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Chapter 4: Q95E (page 193)
Use a statistical software package to construct a normal probability plot of the tensile ultimate-strength data given in Exercise of Chapter and comment.
The normal probability plot is quite linear. Hence, it is very plausible that strength is normally distributed.
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Suppose only \({\rm{75\% }}\) of all drivers in a certain state regularly wear a seat belt. A random sample of 500 drivers is selected. What is the probability that
a. Between 360 and 400 (inclusive) of the drivers in the sample regularly wear a seat belt?
b. Fewer than 400 of those in the sample regularly wear a seat belt?
The article 鈥淎 Model of Pedestrians鈥� Waiting Times for Street Crossings at Signalized Intersections鈥� (Transportation Research, \({\rm{2013: 17--28}}\)) suggested that under some circumstances the distribution of waiting time X could be modelled with the following pdf:
\({\rm{f(x;\theta ,\tau ) = }}\left\{ {\begin{array}{*{20}{c}}{\frac{{\rm{\theta }}}{{\rm{\tau }}}{{{\rm{(1 - x/\tau )}}}^{{\rm{\theta - 1}}}}}&{{\rm{0}} \le {\rm{x < \tau }}}\\{\rm{0}}&{{\rm{ otherwise }}}\end{array}} \right.\)
a. Graph \({\rm{f(x;\theta ,80)}}\) for the three cases \({\rm{\theta = 4,1}}\) and \({\rm{.5}}\) (these graphs appear in the cited article) and comment on their shapes. b. Obtain the cumulative distribution function of X. c. Obtain an expression for the median of the waiting time distribution. d. For the case \({\rm{\theta = 4,\tau = 80}}\) calculate \({\rm{P(50}} \le {\rm{X}} \le {\rm{70)}}\) without at this point doing any additional integration.
The article "'Response of \({\rm{Si}}{{\rm{C}}_{\rm{i}}}{\rm{/S}}{{\rm{i}}_{\rm{3}}}{\rm{\;}}{{\rm{N}}_{\rm{4}}}\) Composites Under Static and Cyclic Loading-An Experimental and Statistical Analysis" (J. of Engr. Materials and Technology, \({\rm{1997: 186 - 193}}\)) suggests that tensile strength (MPa) of composites under specified conditions can be modeled by a Weibull distribution with \({\rm{\alpha = 9}}\) and\({\rm{\beta = 180}}\).
a. Sketch a graph of the density function.
b. What is the probability that the strength of a randomly selected specimen will exceed \({\rm{175}}\)? Will be between \({\rm{150}}\) and \({\rm{175}}\)?
c. If two randomly selected specimens are chosen and their strengths are independent of one another, what is the probability that at least one has a strength between \({\rm{150}}\) and\({\rm{175}}\)?
d. What strength value separates the weakest \({\rm{10\% }}\) of all specimens from the remaining\({\rm{90\% }}\)?
Let Z have a standard normal distribution and define a new rv Y by \[{\text{Y = \sigma Z + \mu }}\]. Show that Y has a normal distribution with parameters \[{\text{\mu }}\] and \[{\text{\sigma }}\]. (Hint: \[{\text{Y\poundsy}}\]if \[{\text{Z\pounds}}\]? Use this to find the cdf of Y and then differentiate it with respect to y.)
Mopeds (small motorcycles with an engine capacity below\({\rm{50\;c}}{{\rm{m}}^{\rm{3}}}\)) are very popular in Europe because of their mobility, ease of operation, and low cost. The article "Procedure to Verify the Maximum Speed of Automatic Transmission Mopeds in Periodic Motor Vehicle Inspections" (J. of Automobile Engr., \({\rm{2008: 1615 - 1623}}\)) described a rolling bench test for determining maximum vehicle speed. A normal distribution with mean value \({\rm{46}}{\rm{.8\;km/h}}\) and standard deviation \({\rm{1}}{\rm{.75\;km/h}}\)is postulated. Consider randomly selecting a single such moped.
a. What is the probability that maximum speed is at most\({\rm{50\;km/h}}\)?
b. What is the probability that maximum speed is at least\({\rm{48\;km/h}}\)?
c. What is the probability that maximum speed differs from the mean value by at most \({\rm{1}}{\rm{.5}}\)standard deviations?
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