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applied statistics and probability for engineers

Probability And Statistics For Engineering And The Sciences 7th Edition Dave Ellis, Jay L Devore - Solutions

=+59. Let x denote the number of trees in a quarter-acre plot within a certain forest. Suppose that x has a Poisson distribution with 5 20 (corresponding to an average density of 80 trees per acre). In what proportion of such plots will there be at least 15 trees?At most 25 trees?
=+60. An article in the Los Angeles Times (Dec. 3, 1993)reports that 1 in 200 people carry the defective gene that causes colon cancer. Let x denote the number of people in a group of size 1000 who carry this defective gene. What is the approximate
=+distribution of x? Use this approximate distribution to determine the proportion of all such groups having at least 8 people who carry the defective gene, as well as the proportion of all such groups for which between 5 and 10 people (inclusive)carry the defective gene.
=+61. 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., August 1997: 59–63).812 832 852 872 892 Class: ,83 ,85 ,87 ,89 ,91
=+Frequency: 6 7 17 30 43 912 932 952 972 Class:,93 ,95 ,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 are at least 85? Less than 95?
=+c. Roughly what proportion of the observations are less than 90?
=+62. The article cited in Exercise 61 presented compelling evidence for assuming that fracture strength(MPa) of ceramic bars fired in a particular kiln is normally distributed (while commenting that the Weibull distribution is traditionally used as a model). Suppose that 5 90 and 5 3.75,
=+a. In the long run, what proportion of bars would have strength values less than 90? Less than 95? At least 95?
=+b. In the long run, what proportion of bars would have strength values between 85 and 95? Between 80 and 100?
=+c. What value is exceeded by 90% of the fracture strengths for all such bars?
=+d. What interval centered at 90 includes 99% of all fracture strength values?
=+63. Once an individual has been infected with a certain disease, let x represent the time (days) that elapses before the individual becomes infectious.The article “The Probability of Containment for Multitype Branching Process Models for Emerging Epidemics” (J. of Applied Probability,
=+a. What proportion of elapsed times exceed 1.5 days?
=+b. What is the 90th percentile of the elapsed time distribution?
=+64. Let x denote the distance (m) that an animal moves from its birth site to the first territorial vacancy it encounters. Suppose that for banner-tailed kangaroo rats, x has an exponential distribution with parameter 5 .01386 (as suggested in the article “Competition and Dispersal from
=+a. What proportion of distances are at most 100 m?At most 200 m? Between 100 m and 200 m?
=+b. What proportion of distances are at least 50 m?
=+c. What is the median distance, that is, the value that separates the smallest 50% of all distances from the largest 50%?
=+65. Suppose the unloading time x (centiminutes) of a forwarder in a harvesting operation could be assumed to be lognormal with 5 6.5 and 5 .75, as suggested in the article “Simulating a HarvesterForwarder Softwood Thinning” (Forest Products J., May 1997: 36–41).
=+a. What proportion of unloading times exceed 1000?2000? 3000?
=+b. What proportion of times are between 2500 and 5000?
=+c. What value characterizes the fastest 10% of all times?
=+d. Sketch a graph of the density function of x. Is the positive skewness quite pronounced?
=+66. In an experiment, 25 laminated glass units configured in a particular way are subjected to an impact test (cf.“Performance of Laminated Glass Units Under Simulated Windborne Debris Impacts,” J. of Architectural Engr., 1996: 95–99). We are interested in the number of units that sustain
=+67. Airlines frequently overbook flights. Suppose that for a plane with 100 seats, an airline takes 110 reservations. Let x represent the number of people with reservations who actually show up for a sold-out flight.From past experience, we know that the distribution of x is as follows:x: 95 96
=+a. For what proportion of such flights is the airline able to accommodate everyone who shows up for the flight?
=+b. For what proportion of all such flights is it not possible to accommodate all passengers?
=+c. For someone who is trying to get a seat on such a flight and is number 1 on the standby list, what proportion of the time is such an individual able to take the flight? Answer the question for individuals who are number 3 on the standby list.
=+68. The accompanying data are observations on shower flow rate for a sample of 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 3.8 11.2
=+ 6.3a. Construct a stem-and-leaf display of the data.
