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applied statistics and probability for engineers

Questions and Answers of Applied Statistics And Probability For Engineers

=+8. A target is located at the point 0 on a horizontal axis.Let x be the landing point of a shot aimed at the target, a continuous variable with density function f(x) 5 .75(1 2 x 2) for 21 # x # 1.
=+9. Let x denote the amount of time for which a book on 2-hour reserve at a college library is checked out by a student, and suppose that x has density function f(x) 5 .5x for 0 , x , 2.
=+a. What is the mean value of x? Why is the mean value not 1, the midpoint of the interval of positive density?
=+b. What is the median of this distribution, and how does it compare to the mean value?
=+c. What proportion of checkout times are within one-half hour of the mean time? What proportion are within one-half hour of the median time?
=+10. Let x have a uniform distribution on the interval from a tob, so the density function of x is f(x) 5 1y(b 2 a)for a # x #b. What is the mean value of x?
=+11. The weekly demand for propane gas (1000s of gallons) at a certain facility is a continuous variable with density function f(x) 5 c 2a1 2 1 x2 b 1 # x # 2 0 otherwise
=+ Determine both the mean value and the median.In the long run, in what proportion of weeks will the value of x be between the mean value and the median?
=+12. Refer to Exercise 27 of Section 1.3, in which x was the number of telephone lines in use at a specified time. If 5 2.64, what are the values of p(5)and p(6)?
=+13. The distribution of the number of underinflated tires x on an automobile is given in Exercise 26a(ii)of Section 1.3. Determine the mean value of x.
=+14. Sometimes, rather than wishing to determine the mean value of x, an investigator wishes to determine the mean value of some function of x. Suppose, for example, that a repairman assesses a
=+a. Refer to Exercise 9. Suppose the library, in a desperate search for revenue to fund its operations, charges a student h(x) 5 x2 dollars to check a book out on 2-hour reserve for x hours. What
=+b. Suppose that h(x) 5 a 1 bx, a linear function of x. Show that h(x) 5 a 1 b (this is true for x continuous or discrete). If the mean value of repair time is .5 hr for the repair situation
=+15. In the article “Mechanical Reliability of Devices Subdermally Implanted into the Young of Long-Lived and Endangered Wildlife” (J. of Materials Engr. and Performance, 2012: 1924–1931),
=+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. Compute the sample standard deviation using a calculator or software function to confirm the accuracy of your answer in (b).
=+16. Return to the puncture test data given in Exercise 15.
=+a. 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 s
=+2 for the original data.
=+b. Consider a sample x1, . . . , xn and let yi 5 xi 2 c for i 5 1, 2, . . . , n, where c is some specified number.Give a general argument to show that the sample variance of the yi’s is identical
=+17. Suppose the following represent quiz scores (out of 15 points) for students in two different study groups:Group 1: 10, 14, 8, 7, 12, 7, 11 Group 2: 5, 8, 9.5, 8.5, 9, 9.5, 13
=+a. Compute the mean and standard deviation for each group.
=+b. Determine the range for each data set.
=+c. Create a dotplot for each data set and ensure you use the same axis scale for each.
=+d. Notice that one group exhibits the smaller standard deviation but the other exhibits the smaller range. Explain how it is possible for a data set to have the smallest standard deviation yet not
=+18. Traumatic knee dislocation often requires surgery to repair ruptured ligaments. One measure of recovery is range of motion (measured as the angle formed when, starting with the leg straight,
=+range of motion appeared in the article “Reconstruction of the Anterior and Posterior Cruciate Ligaments After Knee Dislocation” (Amer. J. Sports Med., 1999: 189–197):154 142 137 133 122 126
=+a. What are the values of the sample mean and sample median?
=+b. An alternative computing formula for the numerator of s 2is:Sxx 5 ^(xi 2 x)2 5 ^xi 2 2 1 n(^xi)2
=+ Using this formula, determine the sample variance of the data.Hint: ^xi 5 1695, ^xi 2 5 222,581.
