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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
=+35. Suppose that the thicknesses of bolts (mm) manufactured by a certain process can be modeled with a normal distribution having 5 10 and 5 1. Note: The density curve here is just the standard normal curve shifted to be centered at 10 rather than 0.
=+a. What is the long-run proportion of bolts whose thicknesses are at most 11 mm? Hint: The corresponding normal curve area is identical to what z curve area?
=+b. In the long run, what proportion of these bolts will have thickness values between 7.5 mm and 12.5 mm?
=+c. In the long run, what proportion of these bolts will have thicknesses that exceed 11.5 mm?
=+36. Suppose the flow of current (milliamps) in wire strips of a certain type under specified conditions can be modeled with a normal distribution having 5 20 and 5 1 (think about how the corresponding density curve relates to the standard normal curve).
=+a. What proportion of strips will have a current flow of between 18.5 and 22 milliamps?
=+b. What proportion of strips will have a current flow exceeding 15 milliamps?
=+c. How large must a current flow be to be among the largest 5% of all flows?
=+37. Mopeds (small motorcycles with an engine capacity below 50 cm3) are popular in Europe because of their mobility, ease of operation, and low cost. The article“Procedure to Verify the Maximum Speed of Automatic Transmission Mopeds in Periodic Motor Vehicle Inspections” (J. of Automobile
=+1623) described a rolling bench test for determining maximum vehicle speed. A normal distribution with 5 46.8 km/h and 5 1.75 km/h is postulated.
=+a. What proportion of mopeds have a maximum speed that is at most 50 km/h?
=+b. What proportion of mopeds have a maximum speed that is at least 48 km/h?
=+c. What speed separates the fastest 75% of all mopeds from the others?
=+38. Spray drift is a constant concern for pesticide applicators and agricultural producers. The inverse relationship between droplet size and drift potential is well known. The paper “Effects of 2,4-D Formulation and Quinclorac on Spray Droplet Size and Deposition” (Weed Technology, 2005:
=+a. What proportion of all droplets have a size that is less than 1500 m? At least 1000 m?
=+b. What proportion of all droplets have a size that is between 1000 and 1500 m?
=+c. How would you characterize the smallest 2% of all droplets?
=+39. The article “Reliability of Domestic-Waste Biofilm Reactors” (J. of Envir. Engr., 1995: 785–790) suggests that substrate concentration (mg/cm3) of influent to a reactor is normally distributed with 5 .30 and 5 .06.
=+a. What proportion of concentration values exceed.25?
=+b. What proportion of concentration values are at most .10?
=+c. How would you characterize the largest 5% of all concentration values?
=+40. Consider babies born in the “normal range” of 37–43 weeks gestational age. Extensive data supports the assumption that for such babies born in the United States, birth weight is normally distributed with 5 3432 g and 5 482 g. [The article “Are Babies Normal?”
=+(The American Statistician, 1999: 298–302) analyzed data from a particular year; for a sensible choice of class intervals, a histogram did not look normal but further investigation revealed that this was because some hospitals measured weight in grams and others measured to the nearest ounce
=+a. For babies of this type, what proportion of all birth weights exceeds 4000 g?
=+b. For babies of this type, what proportion of all birth weights is between 3000 and 4000 g?
=+c. How would you characterize the highest .1% of all birth weights?
=+d. What value c is such that the interval (3432 2 c, 3432 1c) includes 98% of all birth weights?
=+41. Let x denote the number of flaws along a 100-m reel of magnetic tape (values of x are whole numbers). Suppose x has approximately a normal distribution with 5 25 and 5 5.
=+a. What proportion of reels will have between 20 and 40 flaws, inclusive?
=+1.5 Other Continuous Distributions Normal density curves are always bell-shaped and therefore symmetric. Exponential density curves are positively skewed but have their maximum at x 5 0 and decrease as x increases.Many histograms of data encountered in applied work are skewed and unimodal,
=+a maximum and then declining. We now present several useful distributions that have this property. Our survey is not exhaustive. Consult the bibliography at the end of the chapter for information on the gamma, beta, and other distributions not discussed here.The Lognormal Distribution
=+Lognormal distributions are related to normal distributions in exactly the way the name suggests.
=+b. What proportion of reels will have at most 30 flaws? Fewer than 30 flaws?
