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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
=+b. Calculate the square root of each value and then construct a quantile plot based on this transformed data. Does it seem plausible that the square root of precipitation is normally distributed?
=+c. Repeat part (b) after transforming by cube roots.
=+49. The article “A Probabilistic Model of Fracture in Concrete and Size Effects on Fracture Toughness”(Magazine of Concrete Res., 1996: 311–320) gives arguments for why fracture toughness in concrete specimens should have a Weibull distribution and presents several histograms of data that
=+fit by superimposed Weibull curves. Consider the following sample of size n 5 18 observations on toughness for high-strength concrete (consistent
=+with one of the histograms); values of pi5 (i 2 .5)y18 are also given:Obs: .47 .58 .65 .69 .72 .74 pi: .0278 .0833 .1389 .1944 .2500 .3056 Obs: .77 .79 .80 .81 .82 .84 pi: .3611 .4167 .4722 .5278 .5833 .6389 Obs: .86 .89 .91 .95 1.01 1.04 pi: .6944 .7500 .8056 .8611 .9167 .9722 Construct a
=+50. In the article “Weibull Parameter of Oil-Immersed Transformer to Evaluate Insulation Reliability on Temporary Overvoltage” (IEEE Trans. on Dielectrics and Elec. Insul., 2010: 1863–1868), researchers investigated the reliability of oil-immersed transformers under various conditions. In
=+researchers measured the breakdown time of the transformer oil gap under various oil flow velocities and exposure to temporary overvoltage. Consider the following breakdown time data (in s) from their experiment where an oil flow at 16 cm/s and an overvoltage of 81kV were applied.7.2 10.0 18.0
=+ Construct a Weibull plot and comment on the plausibility of breakdown time having a Weibull distribution.
=+51. The accompanying figures show (a) a normal quantile plot of the observations on cell interdivision time(IDT) given in Exercise 16 of Section 1.2 and (b) a normal quantile plot of the logarithms of the IDTs.
=+What do these plots suggest about the distribution of cell interdivision time?–2 –1 10 0 1 2(a)20 30 50 60 40 70 Normal quantile IDT–2 –1 2.5 0 1 2 3.5 4.5 Normal quantile
=+52. A plot to assess the plausibility of an exponential population distribution can be based on quantiles of the exponential distribution having 5 1 (i.e., the exponential distribution with density function f(x) 5 e 2x for x . 0). This is because , like for
=+a normal distribution, is a scale parameter. Consider the following failure time observations (1000s of hours) resulting from accelerated life testing of 16 integrated circuit chips of a certain type:82.8 11.6 359.5 502.5 307.8 179.7 242.0 26.5 244.8 304.3 379.1 212.6 229.9 558.9 366.7 204.6
=+ Construct a quantile plot and comment on the plausibility of failure time having an exponential distribution.
=+53. The article “Families of Distributions for Hourly Median Power and Instantaneous Power of Received Radio Signals” (J. of Research for the National Bureau of Standards, 1963: 753–762) suggests the
=+lognormal distribution for x 5 hourly median power (decibels) of received radio signals transmitted between two cities. Consider the following sample of hourly median power readings:2.7 5.4 9.7 22.8 30.5 55.7 66.2 97.3 186.5 240.0
=+a. Is it plausible that these observations were sampled from a normal distribution?
=+b. Is it plausible that these observations were sampled from a lognormal distribution?
=+54. Anxiety disorders and symptoms can often be effectively treated with benzodiazepine medications.It is known that animals exposed to stress exhibit a decrease in benzodiazepine receptor binding in the frontal cortex. The paper “Decreased Benzodiazepine Receptor Binding in Prefrontal Cortex
=+accompanying data on a receptor binding measure(adjusted distribution volume) was read from a graph in the paper:PTSD: 10 20 25 28 31 35 37 38 38 39 39 42 46 Healthy: 23 39 40 41 43 47 51 58 63 66 67 69 72
=+a. Calculate and interpret the values of the mean, median, and standard deviation for each of the two samples.
=+b. Calculate a trimmed mean for each sample by deleting the smallest and largest observations.
=+What is the trimming percentage? What effect does trimming have?
=+c. Determine the value of the interquartile range for each sample. Does either sample contain any outliers? Any extreme outliers?
=+d. Construct a comparative boxplot, and comment on interesting features.
=+e. Would you recommend estimating the difference between the true average binding measure of PTSD individuals and the true average measure for healthy individuals using a method based on assuming that each sample was selected from a normal population distribution? Explain your reasoning.
