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applied statistics and probability for engineers

Questions and Answers of Applied Statistics And Probability For Engineers

=+b. What proportion of batches in the sample have at most five nonconforming transducers? What proportion have fewer than five? What proportion have at least five nonconforming units?
=+c. Draw a histogram of the data using relative frequency on the vertical scale, and comment on its features.
=+8. In a study of author productivity (“Lotka’s Test,” Collection Mgmt., 1982: 111–118), a large number of authors were classified according to the number of articles they had published
=+a. Construct a histogram corresponding to this frequency distribution. What is the most interesting feature of the shape of the distribution?
=+b. What proportion of these authors published at least five papers? At least ten papers? More than ten papers?
=+c. Suppose the five 15s, three 16s, and three 17s had been lumped into a single category displayed as
=+;$15.< Would you be able to draw a histogram?Explain.
=+d. Suppose that instead of the values 15, 16, and 17 being listed separately, they had been combined into a 15–17 category with frequency 11. Would you be able to draw a histogram? Explain.
=+9. The number of contaminating particles on a silicon wafer prior to a certain rinsing process was determined for each wafer in a sample of size 100, resulting in the following frequencies:Number
=+a. What proportion of the sampled wafers had at least one particle? At least five particles?
=+b. What proportion of the sampled wafers had between five and ten particles, inclusive? Strictly between five and ten particles?
=+c. Draw a histogram using relative frequency on the vertical axis. How would you describe the shape of the histogram?
=+10. The article “Knee Injuries in Women Collegiate Rugby Players” (Amer. J. of Sports Medicine, 1997:360–362) gave the following data on type of injury(A 5 mensical tear, B 5 MCL tear, C 5
=+Construct a Pareto diagram for this data. The three most frequently occurring types of injuries account for what proportion of all injuries?
=+11. The article “Determination of Most Representative Subdivision” (J. of Energy Engr., 1993: 43–55) gave data on various characteristics of subdivisions that could be used in deciding
=+1280 5320 4390 2100 1240 3060 4770 1050 360 3330 3380 340 1000 960 1320 530 3350 540 3870 1250 2400 960 1120 2120 450 2250 2320 2400 3150 5700 5220 500 1850 2460 5850 2700 2730 1670 100 5770 3150
=+a. Construct a stem-and-leaf display using the thousands digit as the stem and the hundreds digit as the leaf, and comment on the various features of the display.
=+b. Construct a histogram using class boundaries 0, 1000, 2000, 3000, 4000, 5000, and 6000. What proportion of subdivisions have total length less than 2000? Between 2000 and 4000? How would you
=+12. The article cited in Exercise 11 also gave the following values of the variables y 5 number of culs-de-sac and z 5 number of intersections:y: 1 0 1 0 0 2 0 1 1 1 2 1 0 0 1 1 0 1 1 1 1 0 0 0 1
=+a. Construct a histogram for the y data. What proportion of these subdivisions had no culs-de-sac?At least one cul-de-sac?
=+b. Construct a histogram for the z data. What proportion of these subdivisions had at most five intersections? Fewer than five intersections?
=+13. The article “Ecological Determinants of Herd Size in the Thorncraft’s Giraffe of Zambia” (Afric. J. Ecol., 2010: 962–971) gave the following data (read from a graph) on herd size for a
=+a. What proportion of the sampled herds had just one giraffe?
=+b. What proportion of the sampled herds had six or more giraffes (characterized in the article as “large herds”)?
=+c. What proportion of the sampled herds had between 5 and 10 giraffes inclusive?
=+d. Draw a histogram using relative frequency on the vertical axis. How would you describe the shape of this histogram?
=+14. The article “Statistical Modeling of the Time Course of Tantrum Anger” (J. of Applied Stats, 2009: 1013–1034) discussed how anger intensity in children’s tantrums could be related to
=+Draw the histogram and then comment on any interesting features.
=+15. Automated electron backscattered diffraction is now being used in the study of fracture phenomena.The following information on misorientation angle(degrees) was extracted from the article
=+a. Is it true that more than 50% of the sampled angles are smaller than 15°, as asserted in the paper?
