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All Textbook Solutions for Probability and Statistics for Engineering and the Sciences
Give one possible sample of size 4 from each of the following populations: a. All daily newspapers published in the United States b. All companies listed on the New York Stock Exchange c. All students at your college or university d. All grade point averages of students at your college or universityFor each of the following hypothetical populations, give a plausible sample of size 4: a. All distances that might result when you throw a football b. Page lengths of books published 5 years from now c. All possible earthquake-strength measurements (Richter scale) that might be recorded in California during the next year d. All possible yields (in grams) from a certain chemical reaction carried out in a laboratoryConsider the population consisting of all computers of a certain brand and model, and focus on whether a computer needs service while under warranty. a. Pose several probability questions based on selecting a sample of 100 such computers. b. What inferential statistics question might be answered by determining the number of such computers in a sample of size 100 that need warranty service?a. Give three different examples of concrete populations and three different examples of hypothetical populations. b. For one each of your concrete and your hypothetical populations, give an example of a probability question and an example of an inferential statistics question.Many universities and colleges have instituted supplemental instruction (SI) programs, in which a student facilitator meets regularly with a small group of students enrolled in the course to promote discussion of course material and enhance subject mastery. Suppose that students in a large statistics course (what else?) are randomly divided into a control group that will not participate in SI and a treatment group that will participate. At the end of the term, each students total score in the course is determined. a. Are the scores from the SI group a sample from an existing population? If so, what is it? If not, what is the relevant conceptual population? b. What do you think is the advantage of randomly dividing the students into the two groups rather thar letting each student choose which group to join? c. Why didnt the investigators put all students in the treatment group? [Note: The article Supplemental Instruction: An Effective Component of Student Affairs Programming (f of College Student Devel 1997: 577-586) discusses the analysis of data from several SI programs.]The California State University (CSU) system consists of 23 campuses, from San Diego State in the south to Humboldt State near the Oregon border. A CSU administrator wishes to make an inference about the average distance between the hometowns of students and their campuses. Describe and discuss several different sampling methods that might be employed. Would this be an enumerative or an analytic study? Explain your reasoning.A certain city divides naturally into ten district neighborhoods. How might a real estate appraiser select a sample of single-family homes that could be used as a basis for developing an equation to predict appraised value from characteristics such as age. size, number of bathrooms, distance to the nearest school, and so on? Is the study enumerative or analytic?The amount of flow through a solenoid valve in an automobiles pollution-control system is an important characteristic. An experiment was carried out to study how flow rate depended on three factors: armature length, spring load, and bobbin depth. Two different levels (low and high) of each factor were chosen, and a single observation on flow was made for each combination of levels. a. The resulting data set consisted of how many observations? b. Is this an enumerative or analytic study? Explain your reasoning.In a famous experiment carried out in 1882, Michelson and Newcomb obtained 66 observations on the time it took for light to travel between two locations in Washington, D.C. A few of the measurements (coded in a certain manner) were 31, 23, 32, 36, 2, 26, 27, and 31. a. Why are these measurements not identical? b. Is this an enumerative study? Why or why not?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 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?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 .46 .46 .47 .48 .48 .48 .51 .54 .54 .55 .58 .62 .66 .66 .67 .68 .75 Construct a stem-and-leaf display using repeated stems, and comment on any interesting features of the display.The accompanying summary data on CeO2 particle sizes (nm) under certain experimental conditions was read from a graph in the article Nanoceria-Energetics of Surfaces, Interfaces and Water Adsorption (J. of the Amer. Ceramic Soc., 2011: 3992-3999): 3.03.5 3.54.0 4.04.5 4.55.0 5.05.5 5 15 27 34 22 5.56.0 6.06.5 6.57.0 7.07.5 7.58.0 14 7 2 4 1 a. What proportion of the observations are less than 5? b. What proportion of the observations are at least 6? c. Construct a histogram with relative frequency on the vertical axis and comment on interesting features. In particular does the distribution of particle sizes appear to be reasonably symmetric or somewhat skewed? [Note: The investigators fit a lognormal distribution to the data; this is discussed in Chapter 4.] d. Construct a histogram with density on the vertical axis and compare to the histogram in (c).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 accompanying data on tensile ultimate strength (ksi) as a basis for addressing the difficulties in developing such a method. 122.2 124.2 124.3 125.6 126.3 126.5 126.5 127.2 127.3 127.5 127.9 128.6 128.8 129.0 129.2 129.4 129.6 130.2 130.4 130.8 131.3 131.4 131.4 131.5 131.6 131.6 131.8 131.8 132.3 132.4 132.4 132.5 132.5 132.5 132.5 132.6 132.7 132.9 133.0 133.1 133.1 133.1 133.1 133.2 133.2 133.2 133.3 133.3 133.5 133.5 133.5 133.8 133.9 134.0 134.0 134.0 134.0 134.1 134.2 134.3 134.4 134.4 134.6 134.7 134.7 134.7 134.8 134.8 134.8 134.9 134.9 135.2 135.2 135.2 135.3 135.3 135.4 135.5 135.5 135.6 135.6 135.7 135.8 135.8 135.8 135.8 135.8 135.9 135.9 135.9 135.9 136.0 136.0 136.1 136.2 136.2 136.3 136.4 136.4 136.6 136.8 136.9 136.9 137.0 137.1 137.2 137.6 137.6 137.8 137.8 137.8 137.9 137.9 138.2 138.2 138.3 138.3 138.4 138.4 138.4 138.5 138.5 138.6 138.7 138.7 139.0 139.1 139.5 139.6 139.8 139.8 140.0 140.0 140.7 140.7 140.9 140.9 141.2 141.4 141.5 141.6 142.9 143.4 143.5 143.6 143.8 143.8 143.9 144.1 144.5 144.5 147.7 147.7 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 1 and 2. a second time for leaves 3 and 4. 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.The accompanying data set consists of observations on shower-flow rate (L/min) for a sample of n = 129 houses in Perth, Australia (An Application of Bayes Methodology to the Analysis of Diary Records in a Water Use Study, J. Amer. Stat. Assoc., 1987: 705-711): 4.6 12.3 7.1 7.0 4.0 9.2 6.7 6.9 11.5 5.1 11.2 10.5 14.3 8.0 8.8 6.4 5.1 5.6 9.6 7.5 7.5 6.2 5.8 2.3 3.4 10.4 9.8 6.6 3.7 6.4 8.3 6.5 7.6 9.3 9.2 7.3 5.0 6.3 13.8 6.2 5.4 4.8 7.5 6.0 6.9 10.8 7.5 6.6 5.0 3.3 7.6 3.9 11.9 2.2 15.0 7.2 6.1 15.3 18.9 7.2 5.4 5.5 4.3 9.0 12.7 11.3 7.4 5.0 3.5 8.2 8.4 7.3 10.3 11.9 6.0 5.6 9.5 9.3 10.4 9.7 5.1 6.7 10.2 6.2 8.4 7.0 4.8 5.6 10.5 14.6 10.8 15.5 7.5 6.4 3.4 5.5 6.6 5.9 15.0 9.6 7.8 7.0 6.9 4.1 3.6 11.9 3.7 5.7 6.8 11.3 9.3 9.6 10.4 9.3 6.9 9.8 9.1 10.6 4.5 6.2 8.3 3.2 4.9 5.0 6.0 8.2 6.3 3.8 6.0 a. Construct a stem-and-leaf display of the data. b. What is a typical, or representative, flow rate? c. Does the display appear to be highly concentrated or spread out? d. Does the distribution of values appear to be reasonably symmetric? If not, how would you describe the departure from symmetry? e. Would you describe any observation as being far from the rest of the data (an outlier)?Do running limes of American movies differ somehow from running times of French movies? The author investigated this question by randomly selecting 25 recent movies of each type, resulting in the following running times: Am: 94 90 95 93 128 95 125 91 104 116 162 102 90 110 92 113 116 90 97 103 95 120 109 91 138 Fr: 123 116 90 158 122 119 125 90 96 94 137 102 105 106 95 125 122 103 96 111 81 113 128 93 92 Construct a comparative stem-and-leaf display by listing stems in the middle of your paper and then placing the Am leaves out to the left and the Fr leaves out to the right. Then comment on interesting features of the display.The article cited in Example 1.2 also gave the accompanying strength observations for cylinders: a. Construct a comparative stem-and-leaf display (see the previous exercise) of the beam and cylinder data, and then answer the questions in parts (b)-(d) of Exercise 10 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? c. Construct a dotplot of the cylinder data.The accompanying data came from a study of collusion in bidding within the construction industry (Detection of Collusive Behavior." J. of Construction Engr. and Mgmnt. 2012:1251-1258). No. Bidders No. Contracts 2 7 3 20 4 26 5 16 6 11 7 9 8 6 9 8 10 3 11 2 a. What proportion of the contracts involved at most five bidders? At least five bidders? b. What proportion of the contracts involved between five and 10 bidders, inclusive? Strictly between five and 10 bidders? c. Construct a histogram and comment on interesting features.Every corporation has a governing board of directors. The number of individuals on a board varies from one corporation to another. One of the authors of the article Does Optimal Corporate Board Size Exist? An Empirical Analysis (J. of Applied Finance, 2010: 57-69) provided the accompanying data on the number of directors on each board in a random sample of 204 corporations. No. directors: 4 5 6 7 8 9 Frequency: 3 12 13 25 24 42 No. directors: 10 11 12 13 14 15 Frequency: 23 19 16 11 5 4 No. directors: 16 17 21 24 32 Frequency: 1 3 1 1 1 a. Construct a histogram of the data based on relative frequencies and comment on any interesting features. b. Construct a frequency distribution in which the last row includes all boards with at least 18 directors. If this distribution had appeared in the cited article, would you be able to draw a histogram? Explain. c. What proportion of these corporations have at most 10 directors? d. What proportion of these corporations have more than 15 directors?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 of particles 0 1 2 3 4 5 6 7 Frequency 1 2 3 12 11 15 18 10 Number of particles 8 9 10 11 12 13 14 Frequency 12 4 5 3 1 2 1 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?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 whether to provide electrical power using overhead lines or underground lines. Here are the values of the variable x = total length of streets within a subdivision: 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 describe the shape of the histogram?The article cited in Exercise 20 also gave the following values of the variables y = number of culs-de-sac and Z = number of intersections: 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?How does the speed of a runner vary over the course of a marathon (a distance of 42.195 km)? Consider determining both the time to run the first 5 km and the time to run between the 35-km and 40-km points, and then subtracting the former time from the latter time. A positive value of this difference corresponds to a runner slowing down toward the end of the race. The accompanying histogram is based on times of runners who participated in several different Japanese marathons (Factors Affecting Runners' Marathon Performance, Chance, Fall, 1993: 24-30). What are some interesting features of this histogram? What is a typical difference value? Roughly what proportion of the runners ran the late distance more quickly than the early distance? Histogram for Exercise 22The article Statistical Modeling of the Time Course of Tantrum Anger (Annals of Applied Stats, 2009: 1013-1034) discussed how anger intensity in childrens tantrums could be related to tantrum duration as well as behavioral indicators such as shouting, stamping, and pushing or pulling. The following frequency distribution was given (and also the corresponding histogram): Draw the histogram and then comment on any interesting features.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 equal-width classes with boundaries 4000. 4200 [The histogram will agree with the one in Comparison of Properties of Joints Prepared by Ultrasonic Welding and Other Means [j. of Aircraft. 1983: 552-556).] Comment on its features.A transformation of data values by means of some mathematical function, such as or 1/x, can often yield a set of numbers that has nicer" statistical properties than the original data. In particular, it may be possible to find a function for which the histogram of transformed values is more symmetric (or, even better, more like a bell-shaped curve) than the original data. As an example, the article Time Lapse Cinematographic Analysis of Beryllium-Lung Fibroblast Interactions (Environ. Research, 1983: 34-43) reported the results of experiments designed to study the behavior of certain individual cells that had been exposed to beryllium. An important characteristic of such an individual cell is its interdivision time (IDT). IDTs were determined for a large number of cells, both in exposed (treatment) and unexposed (control) conditions. The authors of the article used a logarithmic transformation, that is, transformed value = log(original value). Consider the following representative IDT data: IDT log10(IDT) IDT log10(IDT) IDT log10(IDT) 28.1 1.45 60.1 1.78 21.0 1.32 31.2 1.49 23.7 1.37 22.3 1.35 13.7 1.14 18.6 1.27 15.5 1.19 46.0 1.66 21.4 1.33 36.3 1.56 25.8 1.41 26.6 1.42 19.1 1.28 16.8 1.23 26.2 1.42 38.4 1.58 34.8 1.54 32.0 1.51 72.8 1.86 62.3 1.79 43.5 1.64 48.9 1.69 28.0 1.45 17.4 1.24 21.4 1.33 17.9 1.25 38.8 1.59 20.7 1.32 19.5 1.29 30.6 1.49 57.3 1.76 21.1 1.32 55.6 1.75 40.9 1.61 31.9 1.50 25.5 1.41 28.9 1.46 52.1 1.72 Use class intervals 1020, 2030, ... to construct a histogram of the original data. Use intervals 1.11.2, 1.21.3, ... to do the same for the transformed data. What is the effect of the transformation?The accompanying summary data on CeO2 particle sizes (nm) under certain experimental conditions was read from a graph in the article Nanoceria- Energetics of Surfaces, Interfaces and Water Adsorption (J. of the Amer. Ceramic Soc., 2011: 3992-3999): a. What proportion of the observations are less than 5? b. What proportion of the observations are at least 6? c. Construct a histogram with relative frequency on the vertical axis and comment on interesting features. In particular does the distribution of particle sizes appear to be reasonably symmetric or somewhat skewed? [Note: The investigators fit a lognormal distribution to the data; this is discussed in Chapter 4.] d. Construct a histogram with density on the vertical axis and compare to the histogram in (c).The article 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 alloy. 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?The accompanying frequency distribution on deposited energy (mJ) was extracted from the article Experimental Analysis of Laser-Induced Spark Ignition of Lean Turbulent Premixed Flames (Combustion and Flame, 2013: 1414-1427). a. What proportion of these ignition trials resulted in a deposited energy of less than 3 mJ? b. What proportion of these ignition trials resulted in a deposited energy of at least 4 mJ? c. Roughly what proportion of the trials resulted in a deposited energy of at least 3.5 mJ? d. Construct a histogram and comment on its shape.The following categories for type of physical activity involved when an industrial accident occurred appeared in the article Finding Occupational Accident Patterns In the Extractive Industry Using a Systematic Data Mining Approach (Reliability Engr. and System Safety, 2012:108-122): A. Working with handheld tools B. Movement C. Carrying by hand D. Handling of objects E. Operating a machine F. Other Construct a frequency distribution, including relative frequencies, and histogram for the accompanying data from 100 accidents (the percentages agree with those in the cited article):A Pareto diagram is a variation of a histogram for categorical data resulting from a quality control study, Each category represents a different type of product non-conformity or production problem. The categories are ordered so that the one with the largest frequency appears on the far left, then the category with the second largest frequency, and so on. Suppose the following information on nonconformities in circuit packs is obtained: failed component. 126; incorrect component. 210; insufficient solder, 67; excess solder, 54; missing component, 131. Construct a Pareto diagram.The cumulative frequency and cumulative relative frequency for a particular class interval are the sum of frequencies and relative frequencies, respectively, for that interval and all intervals lying below it. If, for example, there are four intervals with frequencies 9, 16, 13, and 12, then the cumulative frequencies are 9, 25, 38, and 50, and the cumulative relative frequencies are .18, .50, .76, and 1.00. Compute the cumulative frequencies and cumulative relative frequencies for the data of Exercise 24.Fire load (MJ/m2) is the heat energy that could be released per square meter of floor area by combustion of contents and the structure itself. The article Fire Loads In Office Buildings (J. of Structural Enter., 1997: 365-368) gave the following cumulative percentages (read from a graph) for lire loads in a sample of 388 rooms: Value 0 150 300 450 600 Cumulative % 0 19.3 37.6 62.7 77.5 Value 750 900 1050 1200 1350 Cumulative % 87.2 93.8 95.7 98.6 99.1 Value 1500 1650 1800 1950 Cumulative % 99.5 99.6 99.8 100.0 a. Construct a relative frequency histogram and comment on interesting features. b. What proportion of fire loads are less than 600? At least 1200? c. What proportion of the loads are between 600 and 1200?The May 1, 2009, issue of The Montclarian reported the following home sale amounts for a sample of homes in Alameda. CA that were sold the previous month (1000s of ): a. Calculate and interpret the sample mean and median. b. Suppose the 6th observation had been 985 rather than 1285. How would the mean and median change? c. Calculate a 20% trimmed mean by first trimming the two smallest and two largest observations. d. Calculate a 15% trimmed mean.Exposure to microbial products, especially endotoxin, may have an impact on vulnerability to allergic diseases. The article Dust Sampling Methods for Endotoxin- An Essential. But Underestimated Issue" (Indoor Air, 2006: 20-27) considered various issues associated with determining endotoxin concentration. The following data on concentration (EU/mg) in settled dust for one sample of urban homes and another of farm homes was kindly supplied by the authors of the cited article. a. Determine the sample mean for each sample. How do they compare? b. Determine the sample median for each sample. How do they compare? Why is the median for the urban sample so different from the mean for that sample? c. Calculate the trimmed mean for each sample by deleting the smallest and largest observation. What are the corresponding trimming percentages? How do the values of these trimmed means compare to the corresponding means and medians?Mercury is a persistent and dispersive environmental contaminant found in many ecosystems around the world. When released as an industrial by-product, it often finds its way into aquatic systems where it can have deleterious effects on various avian and aquatic species. The accompanying data on blood mercury- concentration (g/g) for adult females near contaminated rivers in Virginia was read from a graph in the article Mercury Exposure Effects the Reproductive Success of a Free-Living Terrestrial Songbird, the Carolina Wren (The Auk. 2011:759-769: this is a publication of the American Ornithologists Union). a. Determine the values of the sample mean and sample median and explain why they are different. [Hint: x1 = 18.55.] b. Determine the value of the 10% trimmed mean and compare to the mean and median. c. By how much could the observation .20 be increased without impacting the value of the sample median?A sample of 26 offshore oil workers took part in a simulated escape exercise, resulting in the accompanying data on time (sec) to complete the escape (Oxygen Consumption and Ventilation During Escape from an Offshore Platform, Ergonomics, 1997: 281-292): a. Construct a stem-and-leaf display of the data. How does it suggest that the sample mean and median will compare? b. Calculate the values of the sample mean and median. [Hint: xi = 9638.] c. By how much could the largest time, currently 424, be increased without affecting the value of the sample median? By how much could this value be decreased without affecting the value of the sample median? d. What are the values of x and x when the observations are reexpressed in minutes?The article Snow Cover and Temperature Relationships in North America and Eurasia (J. Climate and Applied Meteorology. 