=+42. Based on extensive data from an urban freeway near Toronto, Canada, “it is assumed that free speeds can best be represented by a normal distribution”(“Impact of Driver Compliance on the Safety and Operational Impacts of Freeway Variable Speed
=+Limit Systems” (J. of Transp. Engr., 2011: 260–268)).The values of and reported in the article were 119 km/h and 13.1 km/h, respectively.
=+a. What percentage of vehicles have speeds that are between 100 and 120 km/hr?
=+b. What speed characterizes the fastest 10% of all speeds?
=+c. The posted speed limit was 100 km/hr. What percentage of vehicles were traveling at speeds exceeding this posted limit?
=+d. What two values, symmetrically placed about 119, capture 90% of all vehicle speeds.
=+e. What values symmetrically placed about 119 separate .1% of the most extreme vehicle speeds from the rest?
=+43. A theoretical justification based on a certain material failure mechanism underlies the assumption that ductile strength of a material has a lognormal distribution. Suppose the values of the parameters are 5 5 and 5 .1.
=+a. What proportion of material specimens have a ductile strength exceeding 120? What proportion have a ductile strength of at least 120?
=+b. What proportion of material specimens have a ductile strength between 110 and 130?
=+c. If the smallest 5% of strength values were unacceptable, what would be the minimum acceptable strength?
=+44. Nonpoint source loads are chemical masses that travel to the main stem of a river and its tributaries in flows that are distributed over relatively long stream reaches in contrast to those that enter at well-defined and regulated points. The article “Assessing Uncertainty in Mass Balance
=+a. What proportion of source loads are at most 15,000 kg/day/km?
=+b. What interval (a,b) is such that 95% of all source loads have values in this interval, 2.5%have values less thana, and 2.5% have values exceeding b?
=+45. The article “Response of SiGf/Si3N4 Composites Under Static and Cyclic Loading—An Experimental and Statistical Analysis” (J. of Engr. Materials and Technology, 1997: 186–193) suggests that tensile strength (MPa) of composites under specified conditions can be modeled by a Weibull
=+a. Sketch a graph of the density function.
=+b. What proportion of specimens of this type have strength values exceeding 175?
=+c. What proportion of specimens of this type have strength values between 150 and 175?
=+d. What strength value separates the weakest 10%of all specimens from the remaining 90%?
=+46. Suppose that fracture strength (MPa) of silicon nitride braze joints under certain conditions has a Weibull distribution with 5 5 and 5 125 (suggested by data in the article “Heat-Resistant Active Brazing of Silicon Nitride: Mechanical Evaluation of Braze Joints,” Welding J., August
=+a. What proportion of such joints have a fracture strength of at most 100? Between 100 and 150?
=+b. What strength value separates the weakest 50% of all joints from the strongest 50%?
=+c. What strength value characterizes the weakest 5%of all joints?
=+47. The Weibull distribution discussed in this section has a positive density function for all x . 0. In some situations, the smallest possible value of x will be some number that exceeds zero. A shifted Weibull distribution, appropriate in such situations, has a density function for x .
=+ 5 4, and 5 5.8 specify an appropriate distribution for diameters of trees in a particular location.
=+a. What proportion of trees have diameters between 2 and 4 cm?
=+b. What proportion of trees have diameters that are at least 5 cm?
=+c. What is the median diameter of trees, that is, the value separating the smallest 50% from the largest 50% of all diameters?
=+48. The paper “Study on the Life Distribution of Microdrills” (J. of Engr. Manufacture, 2002: 301–305) reported the following observations, listed in increasing order, on drill lifetime (number of holes that a drill machines before it breaks) when holes were drilled in a certain brass
=+a. Construct a histogram of the data using class boundaries 0, 50, 100, . . . , and then comment on interesting characteristics.
=+b. Construct a histogram of the natural logarithms of the lifetime observations, and comment on interesting characteristics.11 14 20 23 31 36 39 44 47 50 59 61 65 67 68 71 74 76 78 79 81 84 85 91 93 96 99 101 104 105 105 112 118 123 136 139 141 148 158 161 168 184 206 248 263 289 322 388 513
=+49. The authors of the paper from which the data in the previous exercise was extracted suggested that a reasonable probability model for drill lifetime was a lognormal distribution with 5 4.5 and 5 .8.
=+a. What proportion of lifetime values are at most 100?
