91影视

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/\({\bf{1}}{{\bf{0}}^{\bf{4}}}\)) to reach a given crack size in fastener holes intended for use in military aircraft (鈥淪tatistical Crack Propagation in Fastener Holes Under Spectrum Loading,鈥� J. Aircraft,1983: 1028鈥�1032):

.736 .863 .865 .913 .915 .937 .983 1.007

1.011 1.064 1.109 1.132 1.140 1.153 1.253 1.394

1. Compute and compare the values of the sample mean and median.
2. By how much could the largest sample observation be decreased without affecting the value of the median?

Consider the population consisting of all computers of acertain brand and model, and focus on whether a computerneeds 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 answeredby determining the number of such computers in a sample of size 100 that need warranty service?

Consider the following sample of observations on coating thickness for low-viscosity paint ("Achieving a Target Value for a Manufacturing Process: \({\rm{A}}\)Case Study, \({\rm{97}}\)J. of Quality Technology, \({\rm{1992:22 - 26}}\)):

\(\begin{array}{*{20}{r}}{{\rm{.83}}}&{{\rm{.88}}}&{{\rm{.88}}}&{{\rm{1}}{\rm{.04}}}&{{\rm{1}}{\rm{.09}}}&{{\rm{1}}{\rm{.12}}}&{{\rm{1}}{\rm{.29}}}&{{\rm{1}}{\rm{.31}}}\\{{\rm{1}}{\rm{.48}}}&{{\rm{1}}{\rm{.49}}}&{{\rm{1}}{\rm{.59}}}&{{\rm{1}}{\rm{.62}}}&{{\rm{1}}{\rm{.65}}}&{{\rm{1}}{\rm{.71}}}&{{\rm{1}}{\rm{.76}}}&{{\rm{1}}{\rm{.83}}}\end{array}\)

Assume that the distribution of counting thickness is normal (a normal probability plot strongly supports this assumption).

a. Calculate a point estimate of the mean value of coating thickness, and state which estimator you used.

b. Calculate a point estimate of the median of the coating thickness distribution, and state which estimator you used.

C. Calculate a point estimate of the value that separates the largest of\({\rm{10\% }}\) all values in the thickness distribution from the remaining \({\rm{90\% }}\)and state which estimator you used. (Hint Express what you are trying to estimate in terms of \({\rm{\mu }}\)and \({\rm{\sigma }}\)

d. Estimate \({\rm{P}}\left( {{\rm{X < 1}}{\rm{.5}}} \right){\rm{,}}\)i.e., the proportion of all thickness values less than 1.5. (Hint: If you knew the values of \({\rm{\mu }}\)and \({\rm{\sigma }}\), you could calculate this probability. These values are not available, but they can be estimated.)

e. What is the estimated standard error of the estimator that you used in part (b)?

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.

The article 鈥淢onte Carlo Simulation鈥擳ool for Better Understanding of LRFD鈥� (J. of Structural Engr., \({\rm{1993: 1586 - 1599}}\)) suggests that yield strength (\({\rm{ksi}}\)) for A36 grade steel is normally distributed with \({\rm{\mu = 43}}\) and \({\rm{\sigma = 4}}{\rm{.5 }}\)

a. What is the probability that yield strength is at most \({\rm{40}}\)? Greater than \({\rm{60}}\)?

b. What yield strength value separates the strongest \({\rm{75\% }}\) from the others?

A sample of n=10 automobiles was selected, and eachwas subjected to a 5-mph crash test. Denoting a car withno visible damage by S (for success) and a car with suchdamage by F, results were as follows:

S S F S SS F F S S

1. What is the value of the sample proportion of successes x/n?
2. Replace each S with a 1 and each F with a 0. Then calculate for this numerically coded sample. How does compare to x/n?
3. Suppose it is decided to include 15 more cars in the experiment. How many of these would have to be S鈥檚 to give x/n = .80 for the entire sample of 25 cars?

a. If a constant cis added to each \({x_i}\)in a sample, yielding \({y_i} = {x_i} + c\), how do the sample mean and median of the \({y_i}'s\)relate to the mean and median of the\({x_i}'s\)? Verify your conjectures.

b. If each \({x_i}\)is multiplied by a constant c,yielding \({y_i} = c{x_i}\), answer the question of part (a). Again, verify your conjectures.

An experiment to study the lifetime (in hours) for acertain type of component involved putting ten components into operation and observing them for 100hours. Eight of the components failed during that

period, and those lifetimes were recorded. Denote the lifetimes of the two components still functioning after100 hours by 100+. The resulting sample observations were

48 79 100+ 35 92 86 57 100+ 17 29

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 鈥渃ensored on the right.鈥�)

Poly(3-hydroxybutyrate) (PHB), a semicrystallinepolymer that is fully biodegradable and biocompatible,is obtained from renewable resources. From a sustainabilityperspective, PHB offers many attractive propertiesthough it is more expensive to produce than standardplastics. The accompanying data on melting point(掳C) for each of 12 specimens of the polymer using adifferential scanning calorimeter appeared in the article鈥淭he Melting Behaviour of Poly(3-Hydroxybutyrate)by DSC. Reproducibility Study鈥� (Polymer Testing,2013: 215鈥�220).

180.5 181.7 180.9 181.6 182.6 181.6

181.3 182.1 182.1 180.3 181.7 180.5

Compute the following:

1. The sample range
2. The sample variance \({{\bf{s}}^{\bf{2}}}\)from the definition (Hint:First subtract 180 from each observation.
3. The sample standard deviation
4. \({{\bf{s}}^{\bf{2}}}\)using the shortcut method

The value of Young鈥檚 modulus (GPa) was determined forcast plates consisting of certain intermetallic substrates,resulting in the following sample observations (鈥淪trengthand Modulus of a Molybdenum-Coated Ti-25Al-10Nb-3U-1Mo Intermetallic,鈥� J. of Materials Engr.and Performance, 1997: 46鈥�50):

116.4 115.9 114.6 115.2 115.8

1. Calculate\({\bf{\bar x}}\) and the deviations from the mean.
2. Use the deviations calculated in part (a) to obtain the sample variance and the sample standard deviation.
3. Calculate\({{\bf{s}}^{\bf{2}}}\)by using the computational formula for the numerator \({S_{xx}}\).
4. 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\({{\bf{s}}^{\bf{2}}}\)for the original data.