=+b. What is a typical or representative flow value?
=+Does the data appear to be highly concentrated or quite spread out about this typical value?
=+c. Does the distribution of values appear to be reasonably symmetric? If not, how would you describe the departure from symmetry?
=+d. Does the data set appear to contain any outliers?
=+e. Construct a histogram using class boundaries 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, and 20. From your histogram, approximately what proportion of the observations are at most 11? Compare this with the exact proportion that are at most 11.
=+69. Let x denote the vibratory stress (psi) on a wind turbine blade at a particular wind speed in a wind tunnel. The article “Blade Fatigue Life Assessment with Applications to VAWTS” (J. of Solar Energy Engr., 1982: 107–111) proposes the Rayleigh distribution as a model; the density
=+a. Verify that f(x) is a legitimate density function.
=+b. Suppose that 5 100 (a value suggested by a graph in the cited article). What proportion of vibratory stress values will be at most 200? At least 200? Between 100 and 200?
=+70. The article “Error Distribution in Navigation” (J. Institute of Navigation, 1971: 429–442) suggests thatthe frequency distribution of positive errors (magnitudes of errors) is well approximated by an exponential distribution. Let x denote the lateral position error (nautical miles),
=+a. Sketch the corresponding density curve, and verify that f(x) is a legitimate density function.
=+b. What proportion of errors are negative? At most 2?Between 21 and 2?
=+71. “Time headway” in traffic flow is the elapsed time between the time that one car finishes passing a fixed point and the instant that the next car begins to pass that point. Let x be the time headway (sec) for two consecutive cars on a freeway during a period of heavy flow. The following
=+one suggested in “The Statistical Properties of Freeway Traffic” (Transportation Research, 1977: 221–228):f(x) 5 e.15e 2.15(x2.5)0 x . .5 otherwise
=+a. Sketch the corresponding density curve, and verify that f (x) is a legitimate density function.
=+b. What proportion of time headways are at most 5 sec? Between 5 and 10 sec?
=+c. What value separates the smallest 50% of all time headways from the largest 50%?
=+d. What value characterizes the largest 10% of all time headways?
=+72. A k-out-of-n system is one that will function if and only if at least k out of the n individual components in the system function. If individual components function independently of one another and the long-run proportion of components that function is .9, what is the long-run proportion of
=+73. An insurance company offers its policyholders a number of different premium payment options. Let x denote the number of months between successive payments chosen by a policyholder. For any particular number k, the proportion of x values that are at most k(i.e., # k) is called a cumulative
=+a. Graph this cumulative proportion function, that is, graph (proportion of x values # k) versus k.
=+b. Determine the mass function of x. Hint: The cumulative proportion function jumps only at possible values of x.
=+c. Use the cumulative proportion function to determine the proportion of all policyholders for which 3 # x # 6, and check to see that the mass function gives this same proportion.
=+74. Based on data from a dart-throwing experiment, the article “Shooting Darts” (Chance, Summer 1997,
=+16–19) proposed that the horizontal and vertical errors from aiming at a point target should be independent of one another, each with a normal distribution having parameters 5 0 and . It can then be shown that the density function of the distance from the target to the landing point is
=+a. This pdf is a member of what family introduced in this chapter?
=+b. If 5 20 mm (close to the value suggested in the paper), what proportion of darts will land within 25 mm (roughly 1 in.) of the target?
