
Source:
http://mathnexus.wwu.edu/Archive/problem/detail.asp?ID=174
You are standing in
the rain trying to hail a taxi cab in
a large city.
While waiting, seven taxi cabs pass by that
already have passengers.
The numbers on the taxi cabs are
405, 73, 280, 179,
440, 301, and 218.
Suppose you want to
estimate the number of taxi cabs in the
city while you are
waiting. Assuming that the taxi cabs are
numbered consecutively
from 1 to N and all are still in
service, how can
you use the observed numbers to estimate N,
the total number
of taxi cabs in the city?
How many taxis do
you think there are? How can you test
your method for estimating
N?
Note:
In World War II, the Allies supposedly were able to
estimate the size
of the fleet of German tanks by analyzing
the serial numbers
on the tanks either captured or disabled
in battle.
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Drawing
from tom on 8/21/2009 
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Hint:
To get started, assume that the total number N of taxi
cabs is known (e.g.
500), and then randomly pick seven numbers.
Create different
analysis techniques, test them on the seven
numbers and determine
the strength of your creations. Then,
pick another seven
numbers, and test again, etc.... What do
you learn?
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Finger pointing down
from darrell94590 on 1/2/2006 
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..
THE SOLUTION
Solution Commentary:
The author's of this problem (Goebel
& Teague) offer
the following advice: Student solutions will
vary according to
their mathematical background. Students in
a statistics course
will more likely use techniques from that
course, but our experience
has been that the more creative
solutions often come
from those students who don't have a
strong statistical
background. Students who know a particular
technique often use
that technique without thinking further.
If you don't know
the technique, you have to think more deeply
about the problem
and often come up with a "better" solution
as a result.
In the authors' article,
they share and test eight different
student methods,
with one of the best being perhaps the simplest.
Called the "(n+1)/n
Max" method, the students basically viewed
the seven given numbers
as dividing the number line from 1 to N
into 8 regions.
As the largest number, 440 is considered to be
7/8 of the distance
from a number line representing 1 to N. Thus,
by solving the simple
equation (7/8)N = 440, the predicted number
of taxis is 503!
Again, the authors note that this procedure
combines "a small
average error with the smallest standard
deviation... (being
one of the) minimal variance unbiased
estimators of N."
Try to find a copy
of the article to explore (and enjoy) the
other student methods.
Or, please share your students' methods
with me and I will
try to post them on this web site.
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Drawing
from tom on 8/21/2009 
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My simple method:
Find the average of the seven numbers
and double it.
Adding the seven numbers, I get a total of
1896. Dividing
by seven produced an average of 270.86.
Doubling the average
was 541.7. So my answer was 542 taxis,
which is too high. 