1983: 460-469) used statistical techniques to relate the amount of snow cover on each continent to average continental temperature. Data presented there included the following ten observations on October snow cover for Eurasia during the years 1970-1979 (in million km2): What would you report as a representative, or typical, value of October snow cover for this period, and what prompted your choice?Blood pressure values are often reported to the nearest 5 mmHg (100, 105, 110, etc.). Suppose the actual blood pressure values for nine randomly selected individuals are a. What is the median of the reported blood pressure values? b. Suppose the blood pressure of the second individual is 127.6 rather than 127.4 (a small change in a single value). How does this affect the median of the reported values? What does this say about the sensitivity of the median to rounding or grouping in the data?The propagation of fatigue cracks in various aircraft parts has been the subject of extensive study in recent years. The accompanying data consists of propagation lives (flight hours/104) to reach a given crack size in fastener holes intended for use in military aircraft (Statistical Crack Propagation in Fastener Holes Under Spectrum Loading, J. Aircraft, 1983: 1028-1032): a. Compute and compare the values of the sample mean and median. b. By how much could the largest sample observation be decreased without affecting the value of the median?Compute the sample median. 25% trimmed mean. 10% trimmed mean, and sample mean for the lifetime data given in Exercise 27, and compare these measures.A sample of n = 10 automobiles was selected, and each was subjected to a 5-mph crash test. Denoting a car with no visible damage by S (for success) and a car with such damage by F, results were as follows: a. What is the value of the sample proportion of successes x/n? b. Replace each S with a 1 and each F with a 0. Then calculate x for this numerically coded sample. Mow does x compare to x/n? c. Suppose it is decided to include 15 more cars in the experiment. How many of these would have to be Ss to give x/n = .80 for the entire sample of 25 cars?a. If a constant c is added to each xi in a sample, yielding yi = xi + c, how do the sample mean and median of the yis relate to the mean and median of the xis? Verify your conjectures. b. If each xi is multiplied by a constant c, yielding yi = cxi, answer the question of part (a). Again, verify your conjectures.An experiment to study the lifetime (in hours) for a certain type of component involved putting ten components into operation and observing them Tor 100 hours. Eight of the components failed during that period, and those lifetimes were recorded. Denote the lifetimes of the two components still functioning after 100 hours by 100+. The resulting sample observations were Which of the measures of center discussed in this section can be calculated, and what are the values of those measures? [Note: The data from this experiment is said to be censored on the right]Poly(3-hydroxybutyrate) (PHB), a semicrystalline polymer that is fully biodegradable and biocompatible, is obtained from renewable resources. From a sustainability perspective, PHB offers many attractive properties though it is more expensive to produce than standard plastics. The accompanying data on melting point (C) for each of 12 specimens of the polymer using a differential scanning calorimeter appeared in the article The Melting Behaviour of Poly (3-Hydroxybutyrate) by DSC. Reproducibility Study (Polymer Testing, 2013: 215-220). Compute the following: a. The sample range b. The sample variance s2 from the definition [Hint: First subtract 180 from each observation.] c. The sample standard deviation d. s2 using the shortcut methodThe value of Youngs modulus (GPa) was determined for cast plates consisting of certain intermetallic substrates, resulting in the following sample observations (Strength and Modulus of a Molybdenum-Coated Ti-25AI-10Nb-3U-lMo Intermetallic, J. of Materials Engr. and Performance, 199(7: 46-50): 116.4 115.9 114.6 115.2 115.8 a. Calculate x and the deviations from the mean. b. Use the deviations calculated in part (a) to obtain the sample variance and the sample standard deviation. c. Calculate s2 by using the computational formula for the numerator Sxx. d. Subtract 100 from each observation to obtain a sample of transformed values. Now calculate the sample variance of these transformed values, and compare it to s2 for the original data.The article Effects of Short-Term Warming on Low and High latitude Forest Ant Communities (Ecoshpere, May 2011, Article 62) described an experiment in which observations on various characteristics were made using minichambers of three different types: (1) cooler (PVC frames covered with shade cloth), (2) control (PVC frames only), and (3) warmer (PVC frames covered with plastic). One of die articles authors kindly supplied the accompanying data on the difference between air and soil temperatures (C). Cooler Control Warmer 1.59 1.92 2.57 1.43 2.00 2.60 1.88 2.19 1.93 1.26 1.12 1.58 1.91 1.78 2.30 1.86 1.90 1.84 2.45 0.84 2.65 1.57 2.03 0.12 1.79 1.52 2.74 1.72 0.53 2.53 2.41 1.90 2.13 2.34 2.86 0.83 2.31 1.34 1.91 1.76 a. Compare measures of center for the three different samples. b. Calculate, interpret, and compare the standard deviations for the three different samples. c. Do the fourth spreads for the three samples convey the same message as do the standard deviations about relative variability? d. Construct a comparative boxplot (which was included in the cited article) and comment on any interesting features.Zinfandel is a popular red wine varietal produced almost exclusively in California. It is rather controversial among wine connoisseurs because its alcohol content varies quite substantially from one producer to another. In May 2013, the author went to the website klwines.com. randomly selected 10 zinfandels from among the 325 available, and obtained the following values of alcohol content (%): 14.814.516.114.215.9 13.716.214.613.815.0 a. Calculate and interpret several measures of center. b. Calculate the sample variance using the defining formula. c. Calculate the sample variance using the shortcut formula after subtracting 13 from each observation.Exercise 34 presented the following data on endotoxin concentration in settled dust both for a sample of urban homes and for a sample of farm homes: U: 6.0 5.0 11.0 33.0 4.0 5.0 80.0 18.0 35.0 17.0 23.0 F: 4.0 14.0 11.0 9.0 9.0 8.0 4.0 20.0 5.0 8.9 21.0 9.2 3.0 2.0 0.3 a. Determine the value of the sample standard deviation for each sample, interpret these values, and then contrast variability in the two samples. [Hint: xi = 237.0 for the urban sample and = 128.4 for the farm sample, and xi2=10,079 for the urban sample and 1617.94 for the farm sample.] b. Compute the fourth spread for each sample and compare. Do the fourth spreads convey the same message about variability that the standard deviations do? Explain. c. The authors of the cited article also provided endotoxin concentrations in dust bag dust: U: 34.0 49.0 13.0 33.0 24.0 24.0 35.0 104.0 34.0 40.0 38.0 1.0 F: 2.0 64.0 6.0 17.0 35.0 11.0 17.0 13.0 5.0 27.0 23.0 28.0 10.0 13.0 0.2 Construct a comparative boxplot (as did the cited paper) and compare and contrast the four samples.A study of the relationship between age and various visual functions (such as acuity and depth perception) reported the following observations on the area of scleral lamina (mm2) from human optic nerve heads (Morphometry of Nerve Fiber Bundle Pores in the Optic Nerve Head of the Human, Experimental Eye Research, 1988: 559-568): a. Calculate xi and xi2. b. Use the values calculated in part (a) to compute the sample variance s2 and then the sample standard deviation s.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 injury awarded about 3.5 million for pain and suffering, but the court then set aside that award as being unreasonable compensation. In making this determination, the court identified a normative group of 27 similar cases and specified a reasonable award as one within two standard deviations of 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 = 20,179, xi2=24,657,511. What is the maximum possible amount that could be awarded under the two-standard-deviation rule?The article A Thin-Film Oxygen Uptake Test for the Evaluation of Automotive Crankcase Lubricants (Lubric. Ergr., 1984: 75-83) reported the following data on oxidation-induction time (min) for various commercial oils: a. Calculate the sample variance and standard deviation. b. If the observations were reexpressed in hours, what would be the resulting values of the sample variance and sample standard deviation? Answer without actually performing the reexpression.The first four deviations from the mean in a sample of n = 5 reaction times were .3, .9, 1.0, and 1.3. What is the fifth deviation from the mean? Give a sample for which these are the five deviations from the mean.A mutual fund is a professionally managed investment scheme that pools money from many investors and invests in a variety of securities. Growth funds focus primarily on increasing the value of investments. whereas blended funds seek a balance between current income and growth. Here is data on the expense ratio (expenses as a % of assets, from www.morningstar.com) for samples of 20 large-cap balanced funds and 20 large-cap growth funds (large-cap refers to the sizes of companies in which the funds invest; the population sizes are 825 and 762, respectively): B1 1.03 1.23 1.10 1.64 1.30 1.27 1.25 0.78 1.05 0.64 0.94 2.86 1.05 0.75 0.09 0.79 1.61 1.26 0.93 0.84 Gr 0.52 1.06 1.26 2.17 1.55 0.99 1.10 1.07 1.81 2.05 0.91 0.79 1.39 0.62 1.52 1.02 1.10 1.78 1.01 1.15 a. Calculate and compare the values of x,xo, and s for the two types of funds. b. Construct a comparative boxplot for the two types of funds, and comment on interesting features.Grip is applied to produce normal surface forces that compress the object being gripped. Examples include two people shaking hands, or a nurse squeezing a patients forearm to stop bleeding. The article Investigation of Grip Force, Normal Force, Contact Area, Hand Size, and Handle Size for Cylindrical Handles (Human Factors, 2008: 734-744) included the following data on grip strength (N) for a sample of 42 individuals: a. Construct a stem-and-leaf display based on repeating each stem value twice, and comment on interesting features. b. Determine the values of the fourths and the fourthspread. c. Construct a boxplot based on the five-number summary, and comment on its features. d. How large or small does an observation have to be to qualify as an outlier? An extreme outlier? Are there any outliers? e. By how much could the observation 403, currently the largest, be decreased without affecting fs?Here is a stem-and-leaf display of the escape time data introduced in Exercise 36 of this chapter. 32 55 33 49 34 35 6699 36 34469 37 03345 38 9 39 2347 40 23 41 42 4 a. Determine the value of the fourth spread. b. Are there any outliers in the sample? Any extreme outliers? c. Construct a boxplot and comment on its features. d. By how much could the largest observation, currently 424, be decreased without affecting the value of the fourth spread?The following data on distilled alcohol content (%) for a sample of 35 port wines was extracted from the article A Method for the Estimation of Alcohol in Fortified Wines Using Hydrometer Baume and Refractometer Brix (Amer. J. Enol. Vitic., 2006: 486-490). Each value is an average of two duplicate measurements. Use methods from this chapter, including a boxplot that shows outliers, to describe and summarize the data.A sample of 20 glass bottles of a particular type was selected, and the internal pressure strength of each bottle was determined. Consider the following partial sample information: median = 202.2 lower fourth = 196.0 upper fourth = 216.8 Three smallest observations 125.8 188.1 193.7 Three largest observations 221.3 230.5 250.2 a. Are there any outliers in the sample? Any extreme outliers? b. Construct a boxplot that shows outliers, and comment on any interesting features.A company utilizes two different machines to manufacture parts of a certain type. During a single shift, a sample of n = 20 parts produced by each machine is obtained, and the value of a particular critical dimension for each part is determined. The comparative boxplot at the bottom of this page is constructed from the resulting data. Compare and contrast the two samples. Comparative boxplot for Exercise 58Blood cocaine concentration (mg/L) was determined both for a sample of individuals who had died from cocaine-induced excited delirium (ED) and for a sample of those who had died from a cocaine overdose without excited delirium; survival time for people in both groups was at most 6 hours. The accompanying data was read from a comparative boxplot in the article Fatal Excited Delirium Following Cocaine Use (J. of Forensic Sciences, 1997: 25-31). ED 0 0 0 0 .1 .1 .1 .1 2 .2 .3 .3 .4 .5 .7 .8 1.0 1.5 2.7 2.8 3.5 4.0 8.9 9.2 11.7 21.0 Non-ED 0 0 0 0 0 .1 .1 .1 .1 .2 .2 .2 .3 .3 .3 .4 .5 .5 .6 .8 .9 1.0 1.2 1.4 1.5 1.7 2.0 3.2 3.5 4.1 4.3 4.8 5.0 5.6 5.9 6.0 6.4 7.9 8.3 8.7 9.1 9.6 9.9 11.0 11.5 12.2 12.7 14.0 16.6 17.8 a. Determine the medians, fourths, and fourth spreads for the two samples. b. Are there any outliers in either sample? Any extreme outliers? c. Construct a comparative boxplot, and use it as a basis for comparing and contrasting the HD and non-ED samples.Observations on burst strength (lb/in2) were obtained both for test nozzle closure welds and for production cannister nozzle welds (Proper Procedures Are the Key to Welding Radioactive Waste Cannisters Welding J., Aug. 1997: 61-67). Test 7200 6100 7300 7300 8000 7400 7300 7300 8000 6700 8300 Cannister 5250 5625 5900 5900 5700 6050 5800 6000 5875 6100 5850 6600 Construct a comparative boxplot and comment on interesting features (the cited article did not include such a picture, hut the authors commented that they had looked at one).The accompanying comparative boxplot of gasoline vapor coefficients for vehicles in Detroit appeared in the article Receptor Modeling Approach to VOC Emission Inventory Validation (J. of Envir. Engr., 1995: 483-490). Discuss any interesting features. Comparative boxplot for Exercise 61Consider the following information on ultimate tensile strength (lb/in) for a sample of n = 4 hard zirconium copper wire specimens (from Characterization Methods for Fine Copper Wire, Wire J. Intl., Aug., 1997: 74-80): x = 76,831 s = 180 smallest xi = 76,683 largest xi = 77,048 Determine the values of the two middle sample observations (and dont do it by successive guessing!).A sample of 77 individuals working at a particular office was selected and the noise level (dBA) experienced by each individual was determined, yielding the following data (Acceptable Noise levels for Construction Site Offices, Building Serv. Engr. Research and Technology, 2009: 87-94). Use various techniques discussed in this chapter to organize, summarize, and describe the data.Fretting is a wear process that results from tangential oscillatory movements of small amplitude in machine parts. The article Grease effect on Fretting Wear of Mild Steel (Industrial Lubrication and Tribology, 2008: 67-78) included the following data on volume wear (104mm3) for base oils having four different viscosities. Viscosity Wear 20.4 58.8 30.8 27.3 29.9 17.7 76.5 30.2 44.5 47.1 48.7 41.6 32.8 18.3 89.4 73.3 57.1 66.0 93.8 133.2 81.1 252.6 30.6 24.2 16.6 38.9 28.7 23.6 a. The sample coefficient of variation 100s/x assesses the extent of variability relative to the mean (specifically, the standard deviation as a percentage of the mean). Calculate the coefficient of variation for the sample at each viscosity. Then compare the results and comment. b. Construct a comparative boxplot of the data and comment on interesting features.The accompanying frequency distribution of fracture strength (MPa) observations for ceramic bars fired in a particular kiln appeared in the article Evaluating Tunnel Kiln Performance (Amer. Ceramic Soc. Bull., Aug. 1997: 59-63). Class 81 83 83 85 85 87 87 89 89 91 Frequency 6 7 17 30 43 Class 91 93 93 95 95 97 97 99 Frequency 28 22 13 3 a. Construct a histogram based on relative frequencies, and comment on any interesting features. b. What proportion of the strength observations arc at least 85? Less than 95? c. Roughly what proportion of the observations are less than 90?A deficiency of the (race element selenium in the diet can negatively impact 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 article Effects of Short-Term Supplementation with Selenised Yeast on Milk Production and Composition of Lactating Cows (Australian J. of Dairy Tech.y 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 9-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 8.7 13 9.2 11.6 103.0 12.5 14 10.6 10.9 117.8 9.1 15 10.8 121.5 16 8.2 93.0 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 Sc concentration values in the treatment group differ from those in the control group.Aortic stenosis refers to a narrowing of the aortic valve in the heart. The article 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: 3.8 2.6 3.2 3.0 4.3 3.5 3.1 3.1 3.2 3.0 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 1.3 must he used).a. For what value of c is the quantity (xi c)2 minimized? [Hint: Take the derivative with respect to c, set equal to 0, and solve.] b. Using the result of part (a), which of the two quantities (xi x)2 and (xi )2 will be smaller than the other (assuming that x ?a. Let a and b be constants and let yi = axi + b for i = 1, 2,..., n. What are the relationships between x and y and between sx2 and sy2? b. A sample of temperatures for initiating a certain chemical reaction yielded a sample average (C) of 87.3 and a sample standard deviation of 1.04. What are the sample average and standard deviation measured in F? [Hint: F=93C+32.]Elevated energy consumption during exercise continues after the workout ends. Because calories burned after exercise contribute to weight loss and have other consequences, it is important to understand this process. The article Effect of Weight Training Exercise and Treadmill Exercise on Post-Exercise Oxygen Consumption (Medicine and Science in Sports and Exercise, 1998: 518-522) reported the accompanying data from a study in which oxygen consumption (liters) was measured continuously for 30 minutes for each of 15 subjects both after a weight training exercise and after a treadmill exercise. Subject 1 2 3 4 5 6 7 Weight (x) 14.6 14.4 19.5 24.3 16.3 22.1 23.0 Treadmill (y) 11.3 5.3 9.1 15.2 10.1 19.6 20.8 Subject 8 9 10 II 12 13 14 15 Weight (x) 18.7 19.0 17.0 19.1 19.6 23.2 18.5 15.9 Treadmill (y) 10.3 10.3 2.6 16.6 22.4 23.6 12.6 4.4 a. Construct a comparative boxplot of the weight and treadmill observations, and comment on what you see. b. The data is in the form of (x, y) pairs, with x and y measurements on the same variable under two different conditions, so it is natural to focus on the differences within pairs: d1 = x1 y1,...,dn = xn yn. Construct a boxplot of the sample differences. What does it suggest?Here is a description from Minitab of the strength data given in Exercise 13. Variable N Mean Median TrMean StDev SE Mean strength 153 135.39 135.40 135.41 4.59 0.37 Variable Minimum Maximum Q1 Q3 strength 122.20 147.70 132.95 138.25 a. Comment on any interesting features (the quartiles and fourths are virtually identical here). b. Construct a boxplot of the data based on the quartiles, and comment on what you see.