=+75. The bursting strength of wine bottles of a certain type is normally distributed with parameters 5 250 psi and 5 30 psi. If these bottles are shipped 12 to a carton, in what proportion of cartons will at least one of the bottles have a bursting strength exceeding 300 psi? Hint: Think of a
Let be a number between 0 and 1 and define a sequence W1, W2, W3, . . . by W0 and Wt Xt (1 )Wt1 for t 1, 2, . . . . Substituting for Wt1 its representation in terms of Xt1 and Wt2, then substituting for Wt2, and so on, results in Wt Xt (1 )Xt1(1 )t1 X1 (1 )t The fact
Resistance observations (ohms) for subgroups of a certain type of register gave the following summary quantities:i ni x i si i ni x i si 1 4 430.0 22.5 7 4 420.8 25.4 2 4 418.2 20.6 8 4 431.4 24.0 3 3 435.5 25.1 9 4 428.7 21.2 44427.6 22.3 10 4 440.1 25.8 5 4 444.0 21.5 11 4 445.2 27.3 6 3 431.4
Refer to Example 16.11, in which a single-sample plan with n 50 and c 2 was employed.a. Calculate AOQ for p .01, .02, . . . , .10. What does this suggest about the value of p for which AOQ is a maximum and the corresponding AOQL?b. Determine the value of p for which AOQ is a maximum and the
Refer to Exercise 32 and consider the plan with n 100 and c 2. Calculate P(A) for p .01, .02, . . . , .05, and sketch the two OC curves on the same set of axes. Which of the two plans is preferable (leaving aside the cost of sampling) and why?
When the out-of-control ARL corresponds to a shift of 1 standard deviation in the process mean, what are the characteristics of the CUSUM procedure that has ARLs of 250 and 4.8 for the in-control and out-of-control conditions, respectively?
Construct a control chart for the data of Exercise 23 by using the transformation suggested in the text.
In some situations, the sizes of sampled specimens vary, and larger specimens are expected to have more defects than smaller ones. For example, sizes of fabric samples inspected for flaws might vary over time. Alternatively, the number of items inspected might change with time.Let ui g xi
For what x values will the LCL in a c chart be negative?
The accompanying observations are numbers of defects in 25 1-square-yard specimens of woven fabric of a certain type: 3, 7, 5, 3, 4, 2, 8, 4, 3, 3, 6, 7, 2, 3, 2, 4, 7, 3, 2, 4, 4, 1, 5, 4, 6. Construct a c chart for the number of defects.
Refer to the data of Exercise 20, and construct a control chart using the sin1 transformation as suggested in the text.
When S2 is the sample variance of a normal random sample,(n 1)S2/2 has a chi-squared distribution with n 1 df, so P2.999,n1 (n 2 1)S2 2.001,n1 .998 from which P S2 .998 This suggests that an alternative chart for controlling process variation involves plotting the sample
Calculate control limits for an S chart from the refractive index data of Exercise 9. Does the process appear to be in control with respect to variability? Why or why not?
Calculate control limits for both an S chart and an R chart using the moisture-content data from Exercise 6. Then check for the presence of any out-of-control signals.
Calculate control limits for the data of Exercise 6, using the robust procedure presented in this section.
Apply the supplemental rules suggested in the text to the data of Exercise 6. Are there any out-of-control signals?
Refer to Exercise 9. An assignable cause was found for the unusually high sample average refractive index on day 22.Recompute control limits after deleting the data from this day.What do you conclude?
The accompanying table gives sample means and standard deviations, each based on n 6 observations of the refractive index of fiber-optic cable. Construct a control chart, and comment on its appearance. [Hint: xi 2317.07 and si 30.34.]Day x s Day x s1 95.47 1.30 9 96.63 1.48 2 97.38 .88 10
Refer to Exercises 6 and 7, and now employ control limits based on using the sample ranges to estimate . Does the process appear to be in control?
Refer to the data given in Exercise 6, and construct a control chart with an estimated center line and limits based on using the sample standard deviations to estimate . Is there any evidence that the process is out of control?
The table on page 636 gives data on moisture content for specimens of a certain type of fabric. Determine control limits for a chart with center line at height 13.00 based on .600, construct the control chart, and comment on its appearance.
Refer to Exercise 1 and suppose the ten most recent values of the quality statistic are .0493, .0485, .0490, .0503, .0492,.0486, .0495, .0494, .0493, and .0488. Construct the relevant portion of the corresponding control chart, and comment on its appearance.