=+19. In the article “X-Ray Computed Tomography and Nondestructive Evaluation of Clogging in Porous Concrete Field Samples” (J. of Materials in Civil Engr., 2012: 1103–1109), investigators
=+Calculate and interpret the values of the sample mean and sample standard deviation for this data.
=+20. Use the alternative computing formula for Sxx as shown in Exercise 18 to determine the sample standard deviation for the average porosity measurements presented in Exercise 19.
=+21. Consider the following information on ultimate tensile strength (lb/in.) for a sample of n 5 4 hard zirconium copper wire specimens (from “Characterization Methods for Fine Copper Wire,”
=+ Determine the values of the two middle sample observations (and don’t do it by successive guessing!).Hint: See Exercise 18 part b.
=+22. The federal test procedure (FTP) for determining the levels of various types of vehicle emissions is time-consuming and expensive to perform. According to the article “Motor Vehicle
=+would yield identical (or nearly identical) results.The accompanying data is from one particular vehicle characterized as a high emitter:HC (gm/mi): 13.8 18.3 32.2 32.5 CO (gm/mi): 118 149 232 236
=+a. Compute the sample standard deviations for the HC and CO observations. Does the widespread belief appear to be justified?
=+b. The sample coefficient of variation syx (or 100syx)assesses the extent of variability relative to the mean. Values of this coefficient for several different data sets can be compared to
=+23. Suppose, as in Exercise 57 of Chapter 1, that the number of drivers traveling between a particular origin and destination during a designated time period has a Poisson distribution with 5
=+a. Within 5 of the mean value?
=+b. Within 1 standard deviation of the mean value?
=+24. Suppose that x, the number of flaws on the surface of a boiler of a certain type, has a Poisson distribution with 5 5. For what proportion of such boilers will the number of flaws
=+a. Be within 1 standard deviation of the mean number of flaws?
=+b. Exceed the mean number of flaws by more than 2 standard deviations?
=+25. Let x represent the number of underinflated tires on an automobile of a certain type, and suppose that p(0) 5 .4, p(1) 5 p(2) 5 p(3) 5 .1, and p(4) 5 .3, from which 5 1.8.
=+a. Calculate the standard deviation of x.
=+b. For what proportion of such cars will the number of underinflated tires be within 1 standard deviation of the mean value? More than 3 standard deviations from the mean value?
=+26. Use the fact that (x 2 )2 5 x 2 2 2x 1 2 to show that 2 5 ^x 2p(x) 2 2
=+ for a discrete variable x.Then use this result to compute the variance for the variable whose distribution is given in the previous problem. Hint: Substitute the alternative expression for (x 2
=+ in the definition of 2, and break the summation into three separate terms; the argument in the continuous case involves replacing summation with integration.
=+27. If x has a uniform distribution on the interval from a to b [ f(x) 5 1y(b 2 a)], from which 5 (a 1 b)y2, show that 2 5 (b 2 a)2
=+y12. If task completion time is uniformly distributed with a 5 4 and b 5 6, what proportion of times will be farther than 1 standard deviation from the mean value of completion time?
=+28. Suppose that bearing diameter x has a normal distribution. What proportion of bearings have diameters that are within 1.5 standard deviations of the mean diameter? That exceed the mean
=+29. Historical data implies that 20% of all components of a certain type need service while under warranty.Suppose that whether any particular component needs warranty service is independent of
=+any other component does. If these components are shipped in batches of 25 and x denotes the number of components in a batch that need warranty service, determine the standard deviation of x and
=+30. If the unloading time of a forwarder in a harvesting operation is lognormally distributed with a mean value of 900 and a standard deviation of 725, what are the values of the parameters and
=+31. If component lifetime is exponentially distributed with parameter , obtain an expression for the proportion of components whose lifetime exceeds the mean value by more than 1 standard
=+32. The sample mean and sample standard deviation for the sample of n 5 100 shear strength observations given in Exercise 17 of Section 1.2 are 5049.16 and 351.45, respectively. What percentage
=+2 standard deviations and for 3 standard deviations.
=+33. Reconsider the accompanying data on postsurgical range of motion introduced in Exercise 18 of this chapter:154 142 137 133 122 126 135 135 108 120 127 134 122
=+a. What are the values of the quartiles? What is the value of the IQR?