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 article Decreased Benzodiazepine Receptor Binding in Prefrontal Cortex in Combat-Related Posttraumatic Stress Disorder (Amer. J. of Psychiatry, 2000: 1120-1126) described the first study of benzodiazepine receptor binding in individuals suffering from PTSD. The accompanying data on a receptor binding measure (adjusted distribution volume) was read from a graph in the article. 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 Use various methods from this chapter to describe and summarize the data.The article Can We Really Walk Straight? (Amer. J. of Physical Anthropology, 1992: 19-27) reported on an experiment in which each of 20 healthy men was asked to walk as straight as possible to a target 60 m away at normal speed. Consider the following observations on cadence (number of strides per second): Use the methods developed in this chapter to summarize the data; include an interpretation or discussion wherever appropriate. [Note: The author of the article used a rather sophisticated statistical analysis to conclude that people cannot walk in a straight line and suggested several explanations for this.]The mode of a numerical data set is the value that occurs most frequently in the set. a. Determine the mode for the cadence data given in Exercise 73. b. For a categorical sample, how would you define the modal category?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 2 350 354 359 363 365 368 369 371 373 374 376 380 383 388 392 Type 3 350 361 362 364 364 365 366 371 377 377 377 379 380 380 392 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.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 midfourth, which is the average of the two fourths. Which of these five measures of center are resistant to the effects of outliers and which are not? Explain your reasoning.The authors of the article Predictive Model for Pitting Corrosion in Buried Oil and Gas Pipelines (Corrosion, 2009: 332-342) provided the data on which their investigation was based. a. Consider the following sample of 61 observations on maximum pitting depth (mm) of pipeline specimens buried in clay loam soil. Construct a stem-and-leaf display in which the two largest values are shown in a last row labeled HI. b. Refer back to (a), and create a histogram based on eight classes with 0 as the lower limit of the first class and class widths of .5, .5, .5, .5, 1, 2, 5, and 5, respectively. c. The accompanying comparative boxplot from Minitab shows plots of pitting depth for four different types of soils. Describe its important features.Consider a sample x1 , x2,..., xn and suppose that the values of x, s2, and s have been calculated. a. Let yi = xi - x for i = 1, , n. How do the values of s2 and s for the yis compare to the corresponding values for the xis? Explain. b. Let Zi = (xi - x)/s for i = 1, , n.What are the values of the sample variance and sample standard deviation for the zis?Let xn and sn2 denote the sample mean and variance for the sample x1,...., xn and let xn+1 and sn+12 denote these quantities when an additional observation xn + l is added to the sample. a. Show how xn+1 can be computed from xn and xn + l. b. Show that nsn+12=(n-1)sn2+nn+1(xn+1xn)2 so that sn+12 can be computed fromxn + 1 and sn2. c. 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 ofLengths of bus routes for any particular transit system will typically vary from one route to another. The article Planning of City Bus Routes (J. of the Institution of Engineers, 1995:211-215) gives the following information on lengths (km) for one particular system: Length 68 810 1012 1214 1416 Frequency 6 23 30 35 32 Length 1618 1820 2022 2224 2426 Frequency 48 42 40 28 27 Length 2628 2830 3035 3540 4045 Frequency 26 14 27 11 2 a. Draw a histogram corresponding to these frequencies. b. What proportion of these route lengths are less than 20? What proportion of these routes have lengths of at least 30? c. Roughly what is the value of the 90th percentile of the route length distribution? d. Roughly what is the median route length?A study carried out to investigate the distribution c braking time (reaction time plus accelerator-to-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 Driving, Ergonomics, 1995:1903-1910): mean = 535 median = 500 mode = 500 sd = 96 minimum = 220 maximum = 925 5th percentile = 400 10th percentile = 430 90th percentile = 640 95th percentile = 720 What can you conclude about the shape of a histogram of this data? Explain your reasoning.82SE83SEConsider a sample x1, ... , xn with n even. Let xL and xU denote the average of the smallest n/2 and the largest n/2 observations, respectively. Show that the mean absolute deviation from the median for this sample satisfies xix/n=(xU-xL)/2 Then show that if n is odd and the two averages are calculated after excluding the median from each half, replacing n on the left with n 1 gives the correct result. [Hint: Break the sum into two parts, the first involving observations less than or equal to the median and the second involving observations greater than or equal to the median.]Four universities1, 2, 3, and 4are participating in a holiday basketball tournament. In the first round, 1 will play 2 and 3 will play 4. Then the two winners will play for the championship, and the two losers will also play. One possible outcome can be denoted by 1324 (1 beats 2 and 3 beats 4 in first-round games, and then 1 beats 3 and 2 beats 4). a. List all outcomes in S. b. Let A denote the event that 1 wins the tournament. List outcomes in A. c. Let B denote the event that 2 gets into the championship game. List outcomes in B. d. What are the outcomes in A B and in A B? What are the outcomes in A?Suppose that vehicles taking a particular freeway exit can turn right (R), turn left (L), or go straight (S). Consider observing the direction for each of three successive vehicles. a. List all outcomes in the event A that all three vehicles go in the same direction. b. List all outcomes in the event B that all three vehicles take different directions. c. List all outcomes in the event C that exactly two of the three vehicles turn right. d. List all outcomes in the event D that exactly two vehicles go in the same direction. e. List outcomes in D, C D, and C D.Three components are connected to form a system as shown in the accompanying diagram. Because the components in the 23 subsystem are connected in parallel, that subsystem will function if at least one of the two individual components functions. For the entire system to function, component 1 must function and so must the 23 subsystem. The experiment consists of determining the condition of each component [S (success) for a functioning component and F (failure) for a nonfunctioning component]. a. Which outcomes are contained in the event A that exactly two out of the three components function? b. Which outcomes are contained in the event B that at least two of the components function? c. Which outcomes are contained in the event C that the system functions? d. List outcomes in C, A C, A C, B C, and B C.Each of a sample of four home mortgages is classified as fixed rate (F) or variable rate (V). a. What are the 16 outcomes in S? b. Which outcomes are in the event that exactly three of the selected mortgages are fixed rate? c. Which outcomes are in the event that all four mortgages are of the same type? d. Which outcomes are in the event that at most one of the four is a variable-rate mortgage? e. What is the union of the events in parts (c) and (d), and what is the intersection of these two events? f. What are the union and intersection of the two events in parts (b) and (c)?A family consisting of three personsA, B, and Cgoes to a medical clinic that always has a doctor at each of stations 1, 2, and 3. During a certain week, each member of the family visits the clinic once and is assigned at random to a station. The experiment consists of recording the station number for each member. One outcome is (1, 2, 1) for A to station 1, B to station 2, and C to station 1. a. List the 27 outcomes in the sample space. b. List all outcomes in the event that all three members go to the same station. c. List all outcomes in the event that all members go to different stations. d. List all outcomes in the event that no one goes to station 2.A college library has five copies of a certain text on reserve. Two copies (1 and 2) are first printings, and the other three (3, 4, and 5) are second printings. A student examines these books in random order, stopping only when a second printing has been selected. One possible outcome is 5, and another is 213. a. List the outcomes in S. b. Let A denote the event that exactly one book must be examined. What outcomes are in A? c. Let B be the event that book 5 is the one selected. What outcomes are in B? d. Let C be the event that book 1 is not examined. What outcomes are in C?An academic department has just completed voting by secret ballot for a department head. The ballot box contains four slips with votes for candidate A and three slips with votes for candidate B. Suppose these slips are removed from the box one by one. a. List all possible outcomes. b. Suppose a running tally is kept as slips are removed. For what outcomes does A remain ahead of B throughout the tally?An engineering construction firm is currently working on power plants at three different sites. Let Ai denote the event that the plant at site i is completed by the contract date. Use the operations of union, intersection, and complementation to describe each of the following events in terms of A1, A2, and A3, draw a Venn diagram, and shade the region corresponding to each one. a. At least one plant is completed by the contract date. b. All plants are completed by the contract date. c. Only the plant at site 1 is completed by the contract date. d. Exactly one plant is completed by the contract date. e. Either the plant at site 1 or both of the other two plants are completed by the contract date.Use Venn diagrams to verify the following two relationships for any events A and B (these are called De Morgans laws): a. (A B) = A B b. (A B) = A B [Hint: In each part, draw a diagram corresponding to the left side and another corresponding to the right side.]a. In Example 2.10, identify three events that are mutually exclusive. b. Suppose there is no outcome common to all three of the events A, B, and C. Are these three events necessarily mutually exclusive? If your answer is yes, explain why; if your answer is no, give a counterexample using the experiment of Example 2.10.A mutual fund company offers its customers a variety of funds: a money-market fund, three different bond funds (short, intermediate, and long-term), two stock funds (moderate and high-risk), and a balanced fund. Among customers who own shares in just one fund, the percentages of customers in the different funds are as follows: A customer who owns shares in just one fund is randomly selected. a. What is the probability that the selected individual owns shares in the balanced fund? b. What is the probability that the individual owns shares in a bond fund? c. What is the probability that the selected individual does not own shares in a stock fund?Consider randomly selecting a student at a large university, and let A be the event that the selected student has a Visa card and B be the analogous event for MasterCard. Suppose that P(A) = .6 and P(B) = .4. a. Could it be the case that P(A B) = .5? Why or why not? [Hint: See Exercise 24.] b. From now on, suppose that P(A B) = .3. What is the probability that the selected student has at least one of these two types of cards? c. What is the probability that the selected student has neither type of card? d. Describe, in terms of A and B, the event that the selected student has a Visa card but not a MasterCard, and then calculate the probability of this event. e. Calculate the probability that the selected student has exactly one of the two types of cards.A computer consulting firm presently has bids out on three projects. Let Ai = {awarded project i}, for i = 1, 2, 3, and suppose that P(A1) = .22, P(A2) = .25, P(A3) = .28, P(A1 A2) = .11, P(A1 A3) = .05, P(A2 A3) = .07, P(A1 A2 A3) = .01. Express in words each of the following events, and compute the probability of each event: a. A1 A2 b. A1 A2 [Hint: (A1 A2) = A1 A2] c. A1 A2 A3 d. A1 A2 A3 e. A1 A2 A3 f. (A1 A2 ) A3Suppose that 55% of all adults regularly consume coffee, 45% regularly consume carbonated soda, and 70% regularly consume at least one of these two products. a. What is the probability that a randomly selected adult regularly consumes both coffee and soda? b. What is the probability that a randomly selected adult doesnt regularly consume at least one of these two products?Consider the type of clothes dryer (gas or electric) purchased by each of five different customers at a certain store. a. If the probability that at most one of these purchases an electric dryer is .428, what is the probability that at least two purchase an electric dryer? b. If P(all five purchase gas) = .116 and P(all five purchase electric) = .005, what is the probability that at least one of each type is purchased?An individual is presented with three different glasses of cola, labeled C, D, and P. He is asked to taste all three and then list them in order of preference. Suppose the same cola has actually been put into all three glasses. a. What are the simple events in this ranking experiment, and what probability would you assign to each one? b. What is the probability that C is ranked first? c. What is the probability that C is ranked first and D is ranked last?Let A denote the event that the next request for assistance from a statistical software consultant relates to the SPSS package, and let B be the event that the next request is for help with SAS. Suppose that P(A) = .30 and P(B) = .50. a. Why is it not the case that P(A) + P(B) = 1? b. Calculate P(A). c. Calculate P(A B). d. Calculate P(A B).A wallet contains five 10 bills, four 5 bills, and six 1 bills (nothing larger). If the bills are selected one by one in random order, what is the probability that at least two bills must be selected to obtain a first 10 bill?Human visual inspection of solder joints on printed circuit boards can be very subjective. Part of the problem stems from the numerous types of solder defects (e.g., pad non wetting, knee visibility, voids) and even the degree to which a joint possesses one or more of these defects. Consequently, even highly trained inspectors can disagree on the disposition of a particular joint. In one batch of 10,000 joints, inspector A found 724 that were judged defective, inspector B found 751 such joints, and 1159 of the joints were judged defective by at least one of the inspectors. Suppose that one of the 10,000 joints is randomly selected. a. What is the probability that the selected joint was judged to be defective by neither of the two inspectors? b. What is the probability that the selected joint was judged to be defective by inspector B but not by inspector A?A certain factory operates three different shifts. Over the last year, 200 accidents have occurred at the factory. Some of these can be attributed at least in part to unsafe working conditions, whereas the others are unrelated to working conditions. The accompanying table gives the percentage of accidents falling in each type of accident shift category. Suppose one of the 200 accident reports is randomly selected from a file of reports, and the shift and type of accident are determined. a. What are the simple events? b. What is the probability that the selected accident was attributed to unsafe conditions? c. What is the probability that the selected accident did not occur on the day shift?An insurance company offers four different deductible levelsnone, low, medium, and highfor its homeowners policyholders and three different levelslow, medium, and highfor its automobile policyholders. The accompanying table gives proportions for the various categories of policyholders who have both types of insurance. For example, the proportion of individuals with both low homeowners deductible and low auto deductible is .06 (6% of all such individuals). Suppose an individual having both types of policies is randomly selected. a. What is the probability that the individual has a medium auto deductible and a high homeowners deductible? b. What is the probability that the individual has a low auto deductible? A low homeowners deductible? c. What is the probability that the individual is in the same category for both auto and homeowners deductibles? d. Based on your answer in part (c), what is the probability that the two categories are different? e. What is the probability that the individual has at least one low deductible level? f. Using the answer in part (e), what is the probability that neither deductible level is low?The route used by a certain motorist in commuting to work contains two intersections with traffic signals. The probability that he must stop at the first signal is .4, the analogous probability for the second signal is .5, and the probability that he must stop at at least one of the two signals is .7. What is the probability that he must stop a. At both signals? b. At the first signal but not at the second one? c. At exactly one signal?The computers of six faculty members in a certain department are to be replaced. Two of the faculty members have selected laptop machines and the other four have chosen desktop machines. Suppose that only two of the setups can be done on a particular day, and the two computers to be set up are randomly selected from the six (implying 15 equally likely outcomes; if the computers are numbered 1, 2,, 6, then one outcome consists of computers 1 and 2, another consists of computers 1 and 3, and so on). a. What is the probability that both selected setups are for laptop computers? b. What is the probability that both selected setups are desktop machines? c. What is the probability that at least one selected setup is for a desktop computer? d. What is the probability that at least one computer of each type is chosen for setup?Show that if one event A is contained in another event B (i.e., A is a subset of B), then P(A) P(B). [Hint: For such A and B, A and B A are disjoint and B = A (B A), as can be seen from a Venn diagram.] For general A and B, what does this imply about the relationship among P(A B), P(A) and P(A B)?The three most popular options on a certain type of new car are a built-in GPS (A), a sunroof (B), and an automatic transmission (C). If 40% of all purchasers request A, 55% request B, 70% request C, 63% request A or B, 77% request A or C, 80% request B or C, and 85% request A or B or C, determine the probabilities of the following events. [Hint: A or B is the event that at least one of the two options is requested; try drawing a Venn diagram and labeling all regions.] a. The next purchaser will request at least one of the three options. b. The next purchaser will select none of the three options. c. The next purchaser will request only an automatic transmission and not either of the other two options. d. The next purchaser will select exactly one of these three options.A certain system can experience three different types of defects. Let Ai (i 5 1,2,3) denote the event that the system has a defect of type i. Suppose that P(A1) = .12 P(A2) = .07 P(A3) = .05 P(A1 