A control chart for thickness of rolled-steel sheets is based on an upper control limit of .0520 in. and a lower limit of .0475 in. The first ten values of the quality statistic (in this case X, the sample mean thickness of n 5 sample sheets) are .0506,.0493, .0502, .0501, .0512, .0498, .0485,
The ranking procedure described in Exercise 35 is somewhat asymmetric, because the smallest observation receives rank 1 whereas the largest receives rank 2, and so on.Suppose both the smallest and the largest receive rank 1, the second smallest and second largest receive rank 2, and so on, and let
Suppose we wish to test H0: the X and Y distributions are identical versus Ha: the X distribution is less spread out than the Y distribution The accompanying figure pictures X and Y distributions for which Ha is true. The Wilcoxon rank-sum test is not appropriate in this situation because when Ha
Refer to Exercise 33, and consider a confidence interval associated with the sign test, the sign interval. The relevant hypotheses are now H0:~ ~0 versus Ha:~~0. Let’s use the following rejection region: either Y 15 or Y 5.a. What is the significance level for this test?b. The confidence
The sign test is a very simple procedure for testing hypotheses about a population median assuming only that the underlying distribution is continuous. To illustrate, consider the following sample of 20 observations on component lifetime(hr):We wish to test H0:~ 25.0 versus Ha:~ 25.0. The test
The study reported in “Gait Patterns During Free Choice Ladder Ascents” (Human Movement Sci., 1983: 187–195)was motivated by publicity concerning the increased accident rate for individuals climbing ladders. A number of different gait patterns were used by subjects climbing a portable
Refer to the data of Exercise 30 and compute a 95% CI for the difference between true average concentrations for treatments II and III.
The article “Effects of a Rice-Rich Versus Potato-Rich Diet on Glucose, Lipoprotein, and Cholesterol Metabolism in Noninsulin-Dependent Diabetics” (Amer. J. Clinical Nutr., 1984: 598–606) gives the accompanying data on cholesterolsynthesis rate for eight diabetic subjects. Subjects were fed a
In an experiment to study the way in which different anesthetics affect plasma epinephrine concentration, ten dogs were selected and concentration was measured while they were under the influence of the anesthetics isoflurane, halothane, and cyclopropane (“Sympathoadrenal and Hemodynamic Effects
The accompanying data on cortisol level was reported in the article “Cortisol, Cortisone, and 11-Deoxycortisol Levels in Human Umbilical and Maternal Plasma in Relation to the Onset of Labor” (J. Obstetric Gynaecology British Commonwealth, 1974: 737–745). Experimental subjects were pregnant
The article “Production of Gaseous Nitrogen in Human Steady-State Conditions” (J. Applied Physiology, 1972:155–159) reports the following observations on the amount of nitrogen expired (in liters) under four dietary regimens:(1) fasting, (2) 23% protein, (3) 32% protein, and (4) 67%protein.
Compute a 99% CI for 1 2 using the data in Exercise 11.
Compute the 90% rank-sum CI for 1 2 using the data in Exercise 10.
Compute a CI for D of Example 15.2 using the data given there; your confidence level should be roughly 95%.
Compute the 99% signed-rank interval for true average pH(assuming symmetry) using the data in Exercise 3. [Hint:Try to compute only those pairwise averages having relatively small or large values (rather than all 105 averages).]
The article “The Lead Content and Acidity of Christchurch Precipitation” (New Zealand J. Science, 1980: 311–312)reports the accompanying data on lead concentration ( g/L)in samples gathered during eight different summer rainfalls:17.0, 21.4, 30.6, 5.0, 12.2, 11.8, 17.3, and 18.8. Assuming
Reconsider the situation described in Exercise 79 of Chapter 9 and the accompanying MINITAB output (the Greek letter eta is used to denote a median).Mann-Whitney Confidence Interval and Test good N 8 Median 0.540 poor N 8 Median 2.400 Point estimate for ETA1-ETA2 is 1.155 95.9 Percent CI
The article “Measuring the Exposure of Infants to Tobacco Smoke” (N. Engl. J. Med., 1984: 1075–1078) reports on a study in which various measurements were taken both from a random sample of infants who had been exposed to household smoke and from a sample of unexposed infants.The accompanying
Test the hypotheses suggested in Exercise 13 using the following data:
The accompanying data resulted from an experiment to compare the effects of vitamin C in orange juice and in synthetic ascorbic acid on the length of odontoblasts in guinea pigs over a 6-week period (“The Growth of the Odontoblasts of the Incisor Tooth as a Criterion of the Vitamin C Intake of

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