=+b. Construct a boxplot based on the five-number summary and comment on its features.
=+c. How large or small does an observation have to be to qualify as an outlier? As an extreme outlier?
=+d. By how much could the largest observation be decreased without affecting the IQR?
=+34. Here is a description from the R software of the strength data given in Exercise 4 from Chapter 1.Min. 1st Qu. Median 122.2 133.0 135.4 Mean 3rd Qu. Max.135.4 138.2 147.7
=+a. Comment on any interesting features.b. Construct a boxplot of the data and comment on what you see.
=+35. The diameter length of contact windows used in integrated circuits is normally distributed. About 5%of all lengths exceed 3.75 m, and about 1% of all lengths exceed 3.85 m. What are the mean
=+36. 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
=+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 be decreased without affecting the value of the IQR?
=+37. Grip is applied to produce normal surface forces that compress the object being gripped. Examples include two people shaking hands and a nurse squeezing a patient’s forearm to stop bleeding.
=+a sample of 42 individuals:16 18 18 26 33 41 54 56 66 68 87 91 95 98 106 109 111 118 127 127 135 145 147 149 151 168 172 183 189 190 200 210 220 229 230 233 238 244 259 294 329 403 Construct a
=+38. 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 5
=+a. Are there any outliers in the sample? Any extreme outliers?
=+b. Construct a boxplot that shows outliers, and comment on any interesting features.
=+39. A company utilizes two different machines to manufacture parts of a certain type. During a single shift, a sample of n 5 20 parts produced by each machine is obtained, and the value of a
=+40. Recall from Exercise 2 the data on the concentration (EU/mg) in settled dust for one sample of urban homes and another 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:
=+a. Determine the medians, quartiles, and IQRs 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 two samples.
=+41. The authors of the article cited in Exercise 2 also provided endotoxin concentrations in dust from vacuum-cleaner dust bags: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
=+ Construct a comparative boxplot (which appeared in the cited paper), and compare and contrast the two samples.
=+42. The comparative boxplot (see below) of gasoline vapor coefficients for vehicles in Detroit appeared in the article “Receptor Modeling Approach to VOC Emission Inventory Validation” (J. of
=+43. Exercise 46 from Section 1.5 suggested a Weibull distribution with 5 5 and 5 125 as a model for fracture strength of silicon nitride braze joints.
=+a. What are the quartiles of this distribution, and what is the value of the IQR?
=+b. Suppose that the value of is changed to
=+12.5. Determine the values of the quartiles and the value of the IQR. Note: In essence, this amounts to dividing each observation in the population distribution by 10, because is a “scale”
=+44. Reconsider the lognormal distribution with 5 9.164 and 5.385 proposed in Exercise 44 from Section 1.5 as a model for the distribution of nonpoint source load of total dissolved solids (in
=+a. What are the values of the quartiles?
=+b. What is the value of the 95th percentile of the concentration distribution?
=+c. If were 10.164 rather than 9.164, would the values of the two quartiles simply increase by an identical amount?6 A.M. 8A.M. 12 noon 2 P.M. 10 P.M.10 020 30 40 50 60 70 Time
=+45. The accompanying normal quantile plot was constructed from a sample of 30 readings on tension for mesh screens behind the surface of video display tubes used in computer monitors. Does it
=+46. The following are modulus of elasticity observations for cylinders given in the article cited in Example 1.2:37.0 37.5 38.1 40.0 40.2 40.8 41.0 42.0 43.1 43.9 44.1 44.6 45.0 46.1 47.0 62.0
=+47. A sample of 15 female collegiate golfers was selected, and the clubhead velocity (km/hr) of each golfer while swinging a driver was determined, resulting in the following data (“Hip
=+Construct a normal quantile plot and a dotplot. Is it plausible that the population distribution is normal?
=+48. The accompanying observations are precipitation values during March over a 30-year period in Minneapolis–St. Paul..77 1.20 3.00 1.62 2.81 2.48 1.74 .47 3.09 1.31 1.87 .96.81 1.43 1.51 .32
=+a. Construct and interpret a normal quantile plot for this data set.

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