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## Take a Number and Wait in Line

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.

Source: J. Goebel & D. Teague, "How Many Taxis?" Consortium, pp. 11-14

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?

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.