91影视

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 (鈥淔actors Affecting Runners鈥� Marathon Performance,鈥� Chance, Fall, 1993: 24鈥�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?

The weekly demand for propane gas (in \({\rm{1000s}}\) of gallons) from a particular facility is an \({\rm{rv}}\) \({\rm{X}}\) with pdf

\({\rm{f(x) = }}\left\{ {\begin{array}{*{20}{c}}{{\rm{2}}\left( {{\rm{1 - }}\frac{{\rm{1}}}{{{{\rm{x}}^{\rm{2}}}}}} \right)}&{{\rm{1}} \le {\rm{x}} \le {\rm{2}}}\\{\rm{0}}&{{\rm{ otherwise }}}\end{array}} \right.\)

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, 鈥�. [The histogram will agree with the one in 鈥淐omparison of Properties of Joints Prepared by Ultrasonic Welding and Other Means鈥� (J. of Aircraft, 1983: 552鈥�556).] Comment on its features.

Automated electron backscattered diffraction is now being used in the study of fracture phenomena. The following information on misorientation angle (degrees) was extracted from the article 鈥淥bservations on the Faceted Initiation Site in the Dwell-Fatigue Tested Ti-6242 Alloy: Crystallographic Orientation and Size Effects鈥� (Metallurgical and Materials Trans., 2006: 1507鈥�1518).

Class: 0-<5 5-<10 10-<15 15-<20

Relfreq: .177 .166 .175 .136

Class: 20-<30 30-<40 40-<60 60-<90

Relfreq: .194 .078 .044 .030

a. Is it true that more than 50% of the sampled angles are smaller than 15掳, as asserted in the paper?

b. What proportion of the sampled angles are at least 30掳?

Let \({\rm{X}}\) be the total medical expenses (in \({\rm{1000}}\) s of dollars) incurred by a particular individual during a given year. Although \({\rm{X}}\) is a discrete random variable, suppose its distribution is quite well approximated by a continuous distribution with pdf \({\rm{f(x) = k(1 + x/2}}{\rm{.5}}{{\rm{)}}^{{\rm{ - 7}}}}\) for.

a. What is the value of\({\rm{k}}\)?

b. Graph the pdf of \({\rm{X}}\).

c. What are the expected value and standard deviation of total medical expenses?

d. This individual is covered by an insurance plan that entails a \({\rm{\$ 500}}\) deductible provision (so the first \({\rm{\$ 500}}\) worth of expenses are paid by the individual). Then the plan will pay \({\rm{80\% }}\) of any additional expenses exceeding \({\rm{\$ 500}}\), and the maximum payment by the individual (including the deductible amount) is\({\rm{\$ 2500}}\). Let \({\rm{Y}}\) denote the amount of this individual's medical expenses paid by the insurance company. What is the expected value of\({\rm{Y}}\)?

(Hint: First figure out what value of \({\rm{X}}\) corresponds to the maximum out-of-pocket expense of \({\rm{\$ 2500}}\). Then write an expression for \({\rm{Y}}\) as a function of \({\rm{X}}\) (which involves several different pieces) and calculate the expected value of this function.)

The article 鈥淪tudy on the Life Distribution of Microdrills鈥� (J. of Engr. Manufacture, 2002: 301鈥� 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.

11 14 20 23 31 36 39 44 47 50

59 61 65 67 68 71 74 76 78 79

81 84 85 89 91 93 96 99 101 104

105 105 112 118 123 136 139 141 148 158

161 168 184 206 248 263 289 322 388 513

a. Why can a frequency distribution not be based on the class intervals 0鈥�50, 50鈥�100, 100鈥�150, and so on?

b. Construct a frequency distribution and histogram of the data using class boundaries 0, 50, 100, 鈥� , 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 depositedenergy (mJ) was extracted from the article 鈥淓xperimentalAnalysis of Laser-Induced Spark Ignition of LeanTurbulent Premixed Flames鈥� (Combustion and Flame,2013: 1414鈥�1427).

1.0鈭�< 2.0 5 2.0鈭�<2.4 11

2.4鈭�< 2.6 13 2.6鈭�<2.8 30

2.8鈭�< 3.0 46 3.0鈭�< 3.2 66

3.2鈭�<3.4 133 3.4鈭�< 3.6 141

3.6鈭�< 3.8 126 3.8鈭�< 4.0 92

4.0鈭�< 4.2 73 4.2鈭�< 4.4 38

4.4鈭�< 4.6 19 4.6鈭�< 5.0 11

a. What proportion of these ignition trials resulted in adeposited energy of less than 3 mJ?

b. What proportion of these ignition trials resulted in adeposited energy of at least 4 mJ?

c. Roughly what proportion of the trials resulted in adeposited energy of at least 3.5 mJ?

d. Construct a histogram and comment on its shape.

Let \({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}.....{\rm{,}}{{\rm{X}}_{\rm{n}}}\) represent a random sample from the Rayleigh distribution with density function given in Exercise \({\rm{15}}\). Determine a. The maximum likelihood estimator of \({\rm{\theta }}\), and then calculate the estimate for the vibratory stress data given in that exercise. Is this estimator the same as the unbiased estimator suggested in Exercise \({\rm{15}}\)? b. The mle of the median of the vibratory stress distribution. (Hint: First express the median in terms of \({\rm{\theta }}\).)

The following categories for type of physical activityinvolved when an industrial accident occurred appearedin the article 鈥淔inding Occupational Accident Patternsin the Extractive Industry Using a Systematic DataMining Approach鈥� (Reliability Engr. and System

Safety, 2012: 108鈥�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 B D AA F C A C B E B A C

F D B C D AA C B E B C E A

B A AA B C C D F D B B A F

C B A C B E E D A B C E A A

F C B D DD B D C A F AA B

D E A E D B C A F A C D D A

A B A F D C A C B F D A E A

C D

For 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 laboratory.