A2) = .13 P(A1 A3) = .14 P(A2 A3) = .10 P(A1 A2 A3) = .01 a. What is the probability that the system does not have a type 1 defect? b. What is the probability that the system has both type 1 and type 2 defects? c. What is the probability that the system has both type 1 and type 2 defects but not a type 3 defect? d. What is the probability that the system has at most two of these defects?An academic department with five faculty members Anderson, Box, Cox, Cramer, and Fishermust select two of its members to serve on a personnel review committee. Because the work will be time-consuming, no one is anxious to serve, so it is decided that the representatives will be selected by putting the names on identical pieces of paper and then randomly selecting two. a. What is the probability that both Anderson and Box will be selected? [Hint: List the equally likely outcomes.] b. What is the probability that at least one of the two members whose name begins with C is selected? c. If the five faculty members have taught for 3, 6, 7, 10, and 14 years, respectively, at the university, what is the probability that the two chosen representatives have a total of at least 15 years teaching experience there?In Exercise 5, suppose that any incoming individual is equally likely to be assigned to any of the three stations irrespective of where other individuals have been assigned. What is the probability that a. All three family members are assigned to the same station? b. At most two family members are assigned to the same station? c. Every family member is assigned to a different station?As of April 2006, roughly 50 million .com web domain names were registered (e.g., yahoo.com). a. How many domain names consisting of just two letters in sequence can be formed? How many domain names of length two are there if digits as well as letters are permitted as characters? [Note: A character length of three or more is now mandated.] b. How many domain names are there consisting of three letters in sequence? How many of this length are there if either letters or digits are permitted? [Note: All are currently taken.] c. Answer the questions posed in (b) for four-character sequences. d. As of April 2006, 97,786 of the four-character sequences using either letters or digits had not yet been claimed. If a four-character name is randomly selected, what is the probability that it is already owned?A friend of mine is giving a dinner party. His current wine supply includes 8 bottles of zinfandel, 10 of merlot, and 12 of cabernet (he only drinks red wine), all from different wineries. a. If he wants to serve 3 bottles of zinfandel and serving order is important, how many ways are there to do this? b. If 6 bottles of wine are to be randomly selected from the 30 for serving, how many ways are there to do this? c. If 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety? d. If 6 bottles are randomly selected, what is the probability that this results in two bottles of each variety being chosen? e. If 6 bottles are randomly selected, what is the probability that all of them are the same variety?The composer Beethoven wrote 9 symphonies, 5 piano concertos (music for piano and orchestra), and 32 piano sonatas (music for solo piano). a. How many ways are there to play first a Beethoven symphony and then a Beethoven piano concerto? b. The manager of a radio station decides that on each successive evening (7 days per week), a Beethoven symphony will be played followed by a Beethoven piano concerto followed by a Beethoven piano sonata. For how many years could this policy be continued before exactly the same program would have to be repeated?An electronics store is offering a special price on a complete set of components (receiver, compact disc player, speakers, turntable). A purchaser is offered a choice of manufacturer for each component: Receiver: Kenwood, Onkyo, Pioneer, Sony, Sherwood Compact disc player: Onkyo, Pioneer, Sony, Technics Speakers: Boston, Infinity, Polk Turntable: Onkyo, Sony, Teac, Technics A switchboard display in the store allows a customer to hook together any selection of components (consisting of one of each type). Use the product rules to answer the following questions: a. In how many ways can one component of each type be selected? b. In how many ways can components be selected if both the receiver and the compact disc player are to be Sony? c. In how many ways can components be selected if none is to be Sony? d. In how many ways can a selection be made if at least one Sony component is to be included? e. If someone flips switches on the selection in a completely random fashion, what is the probability that the system selected contains at least one Sony component? Exactly one Sony component?Again consider a Little League team that has 15 players on its roster. a. How many ways are there to select 9 players for the starting lineup? b. How many ways are there to select 9 players for the starting lineup and a batting order for the 9 starters? c. Suppose 5 of the 15 players are left-handed. How many ways are there to select 3 left-handed outfielders and have all 6 other positions occupied by right-handed players?Computer keyboard failures can be attributed to electrical defects or mechanical defects. A repair facility currently has 25 failed keyboards, 6 of which have electrical defects and 19 of which have mechanical defects. a. How many ways are there to randomly select 5 of these key boards for a thorough inspection (without regard to order)? b. In how many ways can a sample of 5 keyboards be selected so that exactly two have an electrical defect? c. If a sample of 5 keyboards is randomly selected, what is the probability that at least 4 of these will have a mechanical defect?A production facility employs 10 workers on the day shift, 8 workers on the swing shift, and 6 workers on the graveyard shift. A quality control consultant is to select 5 of these workers for in-depth interviews. Suppose the selection is made in such a way that any particular group of 5 workers has the same chance of being selected as does any other group (drawing 5 slips without replacement from among 24). a. How many selections result in all 5 workers coming from the day shift? What is the probability that all 5 selected workers will be from the day shift? b. What is the probability that all 5 selected workers will be from the same shift? c. What is the probability that at least two different shifts will be represented among the selected workers? d. What is the probability that at least one of the shifts will be unrepresented in the sample of workers?An academic department with five faculty members narrowed its choice for department head to either candidate A or candidate B. Each member then voted on a slip of paper for one of the candidates. Suppose there are actually three votes for A and two for B. If the slips are selected for tallying in random order, what is the probability that A remains ahead of B throughout the vote count (e.g., this event occurs if the selected ordering is AABAB, but not for ABBAA)?An experimenter is studying the effects of temperature, pressure, and type of catalyst on yield from a certain chemical reaction. Three different temperatures, four different pressures, and five different catalysts are under consideration. a. If any particular experimental run involves the use of a single temperature, pressure, and catalyst, how many experimental runs are possible? b. How many experimental runs are there that involve use of the lowest temperature and two lowest pressures? c. Suppose that five different experimental runs are to be made on the first day of experimentation. If the five are randomly selected from among all the possibilities, so that any group of five has the same probability of selection, what is the probability that a different catalyst is used on each run?A sonnet is a 14-line poem in which certain rhyming patterns are followed. The writer Raymond Queneau published a book containing just 10 sonnets, each on a different page. However, these were structured such that other sonnets could be created as follows: the first line of a sonnet could come from the first line on any of the 10 pages, the second line could come from the second line on any of the 10 pages, and so on (successive lines were perforated for this purpose). a. How many sonnets can be created from the 10 in the book? b. If one of the sonnets counted in part (a) is selected at random, what is the probability that none of its lines came from either the first or the last sonnet in the book?A box in a supply room contains 15 compact fluorescent lightbulbs, of which 5 are rated 13-watt, 6 are rated 18-watt, and 4 are rated 23-watt. Suppose that three of these bulbs are randomly selected. a. What is the probability that exactly two of the selected bulbs are rated 23-watt? b. What is the probability that all three of the bulbs have the same rating? c. What is the probability that one bulb of each type is selected? d. If bulbs are selected one by one until a 23-watt bulb is obtained, what is the probability that it is necessary to examine at least 6 bulbs?Three molecules of type A, three of type B, three of type C, and three of type D are to be linked together to form a chain molecule. One such chain molecule is ABCDABCDABCD, and another is BCDDAAABDBCC. a. How many such chain molecules are there? [Hint: If the three As were distinguishable from one another A1, A2, A3and the Bs, Cs, and Ds were also, how many molecules would there be? How is this number reduced when the subscripts are removed from the As?] b. Suppose a chain molecule of the type described is randomly selected. What is the probability that all three molecules of each type end up next to one another (such as in BBBAAADDDCCC)?An ATM personal identification number (PIN) consists of four digits, each a 0, 1, 2, 8, or 9, in succession. a. How many different possible PINs are there if there are no restrictions on the choice of digits? b. According to a representative at the authors local branch of Chase Bank, there are in fact restrictions on the choice of digits. The following choices are prohibited: (i) all four digits identical (ii) sequences of consecutive ascending or descending digits, such as 6543 (iii) any sequence starting with 19 (birth years are too easy to guess). So if one of the PINs in (a) is randomly selected, what is the probability that it will be a legitimate PIN (that is, not be one of the prohibited sequences)? c. Someone has stolen an ATM card and knows that the first and last digits of the PIN are 8 and 1, respectively. He has three tries before the card is retained by the ATM (but does not realize that). So he randomly selects the 2nd and 3rd digits for the first try, then randomly selects a different pair of digits for the second try, and yet another randomly selected pair of digits for the third try (the individual knows about the restrictions described in (b) so selects only from the legitimate possibilities). What is the probability that the individual gains access to the account? d. Recalculate the probability in (c) if the first and last digits are 1 and 1, respectively.A starting lineup in basketball consists of two guards, two forwards, and a center. a. A certain college team has on its roster three centers, four guards, four forwards, and one individual (X) who can play either guard or forward. How many different starting lineups can be created? [Hint: Consider lineups without X, then lineups with X as guard, then lineups with X as forward.] b. Now suppose the roster has 5 guards, 5 forwards, 3 centers, and 2 swing players (X and Y) who can play either guard or forward. If 5 of the 15 players are randomly selected, what is the probability that they constitute a legitimate starting lineup?In five-card poker, a straight consists of five cards with adjacent denominations (e.g., 9 of clubs, 10 of hearts, jack of hearts, queen of spades, and king of clubs). Assuming that aces can be high or low, if you are dealt a five-card hand, what is the probability that it will be a straight with high card 10? What is the probability that it will be a straight? What is the probability that it will be a straight flush (all cards in the same suit)?Show that (nk)=(nnk). Give an interpretation involving subsets.The population of a particular country consists of three ethnic groups. Each individual belongs to one of the four major blood groups. The accompanying joint probability table gives the proportions of individuals in the various ethnic groupblood group combinations. Suppose that an individual is randomly selected from the population, and define events by A = {type A selected}, B = {type B selected}, and C = {ethnic group 3 selected}. a. Calculate P(A), P(C), and P(A C). b. Calculate both P(A|C) and P(C|A), and explain in context what each of these probabilities represents. c. If the selected individual does not have type B blood, what is the probability that he or she is from ethnic group 1?Suppose an individual is randomly selected from the population of all adult males living in the United States. Let A be the event that the selected individual is over 6 ft in height, and let B be the event that the selected individual is a professional basketball player. Which do you think is larger, P(A|B) or P(B|A)? Why?Return to the credit card scenario of Exercise 12 (Section 2.2), and let C be the event that the selected student has an American Express card. In addition to P(A) = .6, P(B) = .4, and P(A B) = .3, suppose that P(C) = .2, P(A C) = .15, P(B C) = .1, and P(A B C) = .08. a. What is the probability that the selected student has at least one of the three types of cards? b. What is the probability that the selected student has both a Visa card and a MasterCard but not an American Express card? c. Calculate and interpret P(B|A) and also P(A|B). d. If we learn that the selected student has an American Express card, what is the probability that she or he also has both a Visa card and a MasterCard? e. Given that the selected student has an American Express card, what is the probability that she or he has at least one of the other two types of cards?Reconsider the system defect situation described in Exercise 26 (Section 2.2). a. Given that the system has a type 1 defect, what is the probability that it has a type 2 defect? b. Given that the system has a type 1 defect, what is the probability that it has all three types of defects? c. Given that the system has at least one type of defect, what is the probability that it has exactly one type of defect? d. Given that the system has both of the first two types of defects, what is the probability that it does not have the third type of defect?The accompanying table gives information on the type of coffee selected by someone purchasing a single cup at a particular airport kiosk. Small Medium Large Regular 14% 20% 26% Decaf 20% 10% 10% Consider randomly selecting such a coffee purchaser. a. What is the probability that the individual purchased a small cup? A cup of decaf coffee? b. If we learn that the selected individual purchased a small cup, what now is the probability that he/she chose decaf coffee, and how would you interpret this probability? c. If we learn that the selected individual purchased decaf, what now is the probability that a small size was selected, and how does this compare to the corresponding unconditional probability of (a)?A department store sells sport shirts in three sizes (small, medium, and large), three patterns (plaid, print, and stripe), and two sleeve lengths (long and short). The accompanying tables give the proportions of shirts sold in the various category combinations. a. What is the probability that the next shirt sold is a medium, long-sleeved, print shirt? b. What is the probability that the next shirt sold is a medium print shirt? c. What is the probability that the next shirt sold is a short-sleeved shirt? A long-sleeved shirt? d. What is the probability that the size of the next shirt sold is medium? That the pattern of the next shirt sold is a print? e. Given that the shirt just sold was a short-sleeved plaid, what is the probability that its size was medium? f. Given that the shirt just sold was a medium plaid, what is the probability that it was short-sleeved? Long-sleeved?According to a July 31, 2013, posting on cnn.com subsequent to the death of a child who bit into a peanut, a 2010 study in the journal Pediatrics found that 8% of children younger than 18 in the United States have at least one food allergy. Among those with food allergies, about 39% had a history of severe reaction. a. If a child younger than 18 is randomly selected, what is the probability that he or she has at least one food allergy and a history of severe reaction? b. It was also reported that 30% of those with an allergy in fact are allergic to multiple foods. If a child younger than 18 is randomly selected, what is the probability that he or she is allergic to multiple foods?A system consists of two identical pumps, #1 and #2. If one pump fails, the system will still operate. However, because of the added strain, the remaining pump is now more likely to fail than was originally the case. That is, r 5 P(#2 fails | #1 fails). P(#2 fails) = q. If at least one pump fails by the end of the pump design life in 7% of all systems and both pumps fail during that period in only 1%, what is the probability that pump #1 will fail during the pump design life?A certain shop repairs both audio and video components. Let A denote the event that the next component brought in for repair is an audio component, and let B be the event that the next component is a compact disc player (so the event B is contained in A). Suppose that P(A) = .6 and P(B) = .05. What is P(B|A)?In Exercise 13, Ai = {awarded project i}, for i = 1, 2, 3. Use the probabilities given there to compute the following probabilities, and explain in words the meaning of each one. a. P(A2|A1) b. P(A2 A3|A1) c. P(A2 A3|A1) d. P(A1 A2 A3|A1 A2 A3)Deer ticks can be carriers of either Lyme disease or human granulocytic ehrlichiosis (HGE). Based on a recent study, suppose that 16% of all ticks in a certain location carry Lyme disease, 10% carry HGE, and 10% of the ticks that carry at least one of these diseases in fact carry both of them. If a randomly selected tick is found to have carried HGE, what is the probability that the selected tick is also a carrier of Lyme disease?For any events A and B with P(B) 0, show that P(A|B) + P(A|B) = 1.If P(B|A) P(B), show that P(B|A) P(B). [Hint: Add P(B|A) to both sides of the given inequality and then use the result of Exercise 56.]Show that for any three events A, B, and C with P(C) 0, P(A B|C) = P(A|C) + P(B|C) P(A B|C).At a certain gas station, 40% of the customers use regular gas (A1), 35% use plus gas (A2), and 25% use premium (A3). Of those customers using regular gas, only 30% fill their tanks (event B). Of those customers using plus, 60% fill their tanks, whereas of those using premium, 50% fill their tanks. a. What is the probability that the next customer will request plus gas and fill the tank (A2 B)? b. What is the probability that the next customer fills the tank? c. If the next customer fills the tank, what is the probability that regular gas is requested? Plus? Premium?Seventy percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, 60% have an emergency locator, whereas 90% of the aircraft not discovered do not have such a locator. Suppose a light aircraft has disappeared. a. If it has an emergency locator, what is the probability that it will not be discovered? b. If it does not have an emergency locator, what is the probability that it will be discovered?Components of a certain type are shipped to a supplier in batches of ten. Suppose that 50% of all such batches contain no defective components, 30% contain one defective component, and 20% contain two defective components. Two components from a batch are randomly selected and tested. What are the probabilities associated with 0, 1, and 2 defective components being in the batch under each of the following