=+55. A sample of 77 individuals working at a particular office was selected, and the noise level (dBA) experienced by each one was determined, yielding the following data (“Acceptable Noise Levels for Construction Site Offices,” Building Serv. Engr. Res.and Tech., 2009: 87–94).55.3 55.3
=+52. A plot to assess the plausibility of an exponential population distribution can be based on quantiles of the exponential distribution having 5 1 (i.e., the exponential distribution with density function f(x) 5 e 2x
=+ for x . 0). This is because , like for a normal distribution, is a scale parameter. Consider the following failure time observations (1000s of hours) resulting from accelerated life testing of 16 integrated circuit chips of a certain type:82.8 11.6 359.5 502.5 307.8 179.7 242.0 26.5 244.8
=+ Construct a quantile plot and comment on the plausibility of failure time having an exponential distribution.
=+53. The article “Families of Distributions for Hourly Median Power and Instantaneous Power of Received Radio Signals” (J. of Research for the National Bureau of Standards, 1963: 753–762) suggests the lognormal distribution for x 5 hourly median power (decibels) of received radio signals
=+a. Is it plausible that these observations were sampled from a normal distribution?
=+b. Is it plausible that these observations were sampled from a lognormal distribution?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
=+98 chapter 2 Numerical Summary Measures Use various techniques discussed in this chapter to organize, summarize, and describe the data.
=+56. Three different C2F6 flow rates (SCCM) were considered in an experiment to investigate the effect of flow rate on the uniformity (%) of the etch on a silicon wafer used in the manufacture of integrated circuits, resulting in the following data:125: 2.6 2.7 3.0 3.2 3.8 4.6 160: 3.6 4.2 4.2
=+57. Consider a sample x1, . . . , xn, and let xk and sk 2denote the sample mean and variance, respectively, of the first k observations.a. Show that ks2 k11 5 (k 2 1)sk 2 1 kk 1 1(xk11 2 xk)2
=+b. Suppose that a sample of 15 strands of drapery yarn has resulted in a sample mean thread elongation of 12.58 mm and a sample standard deviation of .512 mm. A 16th strand results in an elongation value of 11.8. What are the values of the sample mean and sample standard deviation for all 16
=+58. In 1997 a woman sued a computer keyboard manufacturer, charging that her repetitive stress injuries were caused by the keyboard (Genessy v. Digital Equipment Corp.). The jury awarded about$3.5 million for pain and suffering, but the court then set aside that award as being unreasonable
=+the mean of the awards in the 27 cases. The 27 awards were (in $1000s) 37, 60, 75, 115, 135, 140, 149, 150, 238, 290, 340, 410, 600, 750, 750, 750, 1050, 1100, 1139, 1150, 1200, 1200, 1250, 1576, 1700, 1825, and 2000, from which^xi 5 20,179, ^xi 2 5 24,657,511. What is the maximum possible
=+59. A deficiency of the trace element selenium in the diet can negatively affect growth, immunity, muscle and neuromuscular function, and fertility. The introduction of selenium supplements to dairy cows is justified when pastures have low selenium levels.Authors of the paper “Effects of
=+and Composition of Lactating Cows” (Australian J.of Dairy Tech., 2004: 199–203) supplied the following data on milk selenium concentration (mg/L)for a sample of cows given a selenium supplement and a control sample given no supplement, both initially and after a nine-day period.
=+Obs Init Se Init Cont Final Se Final Cont 1 11.4 9.1 138.3 9.3 2 9.6 8.7 104.0 8.8 3 10.1 9.7 96.4 8.8 4 8.5 10.8 89.0 10.1 5 10.3 10.9 88.0 9.6 6 10.6 10.6 103.8 8.6 7 11.8 10.1 147.3 10.4 8 9.8 12.3 97.1 12.4 9 10.9 8.8 172.6 9.3 10 10.3 10.4 146.3 9.5 11 10.2 10.9 99.0 8.4 12 11.4 10.4 122.3
=+a. Do the initial Se concentrations for the supplement and control samples appear to be similar?Use various techniques from this chapter to summarize the data and answer the question posed.
=+b. Again use methods from this chapter to summarize the data and then describe how the final Se concentration values in the treatment group differ from those in the control group.
=+60. An inequality developed by the Russian mathematician Chebyshev gives information about the percentage of values in any sample or distribution that fall within a specified number of standard deviations of the mean. Let k denote any number satisfying Copyright 2013 Cengage Learning. All
=+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.Supplementary Exercises 99 k $ 1. Then at least 100(1 2
=+a. What does Chebyshev’s inequality say about the percentage of values that are within 2 standard deviations of the mean? Within 3 standard deviations of the mean? Within 5 standard deviations? Within 10 standard deviations?