=+b. What proportion of the sampled angles are at least 30°?
=+c. Roughly what proportion of angles are between 10° and 25°?
=+d. Construct a histogram and comment on any interesting features.
=+16. A transformation of data values by means of some mathematical function, such as 1x or 1yx, can often yield a set of numbers that has “nicer” statistical properties than the original data.
=+For example, the article “Time Lapse Cinematographic Analysis of Beryllium–Lung Ibroblast Interactions”(Envir. Research, 1983: 34–43) reported the results of experiments designed to study
=+19.1 38.4 72.8 48.9 21.4 20.7 57.3 40.9 Construct a histogram of this data based on classes with boundaries 10, 20, 30, . . . . Then calculate log10(x) for each observation, and construct a
=+17. The accompanying data set consists of observations on shear strength (lb) of ultrasonic spot welds made on a certain type of alclad sheet.Construct a relative frequency histogram based on ten
=+J. of Aircraft, 1983: 552–556.) Comment on its features.5434 4948 4521 4570 4990 5702 5241 5112 5015 4659 4806 4637 5670 4381 4820 5043 4886 4599 5288 5299 4848 5378 5260 5055 5828 5218 4859 4780
=+18. 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 lifetimes
=+a. Why can a frequency distribution not be based on the class intervals 0–50, 50–100, 100–150, and so on?
=+b. Construct a frequency distribution and histogram of the data using class boundaries 0, 50, 100, . . . and then comment on interesting characteristics.
=+c. Construct a frequency distribution and histogram of the natural logarithms of the lifetime observations and comment on interesting characteristics.
=+d. What proportion of the lifetime observations in this sample are less than 100? What proportion of the observations are at least 200?
=+19. A continuous variable x is said to have a uniform distribution if the density function is given by f(x) 5 c 1b 2 a a , x , b 0 otherwise The corresponding density “curve” has constant
=+a. Draw the density curve, and verify that the total area under the curve is indeed 1.
=+b. In the long run, what proportion of forms will take between 4.5 min and 5.5 min to process? At least 4.5 min to process?
=+c. What value separates the slowest 50% of all processing times from the fastest 50% (the median of the distribution)?
=+d. What value separates the best 10% of all processing times from the remaining 90%?
=+20. Suppose that the reaction temperature x (°C) in a certain chemical process has a uniform distribution with a 5 25 and b 5 5 (refer to Exercise 19 for a description of a uniform distribution).
=+a. In the long run, what proportion of these reactions will have a negative value of temperature?
=+b. In the long run, what proportion of temperatures will be between 22 and 2? Between 22 and 3?
=+c. For any number k satisfying 25 , k , k 1 4 , 5,
=+what long-run proportion of temperatures will be between k and k 1 4?
=+21. Suppose that your morning waiting time for a bus has a uniform distribution on the interval from 0 to 5 min, and your afternoon waiting time also has this distribution. Then if x denotes the
=+0 for other values of xa. Draw the density curve, and verify that f(x) specifies a legitimate distribution.
=+b. In the long run, what proportion of your total daily waiting times will be at most 3 min? At least 7 min?At least 4 min? Between 4 min and 7 min?
=+c. What value separates the longest 10% of your daily waiting times from the remaining 90%?
=+22. Data collected at Toronto Pearson International Airport suggests that an exponential distribution with 5 .37 is a good model for rainfall duration in hours (Urban Stormwater Management
=+a. What proportion of rainfall durations at this location are at least 2 hours? At most 3 hours?Between 2 and 3 hours?
=+b. What must the duration of a rainfall be to place it among the longest 5% of all times?
=+23. Extensive experience with fans of a certain type used in diesel engines has suggested that the exponential distribution with 5 .00004 provides a good model for time until failure (hr).
=+a. Sketch a graph of the density function.
=+b. What proportion of fans will last at least 20,000 hr? At most 30,000 hr? Between 20,000 and 30,000 hr?
=+c. What must the lifetime of a fan be to place it among the best 1% of all fans? Among the worst 1%?