conditions? a. Neither tested component is defective. b. One of the two tested components is defective. [Hint: Draw a tree diagram with three first-generation branches for the three different types of batches.]Blue Cab operates 15% of the taxis in a certain city, and Green Cab operates the other 85%. After a nighttime hit-and-run accident involving a taxi, an eyewitness said the vehicle was blue. Suppose, though, that under night vision conditions, only 80% of individuals can correctly distinguish between a blue and a green vehicle. What is the (posterior) probability that the taxi at fault was blue? In answering, be sure to indicate which probability rules you are using. [Hint: A tree diagram might help. Note: This is based on an actual incident.]For customers purchasing a refrigerator at a certain appliance store, let A be the event that the refrigerator was manufactured in the U.S., B be the event that the refrigerator had an icemaker, and C be the event that the customer purchased an extended warranty. Relevant probabilities are P(A) = .75 P(B|A) = .9 P(B|A) = .8 P(C|A B) = .8 P(C|A B) = .6 P(C|A B) = .7 P(C|A B) = .3 a. Construct a tree diagram consisting of first-, second-, and third-generation branches, and place an event label and appropriate probability next to each branch. b. Compute P(A B C). c. Compute P(B C). d. Compute P(C). e. Compute P(A|B C), the probability of a U.S. purchase given that an icemaker and extended warranty are also purchased.The Reviews editor for a certain scientific journal decides whether the review for any particular book should be short (12 pages), medium (34 pages), or long (56 pages). Data on recent reviews indicates that 60% of them are short, 30% are medium, and the other 10% are long. Reviews are submitted in either Word or LaTeX. For short reviews, 80% are in Word, whereas 50% of medium reviews are in Word and 30% of long reviews are in Word. Suppose a recent review is randomly selected. a. What is the probability that the selected review was submitted in Word format? b. If the selected review was submitted in Word format, what are the posterior probabilities of it being short, medium, or long?A large operator of timeshare complexes requires anyone interested in making a purchase to first visit the site of interest. Historical data indicates that 20% of all potential purchasers select a day visit, 50% choose a one-night visit, and 30% opt for a two-night visit. In addition, 10% of day visitors ultimately make a purchase, 30% of one-night visitors buy a unit, and 20% of those visiting for two nights decide to buy. Suppose a visitor is randomly selected and is found to have made a purchase. How likely is it that this person made a day visit? A one-night visit? A two-night visit?Consider the following information about travelers on vacation (based partly on a recent Travelocity poll): 40% check work email, 30% use a cell phone to stay connected to work, 25% bring a laptop with them, 23% both check work email and use a cell phone to stay connected, and 51% neither check work email nor use a cell phone to stay connected nor bring a laptop. In addition, 88 out of every 100 who bring a laptop also check work email, and 70 out of every 100 who use a cell phone to stay connected also bring a laptop. a. What is the probability that a randomly selected traveler who checks work email also uses a cell phone to stay connected? b. What is the probability that someone who brings a laptop on vacation also uses a cell phone to stay connected? c. If the randomly selected traveler checked work email and brought a laptop, what is the probability that he/she uses a cell phone to stay connected?There has been a great deal of controversy over the last several years regarding what types of surveillance are appropriate to prevent terrorism. Suppose a particular surveillance system has a 99% chance of correctly identifying a future terrorist and a 99.9% chance of correctly identifying someone who is not a future terrorist. If there are 1000 future terrorists in a population of 300 million, and one of these 300 million is randomly selected, scrutinized by the system, and identified as a future terrorist, what is the probability that he/she actually is a future terrorist? Does the value of this probability make you uneasy about using the surveillance system? Explain.A friend who lives in Los Angeles makes frequent consulting trips to Washington, D.C.; 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. For airline #1, flights are late into D.C. 30% of the time and late into L.A. 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 the two destinations, what are the posterior probabilities of having flown on airlines #1, #2, and #3? Assume that the chance of a late arrival in L.A. is unaffected by what happens on the flight to D.C. [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.]In Exercise 59, consider the following additional information on credit card usage: 70% of all regular fill-up customers use a credit card. 50% of all regular non-fill-up customers use a credit card. 60% of all plus fill-up customers use a credit card. 50% of all plus non-fill-up customers use a credit card. 50% of all premium fill-up customers use a credit card. 40% of all premium non-fill-up customers use a credit card. Compute the probability of each of the following events for the next customer to arrive (a tree diagram might help). a. {plus and fill-up and credit card} b. {premium and non-fill-up and credit card} c. {premium and credit card} d. {fill-up and credit card} e. {credit card} f. If the next customer uses a credit card, what is the probability that premium was requested?Reconsider the credit card scenario of Exercise 47 (Section 2.4), and show that A and B are dependent first by using the definition of independence and then by verifying that the multiplication property does not hold.An oil exploration company currently has two active projects, one in Asia and the other in Europe. Let A be the event that the Asian project is successful and B be the event that the European project is successful. Suppose that A and B are independent events with P(A) = .4 and P(B) = .7. a. If the Asian project is not successful, what is the probability that the European project is also not successful? Explain your reasoning. b. What is the probability that at least one of the two projects will be successful? c. Given that at least one of the two projects is successful, what is the probability that only the Asian project is successful?In Exercise 13, is any Ai independent of any other Aj? Answer using the multiplication property for independent events.If A and B are independent events, show that A and B are also independent. [Hint: First establish a relationship between P(A B), P(B), and P(A B).]The proportions of blood phenotypes in the U.S. population are as follows: A B AB O .40 .11 .04 .45 Assuming that the phenotypes of two randomly selected individuals are independent of one another, what is the probability that both phenotypes are O? What is the probability that the phenotypes of two randomly selected individuals match?One of the assumptions underlying the theory of control charting (see Chapter 16) is that successive plotted points are independent of one another. Each plotted point can signal either that a manufacturing process is operating correctly or that there is some sort of malfunction. Even when a process is running correctly, there is a small probability that a particular point will signal a problem with the process. Suppose that this probability is .05. What is the probability that at least one of 10 successive points indicates a problem when in fact the process is operating correctly? Answer this question for 25 successive points.In October, 1994, a flaw in a certain Pentium chip installed in computers was discovered that could result in a wrong answer when performing a division. The manufacturer initially claimed that the chance of any particular division being incorrect was only 1 in 9 billion, so that it would take thousands of years before a typical user encountered a mistake. However, statisticians are not typical users; some modern statistical techniques are so computationally intensive that a billion divisions over a short time period is not outside the realm of possibility. Assuming that the 1 in 9 billion figure is correct and that results of different divisions are independent of one another, what is the probability that at least one error occurs in one billion divisions with this chip?An aircraft seam requires 25 rivets. The seam will have to be reworked if any of these rivets is defective. Suppose rivets are defective independently of one another, each with the same probability. a. If 15% of all seams need reworking, what is the probability that a rivet is defective? b. How small should the probability of a defective rivet be to ensure that only 10% of all seams need reworking?A boiler has five identical relief valves. The probability that any particular valve will open on demand is .96. Assuming independent operation of the valves, calculate P(at least one valve opens) and P(at least one valve fails to open).Two pumps connected in parallel fail independently of one another on any given day. The probability that only the older pump will fail is .10, and the probability that only the newer pump will fail is .05. What is the probability that the pumping system will fail on any given day (which happens if both pumps fail)?Consider the system of components connected as in the accompanying picture. Components 1 and 2 are connected in parallel, so that subsystem works iff either 1 or 2 works; since 3 and 4 are connected in series, that subsystem works iff both 3 and 4 work. If components work independently of one another and P(component i works) = .9 for i = 1,2 and = .8 for i = 3,4, calculate P(system works).Refer back to the series-parallel system configuration introduced in Example 2.36, and suppose that there are only two cells rather than three in each parallel subsystem [in Figure 2.14(a), eliminate cells 3 and 6, and renumber cells 4 and 5 as 3 and 4]. Using P(Ai) = .9, the probability that system lifetime exceeds t0 is easily seen to be .9639. To what value would .9 have to be changed in order to increase the system lifetime reliability from .9639 to .99? [Hint: Let P(Ai) = p, express system reliability in terms of p, and then let x = p2.]Consider independently rolling two fair dice, one red and the other green. Let A be the event that the red die shows 3 dots, B be the event that the green die shows 4 dots, and C be the event that the total number of dots showing on the two dice is 7. Are these events pairwise independent (i.e., are A and B independent events, are A and C independent, and are B and C independent)? Are the three events mutually independent?Components arriving at a distributor are checked for defects by two different inspectors (each component is checked by both inspectors). The first inspector detects 90% of all defectives that are present, and the second inspector does likewise. At least one inspector does not detect a defect on 20% of all defective components. What is the probability that the following occur? a. A defective component will be detected only by the first inspector? By exactly one of the two inspectors? b. All three defective components in a batch escape detection by both inspectors (assuming inspections of different components are independent of one another)?84EA quality control inspector is examining newly produced items for faults. The inspector searches an item for faults in a series of independent fixations, each of a fixed duration. Given that a flaw is actually present, let p denote the probability that the flaw is detected during any one fixation (this model is discussed in Human Performance in Sampling Inspection, Human Factors, 1979: 99105). a. Assuming that an item has a flaw, what is the probability that it is detected by the end of the second fixation (once a flaw has been detected, the sequence of fixations terminates)? b. Give an expression for the probability that a flaw will be detected by the end of the nth fixation. c. If when a flaw has not been detected in three fixations, the item is passed, what is the probability that a flawed item will pass inspection? d. Suppose 10% of all items contain a flaw [P(randomly chosen item is flawed) = .1]. With the assumption of part (c), what is the probability that a randomly chosen item will pass inspection (it will automatically pass if it is not flawed, but could also pass if it is flawed)? e. Given that an item has passed inspection (no flaws in three fixations), what is the probability that it is actually flawed? Calculate for p = .5.a. A lumber company has just taken delivery on a shipment of 10,000 2 4 boards. Suppose that 20% of these boards (2000) are actually too green to be used in first-quality construction. Two boards are selected at random, one after the other. Let A = {the first board is green} and B = {the second board is green}. Compute P(A), P(B), and P(A B) (a tree diagram might help). Are A and B independent? b. With A and B independent and P(A) = P(B) = .2, what is P(A B)? How much difference is there between this answer and P(A B) in part (a)? For purposes of calculating P(A B), can we assume that A and B of part (a) are independent to obtain essentially the correct probability? c. Suppose the shipment consists of ten boards, of which two are green. Does the assumption of independence now yield approximately the correct answer for P(A B)? What is the critical difference between the situation here and that of part (a)? When do you think an independence assumption would be valid in obtaining an approximately correct answer to P(A B)?Consider randomly selecting a single individual and having that person test drive 3 different vehicles. Define events A1, A2, and A3 by A1 = likes vehicle #1 A2 = likes vehicle #2 A3 = likes vehicle #3 Suppose that P(A1) = .55, P(A2) = .65, P(A3) = .70, P(A1 A2) = .80, P(A2 A3) = .40, and P(A1 A2 A3) = .88. a. What is the probability that the individual likes both vehicle #1 and vehicle #2? b. Determine and interpret P(A2|A3 ). c. Are A2 and A3 independent events? Answer in two different ways. d. If you learn that the individual did not like vehicle #1, what now is the probability that he/she liked at least one of the other two vehicles?The probability that an individual randomly selected from a particular population has a certain disease is .05. A diagnostic test correctly detects the presence of the disease 98% of the time and correctly detects the absence of the disease 99% of the time. If the test is applied twice, the two test results are independent, and both are positive, what is the (posterior) probability that the selected individual has the disease? [Hint: Tree diagram with first-generation branches corresponding to Disease and No Disease, and second- and third-generation branches corresponding to results of the two tests.]Suppose identical tags are placed on both the left ear and the right ear of a fox. The fox is then let loose for a period of time. Consider the two events C1 = {left ear tag is lost} and C2 = {right ear tag is lost}. Let = P(C1) = P(C2), and assume C1 and C2 are independent events. Derive an expression (involving ) for the probability that exactly one tag is lost, given that at most one is lost (Ear Tag Loss in Red Foxes, J. Wildlife Mgmt., 1976: 164167). [Hint: Draw a tree diagram in which the two initial branches refer to whether the left ear tag was lost.]A certain legislative committee consists of 10 senators. A subcommittee of 3 senators is to be randomly selected. a. How many different such subcommittees are there? b. If the senators are ranked 1, 2, ..., 10 in order of seniority, how many different subcommittees would include the most senior senator? c. What is the probability that the selected subcommittee has at least 1 of the 5 most senior senators? d. What is the probability that the subcommittee includes neither of the two most senior senators?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 non conformance, for each of the three lines during a particular time period. 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. a. What is the probability that the can was produced by line 1? That the reason for nonconformance is a crack? b. If the selected can came from line 1, what is the probability that it had a blemish? c. Given that the selected can had a surface defect, what is the probability that it came from line 1?An employee of the records office at a certain university currently has ten forms on his desk awaiting processing. Six of these are withdrawal petitions and the other four are course substitution requests. a. If he randomly selects six of these forms to give to a subordinate, what is the probability that only one of the two types of forms remains on his desk? b. Suppose he has time to process only four of these forms before leaving for the day. If these four are randomly selected one by one, what is the probability that each succeeding form is of a different type from its predecessor?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 off on schedule. If A and B are independent events with P(A) P(B), P(A B) = .626, and P(A B) = .144, determine the values of P(A) and P(B).A transmitter is sending a message by using a binary code, namely, a sequence of 0s and 1s. Each transmitted bit (0 or 1) must pass through three relays to reach the receiver. At each relay, the probability is .20 that the bit sent will be different from the bit received (a reversal). Assume that the relays operate independently of one another. Transmitter Relay 1 Relay 2 Relay 3 Receiver a. If a 1 is sent from the transmitter, what is the probability that a 1 is sent by all three relays? b. If a 1 is sent from the transmitter, what is the probability that a 1 is received by the receiver? [Hint: The eight experimental outcomes can be displayed on a tree diagram with three generations of branches, one generation for each relay.] c. Suppose 70% of all bits sent from the transmitter are 1s. If a 1 is received by the receiver, what is the probability that a 1 was sent?Individual A has a circle of five close friends (B, C, D, E, and F). A has heard a certain rumor from outside the circle and has invited the five friends to a party to circulate the rumor. To begin, A selects one of the five at random and tells the rumor to the chosen individual. That individual then selects at random one of the four remaining individuals and repeats the rumor. Continuing, a new individual is selected from those not already having heard the rumor by the individual who has just heard it, until everyone has been told. a. What is the probability that the rumor is repeated in the order B, C, D, E, and F? b. What is the probability that F is the third person at the party to be told the rumor? c. What is the probability that F is the last person to hear the rumor? d. If at each stage the person who currently has the rumor does not know who has already heard it and selects the next recipient at random from all five possible individuals, what is the probability that F has still not heard the rumor after it has been told ten times at the party?According to the article Optimization of Distribution Parameters for Estimating Probability of Crack Detection (J. of Aircraft, 2009: 20902097), the following Palmberg equation is commonly used to determine the probability Pd(c) of detecting a crack of size c in an aircraft structure: Pd(c)=(c/c)1+(c/c) 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). a. Verify that Pd (c) = .5 b. What is Pd (2c) when = 4? 