=+b. What does Chebyshev’s inequality say about the percentage of values that are more than 2 standard deviations from the mean? More than 3 standard deviations from the mean?
=+c. Suppose the distribution of slot width on a forging has a mean value of 1.000 in. and a standard deviation of .0025 in. What percentage of such forgings have a slot width that is between .995 in. and 1.005 in.? If specifications are 1.000 6 .005 in., what percentage of slot widths will
=+d. Refer to part (c). What percentage of such forgings will have a slot width that is outside the interval from .995 in. to 1.005 in. (i.e., either less than .995 or greater than 1.005)? What can be said about the percentage of widths that exceed 1.005 in.?
=+61. Reconsider Chebyshev’s inequality as stated in the previous exercise.
=+a. Compare what the inequality says about the percentage within 1, 2, or 3 standard deviations of the mean value to the corresponding percentages given by the empirical rule.
=+b. An exponential distribution with parameter has both mean value and standard deviation
=+equal to 1y . If component lifetime is exponentially distributed with a mean value of 100 hr, what percentage of these components have lifetimes within 1 standard deviation of the mean lifetime? Within 2 standard deviations? Within 3 standard deviations? Compare these to the percentages given
=+c. Why do you think the percentages from Chebyshev’s inequality so badly understate the actual percentages in the situations of parts(a) and (b)?
=+62. Consider a sample x1, . . . , xn with mean x and standard deviation s, and let zi 5 (xi 2 x)ys. What are the mean and standard deviation of the zi’s?
=+63. The accompanying observations are carbon monoxide levels (ppm) in air samples obtained from a certain region:9.3 10.7 8.5 9.6 12.2 16.6 9.2 10.5 7.9 13.2 11.0 8.8 13.7 12.1 9.8
=+a. Calculate a trimmed mean by trimming the smallest and largest observations, and give the corresponding trimming percentage. Do the same with the two smallest and two largest values trimmed.
=+b. Using the results of part (a), how would you calculate a trimmed mean with a 10% trimming percentage?
=+c. Suppose there had been 16 sample observations.How would you go about calculating a 10%trimmed mean?
=+64. Specimens of three different types of rope wire were selected, and the fatigue limit (MPa) was determined for each specimen, resulting in the accompanying data:Type 1: 350 350 350 358 370 370 370 371 371 372 372 384 391 391 392 Type 2: 350 354 359 363 365 368 369 371 373 374 376 380 383 388
=+a. Construct a comparative boxplot, and comment on similarities and differences.
=+b. Construct a comparative dotplot (a dotplot for each sample with a common scale). Comment on similarities and differences.
=+c. Does the comparative boxplot of part (a) give an informative assessment of similarities and differences? Explain your reasoning.
=+65. The three measures of center introduced in this chapter are the mean, median, and trimmed mean. Two additional measures of center that are occasionally used are the midrange, which is the average of the smallest and largest observations, and the midhinge, which is the average of the two
=+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
=+66. The capacitance (nf) of multilayer ceramic capacitors supplied by a certain vendor is normally distributed with mean value 98 and standard deviation 2.Specifications for these capacitors are 100 6 5 nf.
=+a. What proportion of these capacitors will conform to specification?
=+b. Suppose that these capacitors are shipped in batches of size 20. Let x denote the number of capacitors in a batch that conform to specification.Provided that capacitances of successive capacitors are independent of one another, what kind of distribution does x have? In the long run, in what
=+67. Aortic stenosis refers to a narrowing of the aortic valve in the heart. The paper “Correlation Analysis
=+of Stenotic Aortic Valve Flow Patterns Using Phase Contrast MRI” (Annals of Biomed. Engr., 2005:878–887) gave the following data on aortic root diameter (cm) and gender for a sample of patients having various degrees of aortic stenosis:M: 3.7 3.4 3.7 4.0 3.9 3.8 3.4 3.6 3.1 4.0 3.4 3.8 3.5 F:
=+a. Compare and contrast the diameter observations for the two genders.
=+b. Calculate a 10% trimmed mean for each of the two samples and compare to other measures of center (for the male sample, the interpolation method mentioned in Section 2.1 must be used).