=+24. The article “Probabilistic Fatigue Evaluation of Riveted Railway Bridges” (J. of Bridge Engr., 2008:237–244) suggested the exponential distribution with 5 1 6 as a model for the
=+a. What proportion of stress ranges are at least 2 MPa? At most 7 MPa? Between 5 and 10 MPa?
=+b. What value separates the highest 2% of the stress ranges from the remaining 98%?
=+25. The actual tracking weight of a stereo cartridge set to track at 3 g can be regarded as a continuous variable with density function f(x) 5 c[1 2 (x 2 3)2] for 2 , x , 4 and f(x) 5 0 otherwise.
=+a. Determine the value of c [you might find it helpful to graph f(x)].
=+b. What proportion of actual tracking weights exceed the target weight?
=+c. What proportion of actual tracking weights are within .25 g of the target weight?
=+26. Let x represent the number of underinflated tires on an automobile.
=+a. Which of the following p(x) functions specifies a legitimate distribution for x, and why are the other two not legitimate?(i) p(0) 5.3, p(1) 5 .2, p(2) 5 .1, p(3) 5 .05, p(4) 5 .05(ii) p(0) 5
=+b. For the legitimate distribution of part (a), determine the long-run proportion of cars having at most two underinflated tires, the proportion having fewer than two underinflated tires, and the
=+27. A mail-order computer business has six telephone lines. Let x denote the number of lines in use at a specified time. Suppose the mass function of x is given by x: 0 1 2 3 4 5 6 p(x): .10 .15
=+a. In the long run, what proportion of the time will at most three lines be in use? Fewer than three lines?
=+b. In the long run, what proportion of the time will at least five lines be in use?
=+c. In the long run, what proportion of the time will between two and four lines, inclusive, be in use?
=+d. In the long run, what proportion of the time will at least four lines not be in use?
=+28. A contractor is required by a county planning department to submit 1, 2, 3, 4, or 5 forms (depending on the nature of the project) when applying for a building permit. Let y denote the number
=+5. Determine the value ofc, as well as the long-run proportion of applications that require at most three forms and the long-run proportion that require between two and four forms, inclusive.
=+29. Many manufacturers have quality control programs that include inspection of incoming materials for defects. Suppose a computer manufacturer receives computer boards in batches of five. Two
=+boards are defective; for convenience, number the defective boards as 1 and 2, and the nondefective boards as 3, 4, and 5. Let x denote the number of defective boards among the two actually
=+2, another of boards 1 and 3, and so on. How many such samples are there, and what is the value of x for each sample?
=+30. Suppose that values are repeatedly chosen from a standard normal distribution.
=+a. In the long run, what proportion of values will be at most 2.15? Less than 2.15?
=+b. What is the long-run proportion of selected values that will exceed 1.50? That will exceed 22.00?
=+c. What is the long-run proportion of values that will be between 21.23 and 2.85?
=+d. What is the long-run proportion of values that will exceed 5? That will exceed 25?
=+e. In the long run, what proportion of selected values z will satisfy |z| , 2.50?
=+31. In the long run, what proportion of values selected from the standard normal distribution will satisfy each of the following conditions?a. Be at most 1.78b. Exceed .55c. Exceed 2.80d. Be
=+32.a. What value z* is such that the area under the standard normal curve to the left of z* is .9082?b. What value z* is such that the area under the standard normal curve to the left of that
=+c. What value z* is such that the area under the standard normal curve to the right of z* is .121?
=+d. What value z* is such that the area under the standard normal curve between 2 z* and z*is .754?
=+e. How far to the right of 0 would you have to go to capture an upper-tail z curve area of .002? How U
=+far to the left of 0 would you have to go to capture this same lower-tail area?33. Suppose that values are successively chosen from the standard normal distribution.
=+a. How large must a value be to be among the largest 15% of all values selected?
=+b. How small must a value be to be among the smallest 25% of all values selected?
=+c. What values are among the 4% that are farthest from 0?
=+34. Determine the following percentiles for the standard normal distribution:a. 91stb. 9thc. 22ndd. 99.9th

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