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 = 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 detected? d. What happens to Pd (c) as ?A chemical engineer is interested in determining whether a certain trace impurity is present in a product. An experiment has a probability of .80 of detecting the impurity if it is present. The probability of not detecting the impurity if it is absent is .90. The prior probabilities of the impurity being present and being absent are .40 and .60, respectively. Three separate experiments result in only two detections. What is the posterior probability that the impurity is present?98SEFasteners used in aircraft manufacturing are slightly crimped so that they lock enough to avoid loosening during vibration. Suppose that 95% of all fasteners pass an initial inspection. Of the 5% that fail, 20% are so seriously defective that they must be scrapped. The remaining fasteners are sent to a recrimping operation, where 40% cannot be salvaged and are discarded. The other 60% of these fasteners are corrected by the recrimping process and subsequently pass inspection. a. What is the probability that a randomly selected incoming fastener will pass inspection either initially or after recrimping? b. Given that a fastener passed inspection, what is the probability that it passed the initial inspection and did not need recrimping?Jay and Maurice are playing a tennis match. In one particular game, they have reached deuce, which means each player has won three points. To finish the game, one of the two players must get two points ahead of the other. For example, Jay will win if he wins the next two points (JJ), or if Maurice wins the next point and Jay the three points after that (MJJJ), or if the result of the next six points is JMMJJJ, and so on. a. Suppose that the probability of Jay winning a point is .6 and outcomes of successive points are independent of one another. What is the probability that Jay wins the game? [Hint: In the law of total probability, let A1 = Jay wins each of the next two points, A2 = Maurice wins each of the next two points, and A3 = each player wins one of the next two points. Also let p = P(Jay wins the game). How does p compare to P(Jay wins the game | A3)?] b. If Jay wins the game, what is the probability that he needed only two points to do so?A system consists of two components. The probability that the second component functions in a satisfactory manner during its design life is .9, the probability that at least one of the two components does so is .96, and the probability that both components do so is .75. Given that the first component functions in a satisfactory manner throughout its design life, what is the probability that the second one does also?The accompanying table categorizing each student in a sample according to gender and eye color appeared in the article Does Eye Color Depend on Gender? It Might Depend on Who or How You Ask (J. of Statistics Educ., 2013, Vol. 21, Num. 2). Suppose that one of these 2026 students is randomly selected. Let F denote the event that the selected individual is a female, and A, B, C, and D represent the events that he or she has blue, brown, green, and hazel eyes, respectively. a. Calculate both P(F) and P(C). b. Calculate P(F C). Are the events F and C independent? Why or why not? c. If the selected individual has green eyes, what is the probability that he or she is a female? d. If the selected individual is female, what is the probability that she has green eyes? e. What is the conditional distribution of eye color for females (i.e., P(A|F), P(B|F), P(C|F), and P(D| F)), and what is it for males? Compare the two distributions.a. A certain company sends 40% of its overnight mail parcels via express mail service E1. Of these parcels, 2% arrive after the guaranteed delivery time (denote the event late delivery by L). If a record of an overnight mailing is randomly selected from the companys file, what is the probability that the parcel went via E1 and was late? b. Suppose that 50% of the overnight parcels are sent via express mail service E2 and the remaining 10% are sent via E3. Of those sent via E2, only 1% arrive late, whereas 5% of the parcels handled by E3 arrive late. What is the probability that a randomly selected parcel arrived late? c. If a randomly selected parcel has arrived on time, what is the probability that it was not sent via E1?A company uses three different assembly linesA1, A2, and A3to manufacture a particular component. Of those manufactured by line A1, 5% need rework to remedy a defect, whereas 8% of A2s components need rework and 10% of A3s need rework. Suppose that 50% of all components are produced by line A1, 30% are produced by line A2, and 20% come from line A3. If a randomly selected component needs rework, what is the probability that it came from line A1? From line A2? From line A3?Disregarding the possibility of a February 29 birthday, suppose a randomly selected individual is equally likely to have been born on any one of the other 365 days. a. If ten people are randomly selected, what is the probability that all have different birthdays? That at least two have the same birthday? b. With k replacing ten in part (a), what is the smallest k for which there is at least a 50-50 chance that two or more people will have the same birthday? c. If ten people are randomly selected, what is the probability that either at least two have the same birthday or at least two have the same last three digits of their Social Security numbers? [Note: The article Methods for Studying Coincidences (F. Mosteller and P. Diaconis, J. Amer. Stat. Assoc., 1989: 853861) discusses problems of this type.]One method used to distinguish between granitic (G) and basaltic (B) rocks is to examine a portion of the infrared spectrum of the suns energy reflected from the rock surface. Let R1, R2, and R3 denote measured spectrum intensities at three different wavelengths; typically, for granite R1 R2 R3, whereas for basalt R3 R1 R2. When measurements are made remotely (using aircraft), various orderings of the Ris may arise whether the rock is basalt or granite. Flights over regions of known composition have yielded the following information: Suppose that for a randomly selected rock in a certain region, P(granite) 5 .25 and P(basalt) 5 .75. a. Show that P(granite | R1 R2 R3) P(basalt | R1 R2 R3). If measurements yielded R1 R2 R3, would you classify the rock as granite or basalt? b. If measurements yielded R1 R3 R2, how would you classify the rock? Answer the same question for R3 R1 R2. c. Using the classification rules indicated in parts (a) and (b), when selecting a rock from this region, what is the probability of an erroneous classification? [Hint: Either G could be classified as B or B as G, and P(B) and P(G) are known.] d. If P(granite) = p rather than .25, are there values of p (other than 1) for which one would always classify a rock as granite?A subject is allowed a sequence of glimpses to detect a target. Let Gi = {the target is detected on the ith glimpse}, with pi 5 P(Gi). Suppose the Gjs are independent events, and write an expression for the probability that the target has been detected by the end of the nth glimpse. [Note: This model is discussed in Predicting Aircraft Detectability, Human Factors, 1979: 277291.]In a Little League baseball game, team As pitcher throws a strike 50% of the time and a ball 50% of the time, successive pitches are independent of one another, and the pitcher never hits a batter. Knowing this, team Bs manager has instructed the first batter not to swing at anything. Calculate the probability that a. The batter walks on the fourth pitch b. The batter walks on the sixth pitch (so two of the first five must be strikes), using a counting argument or constructing a tree diagram c. The batter walks d. The first batter up scores while no one is out (assuming that each batter pursues a no-swing strategy)Four engineers, A, B, C, and D, have been scheduled for job interviews at 10 a.m. on Friday, January 13, at Random Sampling, Inc. The personnel manager has scheduled the four for interview rooms 1, 2, 3, and 4, respectively. However, the managers secretary does not know this, so assigns them to the four rooms in a completely random fashion (what else!). What is the probability that a. All four end up in the correct rooms? b. None of the four ends up in the correct room?A particular airline has 10 a.m. flights from Chicago to New York, Atlanta, and Los Angeles. Let A denote the event that the New York flight is full and define events B and C analogously for the other two flights. Suppose P(A) = .9, P(B) = .7, P(C) = .8 and the three events are independent. What is the probability that a. All three flights are full? That at least one flight is not full? b. Only the New York flight is full? That exactly one of the three flights is full?Consider four independent events A1, A2, A3, and A4, and let pi 5 P(Ai) for i 5 1,2,3,4. Express the probability that at least one of these four events occurs in terms of the pis, and do the same for the probability that at least two of the events occur.A box contains the following four slips of paper, each having exactly the same dimensions: (1) win prize 1; (2) win prize 2; (3) win prize 3; (4) win prizes 1, 2, and 3. One slip will be randomly selected. Let A1 = {win prize 1}, A2 = {win prize 2}, and A3 = {win prize 3}. Show that A1 and A2 are independent, that A1 and A3 are independent, and that A2 and A3 are also independent (this is pairwise independence). However, show that P(A1 A2 A3) P(A1) P(A2) P(A3), so the three events are not mutually independent.Show that if A1, A2, and A3 are independent events, then P(A1 | A2 A3) 5 P(A1).A concrete beam may fail either by shear (S) or flexure (F). Suppose that three failed beams are randomly selected and the type of failure is determined for each one. Let X = the number of beams among the three selected that failed by shear. List each outcome in the sample space along with the associated value of X.Using the experiment in Example 3.3, define two more random variables and list the possible values of each.Let X = the number of nonzero digits in a randomly selected 4-digit PIN that has no restriction on the digits. What are the possible values of X? Give three possible outcomes and their associated X values.If the sample space S is an infinite set, does this necessarily imply that any rv X defined from S will have an infinite set of possible values? If yes, say why. If no, give an example.Starting at a fixed time, each car entering an intersection is observed to see whether it turns left (L), right (R). or goes straight ahead (A). The experiment terminates as soon as a car is observed to turn left. Let X = the number of cars observed. What are possible X values? List five outcomes and their associated X values.For each random variable defined here, describe the set of possible values for the variable, and state whether the variable is discrete. a. X = the number of unbroken eggs in a randomly chosen standard egg carton b. Y = the number of students on a class list for a particular course who are absent on the first day of classes c. U = the number of times a duffer has to swing at a golf ball before hitting it d. X = the length of a randomly selected rattlesnake e. Z = the sales tax percentage for a randomly selected amazon.com purchase f. Y = the pH of a randomly chosen soil sample g. X = the tension (psi) at which a randomly selected tennis racket has been strung h. X = the total number of times three tennis players must spin their rackets to obtain something other than UUU or DDD (to determine which two play next)Each time a component is tested, the trial is a success (S) or failure (F). Suppose the component is tested repeatedly until a success occurs on three consecutive trials. Let Y denote the number of trials necessary to achieve this. List all outcomes corresponding to the five smallest possible values of Y, and state which Y value is associated with each one.An individual named Claudius is located at the point 0 in the accompanying diagram. Using an appropriate randomization device (such as a tetrahedral die, one having four sides), Claudius first moves to one of the four locations B1, B2, B3, B4. Once at one of these locations, another randomization device is used to decide whether Claudius next returns to 0 or next visits one of the other two adjacent points. This process then continues; after each move, another move to one of the (new) adjacent points is determined by tossing an appropriate die or coin. a. Let X = the number of moves that Claudius makes before first returning to 0. What are possible values of X? Is X discrete or continuous? b. If moves are allowed also along the diagonal paths connecting 0 to A1, A2, A3, and A4, respectively, answer the questions in part (a).The number of pumps in use at both a six-pump station and a four-pump station will be determined. Give the possible values for each of the following random variables: a. T = the total number of pumps in use b. X = the difference between the numbers in use at stations 1 and 2 c. U = the maximum number of pumps in use at either station d. Z = the number of stations having exactly two pumps in useLet X be the number of students who show up for a professors office hour on a particular day. Suppose that the pmf of X is p(0) = .20, p(1) = .25, p(2) = .30, p(3) = .15, and p(4) = .10. a. Draw the corresponding probability histogram. b. What is the probability that at least two students show up? More than two students show up? c. What is the probability that between one and three students, inclusive, show up? d. What is the probability that the professor show's up?Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable Y as the number of ticketed passengers who actually show up for the flight. The probability mass function of Y appears in the accompanying table. y 45 46 47 48 49 50 51 52 53 54 55 p(y) .05 .10 .12 .14 .25 .17 .06 .05 .03 .02 .01 a. What is the probability that the flight will accommodate all ticketed passengers who show up? b. What is the probability that not all ticketed passengers who show up can be accommodated? c. If you are the first person on the standby list (which means you will be the first one to get on the plane if there are any seats available after all ticketed passengers have been accommodated), what is the probability that you will be able to take the flight? What is this probability if you are the third person on the standby list?A mail-order computer business has six telephone lines. Let X denote the number of lines in use at a specified time. Suppose the pmf of X is as given in the accompanying table. x 0 1 2 3 4 5 6 p(x) .10 .15 .20 25 .20 .06 .04 Calculate the probability of each of the following events. a. {at most three lines are in use} b. {fewer than three lines are in use} c. {at least three lines are in use} d. {between two and five lines, inclusive, are in use} e. {between two and four lines, inclusive, are not in use} f. {at least four lines are not in use}A contractor is required by a county planning department to submit one, two, three, four, or five forms (depending on the nature of the project) in applying for a building permit. Let Y = the number of forms required of the next applicant. The probability that y forms are required is known to be proportional to ythat is, p(y) = ky for y = 1,...., 5. a. What is the value of k? [Hint:y15p(y)=1 ] b. What is the probability that at most three forms are required? c. What is the probability that between two and four forms (inclusive) are required? d. Could p(y) = y2/50 for y = 1,, 5 be the pmf of Y?Many manufacturers have quality control programs that include inspection of incoming materials for defects. Suppose a computer manufacturer receives circuit boards in batches of five. Two boards are selected from each batch for inspection. We can represent possible outcomes of the selection process by pairs. For example, the pair (1, 2) represents the selection of boards 1 and 2 for inspection. a. List the ten different possible outcomes. b. Suppose that boards 1 and 2 are the only defective boards in a batch. Two boards are to be chosen at random. Define X to be the number of defective boards observed among those inspected. Find the probability distribution of X. c. Let F(x) denote the cdf of X. First determine F(0) = P(X 0). F( 1). and F(2): then obtain F(x) for all other x.Some parts of California are particularly earthquake-prone. Suppose that in one metropolitan area. 25% of all homeowners are insured against earthquake damage. Four homeowners arc to be selected at random: let X denote the number among the four who have earthquake insurance. a. Find the probability distribution of X. [Hint: Let S denote a homeowner who has insurance and F one who does not. Then one possible outcome is SFSS, with probability (.25)(.75)(.25)(.25) and associated X value 3. There are 15 other outcomes.] b. Draw the corresponding probability histogram. c. What is the most likely value for X? d. What is the probability that at least two of the four selected have earthquake insurance?A now batterys voltage may be acceptable (A) or unacceptable (U). A certain flashlight requires two batteries, so batteries will be independently selected and tested until two acceptable ones have been found. Suppose that 90% of all batteries have acceptable voltages. Let Y denote the number of batteries that must be tested. a. What is p(2). that is. P(Y = 2)? b. What is p(3)? [Hint: There are two different out-comes that result in Y = 3.] c. To have Y = 5, what must be true of the fifth battery selected? List the four outcomes for which Y = 5 and then determine p(5). d. Use the pattern in your answers for parts (a)-(c) to obtain a general formula for p(y).Two fair six-sided dice are tossed independently. Let M = the maximum of the two tosses (so M(1,5) = 5, M(3,3) = 3, etc.). a. What is the pmf of M? [Hint: First determine p(1), then p(2), and so on.] b. Determine the cdf of M and graph it.A library subscribes to two different weekly news magazines, each of which is supposed to arrive in Wednesdays mail. In actuality, each one may arrive on Wednesday, Thursday. Friday, or Saturday. Suppose the two arrive independently of one another, and for each one P(Wed.) = .3, P(Thurs.) = .4, P(Fri.) = .2, and P(Sat.) = .1. Let Y = the number of days beyond Wednesday that it takes for both magazines to arrive (so possible Y values are 0, 1, 2, or 3). Compute the pmf of Y. [Hint: There are 16 possible outcomes; Y(W,W) = 0, Y(F,Th) = 2. and soon.]Three couples and two single individuals have been invited to an investment seminar and have agreed to attend. Suppose the probability that any particular couple or individual arrives late is .4 (a couple will travel together in the same vehicle, so either both people will be on time or else both will arrive late). Assume that different couples and individuals are on time or late independently of one another. Let X = the number of people who arrive late for the seminar. a. Determine the probability mass function of X. [Hint: label the three couples #1, #2, and #3 and the two individuals #4 and #5.] b. Obtain the cumulative distribution function of X, and use it to calculate P(2 X 6).Suppose that you read through this years issues of the New York Times and record each number that appears in a news article-the income of a CEO. the number of cases of wine produced by a winery, the total charitable contribution of a politician during the previous tax year, the age of a celebrity, and so on. Now focus on the leading digit of each number, which could be 1.28. or 9. Your first thought might be that the leading digit X of a randomly selected number would be equally likely to be one of the nine possibilities (a discrete uniform distribution). However, much empirical evidence as well as some theoretical arguments suggest an alternative probability distribution called Benfords law: p(x)=P(1stdigitisx)=log10(x+1x)x=1,2,...,9 a. Without computing individual probabilities from this formula, show that it specifies a legitimate pmf. Now compute the individual probabilities and compare to the corresponding discrete uniform distribution. b. Obtain the cdf of X. c. Using the cdf, what is the probability that the leading digit is at most 3? At least 5? [Note: Benfords law is the basis for some auditing procedures used to detect fraud in financial reportingfor example, by the Internal Revenue Service.]Refer to Exercise 13, and calculate and graph the cdf F(x). Then use it to calculate the probabilities of the events given in parts (a)-(d) of that problem. 