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Subj:.....Number Of Taxis (S665)
          From: MathNexus.wwu.edu
          on 10/5/2009
Drawing from MathNexus...

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. 
Drawing from tom on 8/21/2009
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? 

Finger pointing down
from darrell94590 on 1/2/2006
Drawing from Ripleys-Believe It Or Not

              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.

Drawing from tom on 8/21/2009
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.