=+b. What proportion of lifetime values are at least 200? Greater than 200?
=+50. The article cited in Example 1.20 proposed the lognormal distribution with 5 4.5 and 5 .625 as a model for total hydrocarbon emissions (g/gal).
=+a. What proportion of engines emit at least 50 g/gal?Between 50 and 150 g/gal?
=+b. What value c separates the best 1% of engines with respect to THC emissions from the remaining 99%?
=+51. The article “On Assessing the Accuracy of Offshore Wind Turbine Reliability-Based Design Loads from the Environmental Contour Method” (Intl. J. of Offshore and Polar Engr., 2005: 132–140) proposes the Weibull distribution with 5 1.817 and 5 .863 as a model for 1-hour significant
=+a. What proportion of wave heights are at most 0.5 m?
=+b. What proportion of wave heights are between 0.2 and 0.6 m?
=+c. What is the 90th percentile of the wave height distribution? The 10th percentile?
=+52. When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is 5%. Let x denote the number of defective boards in a batch of 25 boards, so that x has a binomial distribution with n 5 25 and 5 .05.
=+a. What proportion of batches have at most 2 defective boards?
=+b. What proportion of batches have at least 5 defective boards?
=+c. What proportion of batches will have all 25 boards free of defects?
=+53. A company packages its crystal goblets in boxes containing six goblets. Suppose that 12% of all its goblets have cosmetic flaws and that the condition of any particular goblet with respect to flaws is independent of the condition of any other goblet.
=+a. What proportion of boxes will contain only one goblet with a cosmetic flaw?
=+b. What proportion of boxes will contain at least two goblets with cosmetic flaws?
=+c. What proportion of boxes will have between one and three goblets, inclusive, with cosmetic flaws?
=+54. On his way to work, a friend of ours must pass through ten traffic signals. Suppose that in the long run, she encounters a red light at 40% of these signals and that whether any particular signal is red is independent of whether any other one is red.
=+a. On what proportion of days will our friend encounter at most two red lights? At least five red lights?
=+b. On what proportion of days will our friend encounter between two and five (inclusive) red lights?
=+55. Suppose that 10% of all bits transmitted through a digital communication channel are erroneously received and that whether any particular bit is erroneously received is independent of whether any other bit is erroneously received. Consider sending
=+a very large number of messages, each consisting of 20 bits.
=+a. What proportion of these messages will have at most 2 erroneously received bits?
=+b. What proportion of these messages will have at least 5 erroneously received bits?
=+c. For what proportion of these messages will more than half the bits be erroneously received?
=+56. Components arrive at a distributor in very large batches. A batch can be characterized as acceptable only if the fraction of defective components in the batch is at most .10. The distributor decides to randomly select ten components from the batch, test each one, and accept the batch only
=+a. If the actual fraction of defectives in each batch is only 5 .01, what proportion of batches will be accepted? Repeat this calculation for the following values of : .05, .10, .20, and .25.
=+b. A graph of the proportion of batches accepted versus the actual fraction of defectives is called the operating characteristic curve. Use the results of part (a) to sketch this curve for 0 # # 1 (proportion of batches accepted is on the vertical axis and is on the horizontal axis).
=+c. Suppose the distributor decides to be more demanding by accepting a batch only if the sample contains at most one defective component. Repeat parts (a) and (b) with this new acceptance sampling plan. Does this plan appear more satisfactory than the original plan?
=+57. Suppose that the number of drivers who travel between a particular origin and destination during a designated time period has a Poisson distribution with parameter 5 20 (suggested in the article“Dynamic Ride Sharing: Theory and Practice,” J.of Transp. Engr., 1997: 308–312). In the
=+in what proportion of time periods will the number of driversa. Be at most 10?b. Exceed 20?c. Be between 10 and 20, inclusive? Be strictly between 10 and 20?
=+58. Let x be the number of material anomalies occurring in a particular region of an aircraft gas-turbine disk. The article “Methodology for Probabilistic Life Prediction of Multiple-Anomaly Materials”(Amer. Inst. of Aeronautics and Astronautics J., 2006: 787–793) proposes a Poisson
=+a. What proportion of gas-turbine disks have exactly one anomaly?
=+b. What proportion of gas-turbine disks have at least three anomalies?
=+c. What proportion of gas-turbine disks have between one and six anomalies inclusive?
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