13. A mail-order computer business has six telephone lines. Let X denote the number of lines in use at a specified time. Suppose the pmf of X is as given in the accompanying table. x 0 1 2 3 4 5 6 p(x) .10 .15 .20 25 .20 .06 .04 Calculate the probability of each of the following events. a. {at most three lines are in use} b. {fewer than three lines are in use} c. {at least three lines are in use} d. {between two and five lines, inclusive, are in use} e. {between two and four lines, inclusive, are not in use} f. {at least four lines are not in use}A branch of a certain bank in New York City has six ATMs. Let X represent the number of machines in use at a particular time of day. The cdf of X is as follows: F(x)={0x1.600x1.191x2.392x3.673x4.924x5.975x616x Calculate the following probabilities directly from the cdf: a. p(2), that is, P(X = 2) b. P(X 3) c. P(2 X 5) d. P(2 X 5)An insurance company offers its policyholders a number of different premium payment options. For a randomly selected policyholder, let X = the number of months between successive payments. The cdf of X is as follows: F(x)={0x1.301x3.403x4.454x6.606x12112x a. What is the pmf of X? b. Using just the cdf, compute P(3 x 6) and P(4 X).In Example 3.12, let Y = the number of girls born before the experiment terminates. With p = P(B) and 1 - p = P(G), what is the pmf of Y? [Hint: First list the possible values of Y, starting with the smallest, and proceed until you see a general formula.]Alvie Singer lives at 0 in the accompanying diagram and has four friends who live at A, B, C, and D. One day Alvie decides to go visiting, so he tosses a fair coin twice to decide which of the four to visit Once at a friends house, he will either return home or else proceed to one of the two adjacent houses (such as 0, A, or C when at B), with each of the three possibilities having probability 1/3. In this way. Alvie continues to visit friends until he returns home. a. Let X = the number of times that Alvie visits a friend. Derive the pmf of X b. Let Y = the number of straight-line segments that Alvie traverses (including those leading to and from 0). What is the pmf of Y? c. Suppose that female friends live at A and C and male friends at B and D. If Z = the number of visits to female friends, what is the pmf of Z?After all students have left the classroom, a statistics professor notices that four copies of the text were left under desks. At the beginning of the next lecture, the professor distributes the four books in a completely random fashion to each of the four students (1, 2, 3, and 4) who claim to have left books. One possible outcome is that 1 receives 2s book, 2 receives 4s book. 3 receives his or her own book, and 4 receives ls book. This out-come can be abbreviated as (2.4,3, 1). a. List the other 23 possible outcomes. b. Let X denote the number of students who receive their own book. Determine the pmf of X.Show that the cdf F(x) is a nondecreasing function; that is, x1 x2 implies that F(x1) F(x2). Under what condition will F(x1) = F(x2)?The pmf of the amount of memory X(GB) in a purchased flash drive was given in Example 3.13 as X 1 2 4 8 16 P(x) .05 .10 .35 .40 .10 Compute the following: a. E(X) b. V(X) directly from the definition c. The standard deviation of X d. V(X) using the shortcut formulaAn individual who has automobile insurance from a certain company is randomly selected Let Y be the number of moving violations for which the individual was cited during the last 3 years. The pmf of Y is y 0 l 2 3 p(y) .60 .25 .10 .05 a. Compute E(Y). b. Suppose an individual with Y violations incurs a surcharge of 100Y2. Calculate the expected amount of the surcharge.Refer to Exercise 12 and calculate V(Y) and Y. Then determine the probability that Y is within 1 standard deviation of its mean value.A certain brand of upright freezer is available in three different rated capacities: 16 ft3, 18 ft3, and 20 ft3. Let X = the rated capacity of a freezer of this brand sold at a certain store. Suppose that X has pmf x 16 18 20 p(x) .2 .5 3 a. Compute E(X), E(X2), and V(X). b. If the price of a freezer having capacity X is 70X 650, what is the expected price paid by the next customer to buy a freezer? c. What is the variance of the price paid by the next customer? d. Suppose that although the rated capacity of a freezer is X, the actual capacity is h(X) = X - .008X2. What is the expected actual capacity of the freezer purchased by the next customer?Let X be a Bernoulli rv with pmf as in Example 3.18. a. Compute E(X2). b. Show that V(X) = p(1 - p). c. Compute E(X79).Suppose that the number of plants of a particular type found in a rectangular sampling region (called a quadrat by ecologists) in a certain geographic area is an rv X with pmf p(x)={c/x3x=1,2,3,...0otherwise Is E(X) finite? Justify your answer (this is another distribution that statisticians would call heavy-tailed).A small market orders copies of a certain magazine for its magazine rack each week. Let X = demand for the magazine, with pmf x 1 2 3 4 5 6 p(x) 115 215 315 415 315 215 Suppose the store owner actually pays 2.00 for each copy of the magazine and the price to customers is 4.00. If magazines left at the end of the week have no salvage value, is it better to order three or four copies of the magazine? [Hint: For both three and four copies ordered, express net revenue as a function of demand X, and then compute the expected revenue.]Let X be the damage incurred (in ) in a certain type of accident during a given year. Possible X values are 0, 1000, 5000, and 10000, with probabilities .8, .1, .08, and .02, respectively. A particular company offers a 500 deductible policy. If the company wishes its expected profit to be 100, what premium amount should it charge?The n candidates for a job have been ranked 1, 2, 3,, n. Let X = the rank of a randomly selected candidate, so that X has pmf p(x)={1/nx=1,2,3,...,n0otherwise (this is called the discrete uniform distribution). Compute E(X) and V(X) using the shortcut formula. [Hint: The sum of the first n positive integers is n(n + 1)/2, whereas the sum of their squares is n(n + 1 )(2n + 1 )/6.]Possible values of X, the number of components in a system submitted for repair that must be replaced, are 1, 2, 3, and 4 with corresponding probabilities .15, .35, .35, and .15, respectively. a. Calculate E(X) and then E(5 - X). b. Would the repair facility be better off charging a flat fee of 75 or else the amount [150/(5 - X)]? [Note: It is not generally true that E(c/Y) = c/E(Y).]A chemical supply company currently has in stock 100 lb of a certain chemical, which it sells to customers in 5-lb batches. Let X = the number of batches ordered by a randomly chosen customer, and suppose that X has pmf x 1 2 3 4 p(x) .2 .4 .3 .1 Compute E(X) and V(X). Then compute the expected number of pounds left after the next customer s order is shipped and the variance of the number of pounds left. [Hint: The number of pounds left is a linear function of X. ]a. Draw a line graph of the pmf of X in Exercise 35. Then determine the pmf of X and draw its line graph. From these two pictures, what can you say about V(X) and V(X)? b. Use the proposition involving V(aX + b) to establish a general relationship between V(X) and V(-X).Use the definition in Expression (3.13) to prove that V(aX + b) = a2 x2. [Hint: With h(X) = aX + b, E[h(X)] = a + b where = E(X).]Suppose E(X) = 5 and E[X(X - 1)] = 27.5. What is a. E(X2)? [Hint: First verify that E[X(X 1)] = E(X2) - E(X)]? b. V(X)? c. The general relationship among the quantities E(X), E[X(X 1)], and V(X)?Write a general rule for E(X c) where c is a constant. What happens when c = , the expected value of X?A result called Chebyshevs inequality states that for any probability distribution of an rv X and any number k that is at least 1, P(|X - | k) 1/k2. In words, the probability that the value of X lies at least k standard deviations from its mean is at most 1/x2. a. What is the value of the upper bound for k = 2? k = 3? k = 4? k = 5? k = 10? b. Compute and for the distribution of Exercise 13. Then evaluate P(|X - | k) for the values of k given in part (a). What does this suggest about the upper bound relative to the corresponding probability? c. Let X have possible values 1,0, and 1, with probabilities 118,89 and 118, respectively What is P( |X | 3), and how does it compare to the corresponding bound? d. Give a distribution for which P(|X | 5) = .04.If a X b, show that a E(X) b.Compute the following binomial probabilities directly from the formula for b(x, n, p): a. b(3; 8, .35) b. b(5; 8, .6) c. P(3 X 5) when n = 7and p = .6 d. P(1 X) when n = 9 and p = .1The article Should You Report That Fender-Bender? (Consumer Reports, Sept. 2013:15) reported that 7 in 10 auto accidents involve a single vehicle (the article recommended always reporting to the insurance company an accident involving multiple vehicles). Suppose 15 accidents are randomly selected. Use Appendix Table A.1 to answer each of the following questions. a. What is the probability that at most 4 involve a single vehicle? b. What is the probability that exactly 4 involve a single vehicle? c. What is the probability that exactly 6 involve multiple vehicles? d. What is the probability that between 2 and 4, inclusive, involve a single vehicle? e. What is the probability that at least 2 involve a single vehicle? f. What is the probability that exactly 4 involve a single vehicle and the other 11 involve multiple vehicles?NBC News reported on May 2. 2013. that 1 in 20 children in the United States have a food allergy of some sort. Consider selecting a random sample of 25 children and let X be the number in the sample who have a food allergy. Then X Bin(25, .05). a. Determine both P(X 3) and P(X 3). b. Determine P(X 4). c. Determine P( 1 X 3). d. What are E(X) and x? e. In a sample of 50 children, what is the probability that none has a food allergy?A company that produces fine crystal knows from experience that 10% of its goblets have cosmetic flaws and must be classified as seconds. a. Among six randomly selected goblets, how likely is it that only one is a second? b. Among six randomly selected goblets, what is the probability that at least two are seconds? c. If goblets are examined one by one, what is the probability that at most five must be selected to find four that are not seconds?A particular telephone number is used to receive both voice calls and fax messages. Suppose that 25% of the incoming calls involve fax messages, and consider a sample of 25 incoming calls. What is the probability that a. At most 6 of the calls involve a fax message? b. Exactly 6 of the calls involve a fax message? c. At least 6 of the calls involve a fax message? d. More than 6 of the calls involve a fax message?Refer to the previous exercise. a. What is the expected number of calls among the 25 that involve a fax message? b. What is the standard deviation of the number among the 25 calls that involve a fax message? c. What is the probability that the number of calls among the 25 that involve a fax transmission exceeds the expected number by more than 2 standard deviations?Suppose that 30% of all students who have to buy a text for a particular course want a new copy (the successes!), whereas the other 70% want a used copy. Consider randomly selecting 25 purchasers. a. What are the mean value and standard deviation of the number who want a new copy of the book? b. What is the probability that the number who want new copies is more than two standard deviations away from the mean value? c. The bookstore has 15 new copies and 15 used copies in stock. If 25 people come in one by one to purchase this text, what is the probability that all 25 will get the type of book they want from current stock? [Hint: Let X = the number who want a new copy. For what values of X will all 25 get what they want?] d. Suppose that new copies cost 100 and used copies cost 70. Assume the bookstore currently has 50 new copies and 50 used copies. What is the expected value of total revenue from the sale of the next 25 copies purchased? Be sure to indicate what rule of expected value you are using. [Hint: Let h(X) = the revenue when X of the 25 purchasers want new copies. Express this as a linear function.]Exercise 30 (Section 3.3) gave the pmf of Y, the number of traffic citations for a randomly selected individual insured by a particular company. What is the probability that among 15 randomly chosen such individuals a. At least 10 have no citations? b. Fewer than half have at least one citation? c. The number that have at least one citation is between 5 and 10, inclusive?A particular type of tennis racket comes in a midsize version and an oversize version. Sixty percent of all customers at a certain store want the oversize version. a. Among ten randomly selected customers who want this type of racket, what is the probability that at least six want the oversize version? b. Among ten randomly selected customers, what is the probability that the number who want the oversize version is within 1 standard deviation of the mean value? c. The store currently has seven rackets of each version. What is the probability that all of the next ten customers who want this racket can get the version they want from current stock?Twenty percent of all telephones of a certain type are submitted for service while under warranty. Of these, 60% can be repaired, whereas the other 40% must be replaced with new units. If a company purchases ten of these telephones, what is the probability that exactly two will end up being replaced under warranty?The College Board reports that 2% of the 2 million high school students who take the SAT each year receive special accommodations because of documented disabilities (Los Angeles Times. July 16. 2002). Consider a random sample of 25 students who have recently taken the test. a. What is the probability that exactly 1 received a special accommodation? b. What is the probability that at least 1 received a special accommodation? c. What is the probability that at least 2 received a special accommodation? d. What is the probability that the number among the 25 who received a special accommodation is within 2 standard deviations of the number you would expect to be accommodated? e. Suppose that a student who does not receive a special accommodation is allowed 3 hours for the exam, whereas an accommodated student is allowed 4.5 hours. What would you expect the average time allowed the 25 selected students to be?A certain type of flashlight requires two type-D batteries, and the flashlight will work only if both its batteries have acceptable voltages. Suppose that 90% of all batteries from a certain supplier have acceptable voltages. Among ten randomly selected flashlights, what is the probability that at least nine will work? What assumptions did you make in the course of answering the question posed?A very large batch of components has arrived at a distributor. The batch can be characterized as accept-able only if the proportion of defective components is at most .10. The distributor decides to randomly select 10 components and to accept the batch only if the number of defective components in the sample is at most 2. a. What is the probability that the batch will be accepted when the actual proportion of defectives is .01? .05? .10? .20? .25? b. Let p denote the actual proportion of defectives in the batch. A graph of P(batch is accepted) as a function of p, with p on the horizontal axis and P(batch is accepted) on the vertical axis, is called the operating characteristic curve for the acceptance sampling plan. Use the results of part (a) to sketch this curve for 0 p 1. c. Repeat parts (a) and (b) with 1 replacing 2 in the acceptance sampling plan. d. Repeat parts (a) and (b) with 15 replacing 10 in the acceptance sampling plan. e. Which of the three sampling plans, that of part (a), (c), or (d), appears most satisfactory, and why?An ordinance requiring that a smoke detector be installed in all previously constructed houses has been in effect in a particular city for 1 year. The fire department is concerned that many houses remain without detectors. Let p = the true proportion of such houses having detectors, and suppose that a random sample of 25 homes is inspected. If the sample strongly indicates that fewer than 80% of all houses have a detector, the fire department will campaign for a mandatory inspection program. Because of the costliness of the program, the department prefers not to call for such inspections unless sample evidence strongly argues for their necessity. Let X denote the number of homes with detectors among the 25 sampled. Consider rejecting the claim that p .8 if x 15. a. What is the probability that the claim is rejected when the actual value of p is .8? b. What is the probability of not rejecting the claim when p = .7? When p = .6? c. How do the error probabilities of parts (a) and (b) change if the value 15 in the decision rule is replaced by 14?A toll bridge charges 1.00 for passenger cars and 2.50 for other vehicles. Suppose that during daytime hours, 60% of all vehicles are passenger cars. If 25 vehicles cross the bridge during a particular daytime period, what is the resulting expected toll revenue? [Hint: Let X = the number of passenger cars: then the toll revenue h(X) is a linear function of X.]A student who is trying to write a paper for a course has a choice of two topics, A and B. If topic A is chosen, the student will order two books through interlibrary loan, whereas if topic B is chosen, the student will order four books. The student believes that a good paper necessitates receiving and using at least half the books ordered for either topic chosen. If the probability that a book ordered through interlibrary loan actually arrives in time is .9 and books arrive independently of one another, which topic should the student choose to maximize the probability of writing a good paper? What if the arrival probability is only .5 instead of .9?a. For fixed n, are there values of p(0 p 1) for which V(X) = 0? Explain why this is so. b. For what value of p is V(X) maximized? [Hint: Either graph V(X) as a function of p or else take a derivative.]a. Show that b(x; n, 1 p) = b(n x; n, p). b. Show that B(x; n, 1 p) = 1 B(n x 1; n, p). [Hint: At most x Ss is equivalent to at least (n x) Fs.] c. What do parts (a) and (b) imply about the necessity of including values of p greater than .5 in Appendix Table A. 1?Show that E(X) = np when X is a binomial random variable. [Hint: First express E(X) as a sum with lower limit x = 1. Then factor out np, let y = x 1 so that the sum is from y = 0 to y = n 1, and show that the sum equals 1.]Customers at a gas station pay with a credit card (A), debit card (B), or cash (C). Assume that successive customers make independent choices, with P(A) = .5, P(B) = .2, and P(C) = .3. a. Among the next 100 customers, what are the mean and variance of the number who pay with a debit card? Explain your reasoning. b. Answer part (a) for the number among the 100 who dont pay with cash.An airport limousine can accommodate up to four passengers on any one trip. The company will accept a maximum of six reservations for a trip, and a passenger must have a reservation. From previous records, 20% of all those making reservations do not appear for the trip. Answer the following questions, assuming independence wherever appropriate. a. If six reservations are made, what is the probability that at least one individual with a reservation cannot be accommodated on the trip? b. If six reservations are made, what is the expected number of available places when the limousine departs? c. Suppose the probability distribution of the number of reservations made is given in the accompanying table. Number of reservations 3 4 5 6 Probability .1 .2 .3 .4 Let X denote the number of passengers on a randomly selected trip. Obtain the probability mass function of X.Refer to Chebyshevs inequality given in Exercise 