A leading bicycle company has engineered a revolutionary lightweight carbon fiber frame for its new racing bicycle, but is concerned that the main tube (called the ‘down tube’) might bend too much under strong pedaling forces. Too much, according to the bicycle company’s specification, is more than a 10-degree bend when applying 600 newtons of force—that’s more than twice the force that most racing cyclists can produce. Carbon fiber is laid by hand, so some variation is expected. The bicycle company’s engineers predict that the new frame design can be manufactured to meet the bending specification only 85% of the time. However, the company’s favored manufacturer has historically made carbon fiber frames that meet the bending specification 95% of the time. The bicycle company asked this manufacturer to supply 12 frames with the new lightweight design in a range of sizes. Let the random variable X be the number of frames in the sample of 12 that meet the bending specification. a. Assume that the engineers are right about the probability of manufacturing to specification, what is the expected number of frames that meet the specification? b. Assuming again that the engineers are right about the probability of manufacturing to specification, what is the probability that all 12 of the frames meet the specification? c. Assume now that the favored manufacturer’s historical probability of making to specification is right, what is the probability that all 12 of the frames meet the specification? d. Assuming again that the manufacturer’s historical probability of making to specification is right, what is the probability that more than 10 frames meet the specification? e. If all 12 of the frames supplied by the manufacturer are found to meet the bending specification, what if anything can we say about whether the engineers’ predicted probability or the manufacturer’s historical probability is more likely to be right for the new frame design?
A leading bicycle company has engineered a revolutionary lightweight
carbon fiber frame for its new racing bicycle, but is concerned that the main tube (called the
‘down tube’) might bend too much under strong pedaling forces. Too much, according to the
bicycle company’s specification, is more than a 10-degree bend when applying 600 newtons of
force—that’s more than twice the force that most racing cyclists can produce. Carbon fiber is
laid by hand, so some variation is expected. The bicycle company’s engineers predict that the
new frame design can be manufactured to meet the bending specification only 85% of the time.
However, the company’s favored manufacturer has historically made carbon fiber frames that
meet the bending specification 95% of the time. The bicycle company asked this manufacturer
to supply 12 frames with the new lightweight design in a range of sizes. Let the random
variable X be the number of frames in the sample of 12 that meet the bending specification.
a. Assume that the engineers are right about the
specification, what is the expected number of frames that meet the specification?
b. Assuming again that the engineers are right about the probability of manufacturing to
specification, what is the probability that all 12 of the frames meet the specification?
c. Assume now that the favored manufacturer’s historical probability of making to
specification is right, what is the probability that all 12 of the frames meet the
specification?
d. Assuming again that the manufacturer’s historical probability of making to
specification is right, what is the probability that more than 10 frames meet the
specification?
e. If all 12 of the frames supplied by the manufacturer are found to meet the bending
specification, what if anything can we say about whether the engineers’ predicted
probability or the manufacturer’s historical probability is more likely to be right for the
new frame design?
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