the probability that a randomly chosen tree comes from one of the first three parcels of land?
=+ Because crop-bearing trees are uniformly planted within each parcel, the probability that a randomly sampled tree from the farm comes from a particular parcel is assumed to be proportional to the size of the parcel.
=+61. A large farming area is divided into five parcels of land of different sizes, as follows:Parcel: B1 B2 B3 B4 B5 Size (acres): 15 20 25 10 20
=+60. Figure 5.5 shows how a tree diagram can be used to verify that {A or B}=5 {A= and B=}. Use a Venn diagram to prove this fact.
=+b. What is the probability that an atmospheric particle will have a radius exceeding .12 m?
=+a. Find the mean radius (in m) of the atmospheric particles.
=+parameters 5 22.62 and 5 .788 (Crow, E.L., and K. Shimizu, Lognormal Distributions: Theory and Applications, Marcel Dekker, New York, 1988: 337).
=+and standard deviation . The successive breaking of particles into finer and finer pieces, a process that can be modeled as a product of positive random variables, leads to lognormal particle size distributions. In particular, small particles suspended in the atmosphere (called aerosols) have
=+Similarly, it can be shown that products of independent positive random variables tend to have lognormal distributions. Recall from Section 1.5 that a random variable x is said to have a lognormal distribution with parameters and if the random variable y 5 ln(x) is normal with mean
=+59. Roughly speaking, the Central Limit Theorem says that sums of independent random variables tend to have (approximately) normal distributions.
=+58. In Exercise 36, what is the probability that the average of two measurements will lie within 2 mm of the true length of the object?
=+b. What is the probability that the proportion of resistors with resistances exceeding 105 ohms in a random sample of 100 will be less than 3%?
=+a. For samples of size 100 from this population, describe the sampling distribution of the sample proportion of resistors that have resistances in excess of 105 ohms.
=+57. Only 2% of a large population of 100-ohm gold-band resistors have resistances that exceed 105 ohms.
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