44. Calculate P(|X - | k) for k = 2 and k = 3 when X Bin(20, .5), and compare to the corresponding upper bound. Repeat for X Bin(20, .75).Eighteen individuals are scheduled to take a driving test at a particular DMV office on a certain day, eight of whom will be taking the test for the first time. Suppose that six of these individuals are randomly assigned to a particular examiner, and let X be the number among the six who are taking the test for the first time. a. What kind of a distribution does X have (name and values of all parameters)? b. Compute P(X = 2), P(X 2), and P(X 2). c. Calculate the mean value and standard deviation of X.Each of 12 refrigerators of a certain type has been returned to a distributor because of an audible, high-pitched, oscillating noise when the refrigerators are running. Suppose that 7 of these refrigerators have a defective compressor and the other 5 have less serious problems. If the refrigerators are examined in random order, let X be the number among the first 6 examined that have a defective compressor. a. Calculate P(X = 4) and P(X 4) b. Determine the probability that X exceeds its mean value by more than 1 standard deviation. c. Consider a large shipment of 400 refrigerators, of which 40 have defective compressors. If X is the number among 15 randomly selected refrigerators that have defective compressors, describe a less tedious way to calculate (at least approximately) P(X 5) than to use the hypergeometric pmf.An instructor who taught two sections of engineering statistics last term, the first with 20 students and the second with 30, decided to assign a term project. After all projects had been turned in, the instructor randomly ordered them before grading. Consider the first 15 graded projects. a. What is the probability that exactly 10 of these are from the second section? b. What is the probability that at least 10 of these are from the second section? c. What is the probability that at least 10 of these are from the same section? d. What are the mean value and standard deviation of the number among these 15 that are from the second section? e. What are the mean value and standard deviation of the number of projects not among these first 15 that are from the second section?A geologist has collected 10 specimens of basaltic rock and 10 specimens of granite. The geologist instructs a laboratory assistant to randomly select 15 of the specimens for analysis. a. What is the pmf of the number of granite specimens selected for analysis? b. What is the probability that all specimens of one of the two types of rock are selected for analysis? c. What is the probability that the number of granite specimens selected for analysis is within 1 standard deviation of its mean value?A personnel director interviewing 11 senior engineers for four job openings has scheduled six interviews for the first day and five for the second day of interviewing. Assume that the candidates are interviewed in random order. a. What is the probability that x of the top four candidates are interviewed on the first day? b. How many of the top four candidates can be expected to be interviewed on the first day?Twenty pairs of individuals playing in a bridge tournament have been seeded 1,, 20. In the first part of the tournament, the 20 are randomly divided into 10 east-west pairs and 10 north-south pairs. a. What is the probability that x of the top 10 pairs end up playing east-west? b. What is the probability that all of the top five pairs end up playing the same direction? c. If there are 2n pairs, what is the pmf of X = the number among the top n pairs who end up playing east-west? What are E(X) and V(X)?A second-stage smog alert has been called in a certain area of Los Angeles County in which there are 50 industrial firms. An inspector will visit 10 randomly selected firms to check for violations of regulations. a. If 15 of the firms are actually violating at least one regulation, what is the pmf of the number of firms visited by the inspector that are in violation of at least one regulation? b. If there are 500 firms in the area, of which 150 are in violation, approximate the pmf of part (a) by a simpler pmf. c. For X = the number among the 10 visited that are in violation, compute E(X) and V(X) both for the exact pmf and the approximating pmf in part (b).The probability that a randomly selected box of a certain type of cereal has a particular prize is .2. Suppose you purchase box after box until you have obtained two of these prizes. a. What is the probability that you purchase x boxes that do not have the desired prize? b. What is the probability that you purchase four boxes? c. What is the probability that you purchase at most four boxes? d. How many boxes without the desired prize do you expect to purchase? How many boxes do you expect to purchase?A family decides to have children until it has three children of the same gender. Assuming P(B) = P(G) = .5, what is the pmf of X = the number of children in the family?Three brothers and their wives decide to have children until each family has two female children. What is the pmf of X = the total number of male children born to the brothers? What is E(X), and how does it compare to the expected number of male children born to each brother?According to the article Characterizing the Severity and Risk of Drought in the Poudre River, Colorado (J. of Water Res. Planning and Algmnt., 2005: 383-393), the drought length Y is the number of consecutive time intervals in which the water supply remains below a critical value y0 (a deficit), preceded by and followed by periods in which the supply exceeds this critical value (a surplus). The cited paper proposes a geometric distribution with p = .409 for this random variable. a. What is the probability that a drought lasts exactly 3 intervals? At most 3 intervals? b. What is the probability that the length of a drought exceeds its mean value by at least one standard deviation?The article Expectation Analysis of the Probability of Failure for Water Supply Pipes (J. of Pipeline Systems Engr. and Practice, May 2012: 36-46) proposed using the Poisson distribution to model the number of failures in pipelines of various types. Suppose that for cast-iron pipe of a particular length, the expected number of failures is 1 (very close to one of the cases considered in the article). Then X, the number of failures, has a Poisson distribution with = 1. a. Obtain P(X 5) by using Appendix Table A.2. b. Determine P(X = 2) first from the pmf formula and then from Appendix Table A.2. c. Determine P(2 X 4). d. What is the probability that X exceeds its mean value by more than one standard deviation?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 distribution for X. Suppose that = 4. a. Compute both P(X 4) and P(X 4). b. Compute P(4 X 8). c. Compute P(8 X). d. What is the probability that the number of anomalies exceeds its mean value by no more than one standard deviation?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 = 20 (suggested in the article Dynamic Ride Sharing: Theory and Practice, J. of Transp. Engr., 1997: 308-312). What is the probability that the number of drivers will a. Be at most 10? b. Exceed 20? c. Be between 10 and 20, inclusive? Be strictly between 10 and 20? d. Be within 2 standard deviations of the mean value?Consider writing onto a computer disk and then sending it through a certifier that counts the number of missing pulses. Suppose this number X has a Poisson distribution with parameter = .2. (Suggested in Average Sample Number for Semi-Curtailed Sampling Using the Poisson Distribution, J. Quality Technology, 1983: 126-129.) a. What is the probability that a disk has exactly one missing pulse? b. What is the probability that a disk has at least two missing pulses? c. If two disks are independently selected, what is the probability that neither contains a missing pulse?An article in the Los Angeles Times (Dec. 3. 1993) reports that 1 in 200 people carry the defective gene that causes inherited colon cancer. In a sample of 1000 individuals. what is the approximate distribution of the number who carry this gene? Use this distribution to calculate the approximate probability that a. Between 5 and 8 (inclusive) carry the gene. b. At least 8 carry the gene.The Centers for Disease Control and Prevention reported in 2012 that 1 in 88 American children had been diagnosed with an autism spectrum disorder (ASD). a. If a random sample of 200 American children is selected, what are the expected value and standard deviation of the number who have been diagnosed with ASD? b. Referring back to (a), calculate the approximate probability that at least 2 children in the sample have been diagnosed with ASD? c. If the sample size is 352, what is the approximate probability that fewer than 5 of the selected children have been diagnosed with ASD?Suppose small aircraft arrive at a certain airport according to a Poisson process with rate = 8 per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter = 8t. a. What is the probability that exactly 6 small aircraft arrive during a 1-hour period? At least 6? At least 10? b. What are the expected value and standard deviation of the number of small aircraft that arrive during a 90-min period? c. What is the probability that at least 20 small air-craft arrive during a 2.5-hour period? That at most 10 arrive during this period?Organisms are present in ballast water discharged from a ship according to a Poisson process with a concentration of 10 organisms/m3 [the article Counting at Low Concentrations: The Statistical Challenges of Verifying Ballast Water Discharge Standards (Ecological Applications, 2013: 339-351) considers using the Poisson process for this purpose]. a. What is the probability that one cubic meter of discharge contains at least 8 organisms? b. What is the probability that the number of organisms in 1.5 m3 of discharge exceeds its mean value by more than one standard deviation? c. For what amount of discharge would the probability of containing at least 1 organism be .999?The number of requests for assistance received by a towing service is a Poisson process with rate = 4 per hour. a. Compute the probability that exactly ten requests are received during a particular 2-hour period. b. If the operators of the towing service take a 30-min break for lunch, what is the probability that they do not miss any calls for assistance? c. How many calls would you expect during their break?In proof testing of circuit boards, the probability that any particular diode will fail is .01. Suppose a circuit board contains 200 diodes. a. How many diodes would you expect to fail, and what is the standard deviation of the number that are expected to fail? b. What is the (approximate) probability that at least four diodes will fail on a randomly selected board? c. If five boards are shipped to a particular customer, how likely is it that at least four of them will work properly? (A board works properly only if all its diodes work.)The article Reliability-Based Service-Life Assessment of Aging Concrete Structures" (J. Structural Engr., 1993: 1600-1621) suggests that a Poisson process can be used to represent the occurrence of structural loads over time. Suppose the mean time between occurrences of loads is .5 year. a. How many loads can be expected to occur during a 2-ycar period? b. What is the probability that more than five loads occur during a 2-year period? c. How long must a time period be so that the probability of no loads occurring during that period is at most .1?Let X have a Poisson distribution with parameter . Show that E(X) = directly from the definition of expected value. [Hint: The first term in the sum equals 0, and then x can be canceled. Now factor out and show that what is left sums to 1.]Suppose that trees are distributed in a forest according to a two-dimensional Poisson process with parameter , the expected number of trees per acre, equal to 80. a. What is the probability that in a certain quarter-acre plot, there will be at most 16 trees? b. If the forest covers 85.000 acres, what is the expected number of trees in the forest? c. Suppose you select a point in the forest and construct a circle of radius .1 mile. Let X = the number of trees within that circular region. What is the pmf of X? [Hint: 1 sq mile = 640 acres.)Automobiles arrive at a vehicle equipment inspection station according to a Poisson process with rate = 10 per hour. Suppose that with probability .5 an arriving vehicle will have no equipment violations. a. What is the probability that exactly ten arrive during the hour and all ten have no violations? b. For any fixed y 10, what is the probability that y arrive during the hour, of which ten have no violations? c. What is the probability that ten no-violation cars arrive during the next hour? [Hint: Sum the probabilities in part (b) from y = 10 to .]a. In a Poisson process, what has to happen in both the time interval (0, t) and the interval (t, t + t) so that no events occur in the entire interval (0, t + t)? Use this and Assumptions 1-3 to write a relationship between P0(t + t) and P0(t). b. Use the result of part (a) to write an expression for the difference P0(t + t) P0(t). Then divide by t and let t 0 to obtain an equation involving (d/dt)P0(t), the derivative of P0(t) with respect to t. c. Verify that P0(t) = et satisfies the equation of part (b). d. It can be shown in a manner similar to parts (a) and (b) that the Pk(t)s must satisfy the system of differential equations ddtPk(t)=Pk1(t)-Pk(t)k=1,2,3,... Verify that Pk(t) = et(t)k/k! satisfies the system. (This is actually the only solution.)Consider a deck consisting of seven cards, marked 1, 2,..., 7. Three of these cards are selected at random. Define an rv W by W = the sum of the resulting numbers, and compute the pmf of W. Then compute and 2. [Hint: Consider outcomes as unordered, so that (1, 3, 7) and (3, 1, 7) are not different outcomes. Then there are 35 outcomes. and they can be listed. (This type of rv actually arises in connection with a statistical procedure called Wilcoxons rank-sum test, in which there is an x sample and a y sample and W is the sum of the ranks of the x's in the combined sample; see Section 15.2.)After shuffling a deck of 52 cards, a dealer deals out 5. Let X = the number of suits represented in the five-card hand. a. Show that the pmf of X is x 1 2 3 4 p(x) .002 .146 .588 .264 [Hint: p(1) = 4P(all are spades), p(2) = 6P(only spades and hearts with at least one of each suit), and p(4) = 4P(2 spades one of each other suit).] b. Compute , 2. and .The negative binomial rv X was defined as the number of Fs preceding the rth S. Let Y = the number of trials necessary to obtain the rth S. In the same manner in which the pmf of X was derived, derive the pmf of Y.Of all customers purchasing automatic garage-door openers, 75% purchase a chain-driven model. Let X = the number among the next 15 purchasers who select the chain-driven model. a. What is the pmf of X? b. Compute P(X 10). c. Compute P(6 X 10). d. Compute and 2. e. If the store currently has in stock 10 chain-driven models and 8 shaft-driven models, what is the probability that the requests of these 15 customers can all be met from existing stock?In some applications the distribution of a discrete rv X resembles the Poisson distribution except that zero is not a possible value of X. For example, let X = the number of tattoos that an individual wants removed when she or he arrives at a tattoo-removal facility. Suppose the pmf of X is p(x)=kexxx=1,2,3,... a. Determine the value of k. Hint: The sum of all probabilities in the Poisson pmf is 1, and this pmf must also sum to 1. b. If the mean value of X is 2.313035, what is the probability that an individual wants at most 5 tattoos removed? c. Determine the standard deviation of X when the mean value is as given in (b). [Note: The article An Exploratory Investigation of Identity Negotiation and Tattoo Removal (Academy of Marketing Science Review, vol. 12, no. 6, 2008) gave a sample of 22 observations on the number of tattoos people wanted removed; estimates of and calculated from the data were 2.318182 and 1.249242, respectively.]A k-out-of-n system is one that will function if and only if at least k of the n individual components in the system function. If individual components function independently of one another, each with probability .9, what is the probability that a 3-out-of-5 system functions?A manufacturer of integrated circuit chips wishes to control the quality of its product by rejecting any batch in which the proportion of defective chips is too high. To this end. out of each batch (10,000 chips), 25 will be selected and tested. If at least 5 of these 25 are defective, the entire batch will be rejected. a. What is the probability that a batch will be rejected if 5% of the chips in the batch are in fact defective? b. Answer the question posed in (a) if the percentage of defective chips in the batch is 10%. c. Answer the question posed in (a) if the percentage of defective chips in the batch is 20%. d. What happens to the probabilities in (a)-(c) if the critical rejection number is increased from 5 to 6?Of the people passing through an airport metal detector, .5% activate it; let X = the number among a randomly selected group of 500 who activate the detector. a. What is the (approximate) pmf of X? b. Compute P(X = 5). c. Compute P(5 X).An educational consulting firm is trying to decide whether high school students who have never before used a hand-held calculator can solve a certain type of problem more easily with a calculator that uses reverse Polish logic or one that does not use this logic. A sample of 25 students is selected and allowed to practice on both calculators. Then each student is asked to work one problem on the reverse Polish calculator and a similar problem on the other. Let p = P(S), where S indicates that a student worked the problem more quickly using reverse Polish logic than without, and let X = number of S's. a. If p = .5, what is P(7 X 18)? b. If p = .8. what is P(7 X 18)? c. If the claim that p = .5 is to be rejected when either x 7 or x 18, what is the probability of rejecting the claim when it is actually correct? d. If the decision to reject the claim p = .5 is made as in part (c), what is the probability that the claim is not rejected when p = .6? When p = .8? e. What decision rule would you choose for rejecting the claim p = .5 if you wanted the probability in part (c) to be at most .01?Consider a disease whose presence can be identified by carrying out a blood test. Let p denote the probability that a randomly selected individual has the disease. Suppose n individuals are independently selected for testing. One way to proceed is to carry out a separate test on each of the n blood samples. A potentially more economical approach, group testing, was introduced during World War II to identify syphilitic men among army inductees. First, take a part of each blood sample, combine these specimens, and carry out a single test. If no one has the disease, the result will be negative, and only the one test is required. If at least one individual is diseased, the test on the combined sample will yield a positive result, in which case the n individual tests are then carried out. If p = .1 and n = 3, what is the expected number of tests using this procedure? What is the expected number when n = 5? [The article Random Multiple-Access Communication and Group Testing (IEEE Trans, on Commun., 1984: 769-774 ) applied these ideas to a communication system in which the dichotomy was active/ idle user rather than diseased/nondiseased.]Let p1 denote the probability that any particular code symbol is erroneously transmitted through a communication system. Assume that on different symbols, errors occur independently of one another. Suppose also that with probability p2 an erroneous symbol is corrected upon receipt. Let X denote the number of correct symbols in a message block consisting of n symbols (after the correction process has ended). What is the probability distribution of X?