=+68. A study carried out to investigate the distribution of total braking time (reaction time plus acceleratorto-brake movement time, in ms) during real driving conditions at 60 km/hr gave the following summary information on the distribution of times (“A Field Study on Braking Responses during
=+69. Let x denote the maximum physical stress that a unit of a certain product encounters during its lifetime. Suppose that x is normally distributed with 99th percentile 5 5.33 and 10th percentile 5 1.72(suggested in the article “A Formulation of Product Reliability through Environmental
=+70. The indoor thermal climate is an important characteristic affecting the health and productivity of workers in buildings. The paper“Adaptive Comfort Temperature Model of AirConditioned Buildings in Hong Kong” (Building and Environment, 2003: 837–852) reported data on a number of
=+winter. Consider the accompanying values of relative humidity.Summer: 57.18 58.11 56.53 58.61 57.40 62.64 61.72 57.26 53.43 53.71 58.64 45.12 47.52 54.47 55.88 51.08 53.69 54.37 54.36 61.01 52.66 56.20 48.40 46.99 50.63 52.40 52.20 55.95 53.77 Winter: 52.20 41.83 55.63 54.18 54.56 56.20 58.09
=+ Use methods from this and the previous chapter to describe, summarize, compare, and contrast the summer and winter relative humidity data.
=+1. Consider the strength data for beams given in Example 1.2.
=+a. Construct a stem-and-leaf display of the data.What appears to be a representative strength value?
=+Do the observations appear to be highly concentrated about the representative value or rather spread out?
=+b. Does the display appear to be reasonably symmetric about a representative value, or would you Unless otherwise noted, all content on this page is © Cengage Learning.describe its shape in some other way?
=+c. Do there appear to be any outlying strength values?
=+d. What proportion of strength observations in this sample exceed 10 MPa?
=+2. The article cited in Example 1.2 also gave the accompanying strength observations for cylinders:6.1 5.8 7.8 7.1 7.2 9.2 6.6 8.3 7.0 8.3 7.8 8.1 7.4 8.5 8.9 9.8 9.7 14.1 12.6 11.2
=+a. Construct a comparative stem-and-leaf display of the beam and cylinder data, then answer the questions in parts (b)–(d) of Exercise 1 for the observations on cylinders.
=+b. In what ways are the two sides of the display similar? Are there any obvious differences between the beam observations and the cylinder observations?
=+3. The accompanying specific gravity values for various wood types used in construction appeared in the article “Bolted Connection Design Values Based on European Yield Model” (J. of Structural Engr., 1993:2169–2186):.31 .35 .36 .36 .37 .38 .40 .40 .40.41 .41 .42 .42 .42 .42 .42 .43 .44.45
=+4. Allowable mechanical properties for structural design of metallic aerospace vehicles requires an approved method for statistically analyzing empirical test data. The article “Establishing Mechanical Property Allowables for Metals” (J. of Testing and Evaluation, 1998: 293–299) used the
=+a. Construct a stem-and-leaf display of the data by first deleting (truncating) the tenths digit and then repeating each stem value five times (once for leaves 0 and 1, a second time for leaves 2 and 3, etc.). Why is it relatively easy to identify a representative strength value?
=+b. Construct a histogram using equal-width classes with the first class having a lower limit of 122 and an upper limit of 124. Then comment on any interesting features of the histogram.
=+5. Consider the accompanying values of golf course lengths (yards) for a sample of courses designated by Golf Magazine as being among the most challenging in the United States:6433 6435 6464 6470 6506 6526 6527 6583 6605 6614 6694 6700 6713 6745 6770 6770 6790 6798 6850 6870 6873 6890 6900 6904
=+a. Would it be best to use one-digit, two-digit, or three-digit stems as a basis for a stem-and-leaf display? Explain your reasoning.
=+b. Construct a stem-and-leaf display based on two-digit stems and two-digit leaves, with successive leaves separated by either a comma or a space.
=+c. Construct a stem-and-leaf display in which the leaf of each observation is its tens digit (so the ones digit is truncated). Does this display appear
=+to be significantly less informative about course lengths than the display of part (b)?
=+ What advantage would this display have over the one in part (b) if there had been 200 courses in the sample?
=+6. Construct two stem-and-leaf displays for the accompanying set of exam scores, one in which each stem value appears just once and the other in which stem values are repeated:74 89 80 93 64 67 72 70 66 85 89 81 81 71 74 82 85 63 72 81 81 95 84 81 80 70 69 66 60 83 85 98 84 68 90 82 69 72 87 88
=+ What feature of the data is revealed by the display with repeated stems that is not so readily apparent in the first display?
=+7. Temperature transducers of a certain type are shipped in batches of 50. A sample of 60 batches was selected, and the number of transducers in each batch not conforming to design specifications was determined, resulting in the following data:2 1 2 4 0 1 3 2 0 5 3 3 1 3 2 4 7 0 2 3 0 4 2 1 3 1 1
=+a. Determine frequencies and relative frequencies for the observed values of x 5 number of nonconforming transducers in a batch.
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