- Flying the not-so-friendly skies
Bernoulli Airways flies 2 round-trip shuttle flights per
day on the ever-popular Harford-Buffalo route. One flight
is in a 2-engine propeller plane; the other in a 4-engine
prop plane. Suppose that each engine on each plane will
fail independently with probability p,
and that either plane
arrives safely at its destination only if at least one-half
of its engines are in working order.
Assuming you actually want to survive the flight, in which
plane would you rather travel? To answer this question,
follow the steps below:
- Using the
BINOMDIST function in Excel,
calculate the probability that the 2-engine plane
will arrive safely at its destination for each
value of p between 0 and 1, in 0.01 increments.
(That is, compute the probability of survival for
p = 0.00, p = 0.01, p = 0.02, up to
p = 0.98, p = 0.99, p = 1.00.)
- Calculate similar values for the 4-engine plane,
in the same manner.
- Plot the two results of parts (a) and (b) above
as two line graphs. The horizontal (X) axis should
have the probability p of an engine failing, and
the vertical (Y) axis should have the probability of
a safe flight.
Hints: If the engines never fail (p = 0),
then the plane arrives safely with probability 1. On
the other hand, if the engines fail with certainty
(p = 1), then probabilty of a safe flight must be
zero! This information should help you check that your
calculations and graph are correct.
- Based on this graph, over what ranges of p (if any)
would you prefer to fly in the 2-engine plane instead
of the 4-engine plane?
- Provide an intuitive explanation for your answer to
part (d) above.
- Judging the Olympic Judges
Presumably, all of you have heard about "Skategate": the judging scandal in
pairs figurestaking at the 2002 Winter Olympic Games in Salt Lake City.
Many commentators thought a Canadian couple skated better than a Russian
couple, but by a 5-to-4 margin the Russians won the "Free Program" competition
and thus the gold medal. However, after the judge from France admitted to
being influenced to vote for the Russians, the Canadians were awarded a gold
medal as well. (The French judge has since recanted on her confession.)
In this assignment, you will see if you can ascertain whether any of the judges'
ranking patterns in the competition in question are themselves questionable. (As
it turns out, the ranking of at least one other judge has been questioned in this
case, and the International Skating Association has decided to completely revamp
their procedures for evaluating figure skating.)
First, download the Excel spreadsheet judging.xls,
which contains the individual and overall rankings by skating pair and judge.
(Click on the link to download. You may need to option-click (Mac) or right-click
(Windows) to save the speadsheet to a file. If you have trouble with the file,
chances are your browser preferences are not set correctly. See an ITS lab tech
for assistance.) Then answer the following questions:
- For each couple, S1 through S20, compute the mean absolute deviations
(MAD) from that couple's overall ranking in the free program.
Then resort the couples by this measure. Interpret this new ranking.
What does it tell you about the degree of agreement among the judges?
- For each couple, repeat all the above for the maan squared deviations
(MSD) from that couple's overall ranking in the free program.
Explain why the resorted rankings by MAD and MSD differ, and interpret.
(Hint: does this tell you something about the quality of the
judging or the quality of the skaters -- or both?)
- Now for each judge, J1 through J9, compute that judge's MAD and MSD from
the overall ranking. Sort the judges by these measures; do they differ?
Interpret the sorted list of judges: does this indicate which judges
were "questionable" in their ranking of the skaters? Why or why not?
- Repeat the above MAD and MSD calculation by judge, except only include
the top 7 couples in this calculation. (Those were the only couples
believed to be in serious medal contention.) How do these statistics
differ from the ones you just computed above? Interpret the differences.
For the purposes of identifying "questionable" or "tainted" judges, which
measure -- one based on all 20 couples or one based only on the top 7 --
is more informative? Explain.
- Now, compute the correlation of rankings between judges. That is, find
the correlation of J1 with J2, J3, ..., J9; then J2 with J3, ..., J9;
and so on. (Use all 20 couples in the calculation.) Are the rankings by
the judges with low MAD/MSD highly correlated? What about those judges
with high MAD/MSD? Interpret in the context of identifying the "questionable"
judges.
- Based on your above analysis, which judge (J1 - J9) is "most questionable"
in his or her rankings? Which judge is "least questionable"? Explain
your reasons behind your answers.
- Submitting this assignment for credit:
-
In order to receive credit for this assignment,
you must e-mail to me your Excel spreadsheet as
an attachment.
Please name the Excel file as your e-mail name
-- e.g.
mshanson.xls -- and include
your name in a cell of the worksheet as well.
I cannot give credit if I cannot find a worksheet
named after you.
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