91影视

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 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.

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

The National Health and Nutrition Examination Survey (NHANES) collects demographic, socioeconomic, dietary, and health related information on an annual basis. Here is a sample of \({\rm{20}}\) observations on HDL cholesterol level \({\rm{(mg/dl)}}\) obtained from the \({\rm{2009 - 2010}}\) survey (HDL is 鈥済ood鈥� cholesterol; the higher its value, the lower the risk for heart disease):

\(\begin{array}{l}{\rm{35 49 52 54 65 51 51}}\\{\rm{47 86 36 46 33 39 45}}\\{\rm{39 63 95 35 30 48}}\end{array}\)

a. Calculate a point estimate of the population mean HDL cholesterol level.

b. Making no assumptions about the shape of the population distribution, calculate a point estimate of the value that separates the largest \({\rm{50\% }}\) of HDL levels from the smallest \({\rm{50\% }}\).

c. Calculate a point estimate of the population standard deviation.

d. An HDL level of at least \({\rm{60}}\) is considered desirable as it corresponds to a significantly lower risk of heart disease. Making no assumptions about the shape of the population distribution, estimate the proportion \({\rm{p}}\) of the population having an HDL level of at least \({\rm{60}}\).

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.

A Pareto diagram is a variation of a histogram forcategorical data resulting from a quality control study.Each category represents a different type of product non-conformity or production problem. The categories areordered so that the one with the largest frequencyappears on the far left, then the category with the secondlargest frequency, and so on. Suppose the following information on nonconformities in circuit packs isobtained: failed component, 126; incorrect component,210; insufficient solder, 67; excess solder, 54; missingcomponent, 131. Construct a Pareto diagram.

The cumulative frequency and cumulative relativefrequency for a particular class interval are the sum offrequencies and relativefrequencies, respectively, forthat interval and all intervals lying below it. If, forexample, 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 frequenciesare .18, .50, .76, and 1.00. Compute the cumulativefrequencies and cumulative relative frequencies for thedata of Exercise 24.

Fire load (MJ/m2) is the heat energy that could bereleased per square meter of floor area by combustionof contents and the structure itself. The article 鈥淔ireLoads in Office Buildings鈥� (J. of Structural Engr.,

1997: 365鈥�368) gave the following cumulative percentages(read from a graph) for fire loads in a sample of388 rooms:

Value0 150 300 450 600

Cumulative %0 19.3 37.6 62.7 77.5

Value750 900 1050 1200 1350

Cumulative %87.2 93.8 95.7 98.6 99.1

Value1500 1650 1800 1950

Cumulative %99.5 99.6 99.8 100.0

a. Construct a relative frequency histogram and commenton interesting features.

b. What proportion of fire loads are less than 600? At least 1200?

c. What proportion of the loads are between 600 and1200?

The May 1, 2009, issue of the Mont clarian reported the following home sale amounts for a sample of homes in Alameda, CA that were sold the previous month (1000s of $):

590 815 575 608 350 1285 408 540 555 679

1. Calculate and interpret the sample mean and median.
2. Suppose the 6th observation had been 985 rather than 1285. How would the mean and median change?
3. Calculate a 20% trimmed mean by first trimming the two smallest and two largest observations.
4. Calculate a 15% trimmed mean.

Exposure to microbial products, especially endotoxin, may have an impact on vulnerability to allergic diseases. The article 鈥淒ust Sampling Methods for Endotoxin鈥擜n Essential, But Underestimated Issue鈥� (Indoor Air,2006: 20鈥�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.

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

1. Determine the sample mean for each sample. How do they compare?
2. 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?
3. 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?