Teach 180: The German Tank Problem (Day 40)

I spent about two hours on Friday finalizing what I needed to do to prep for a talk that I was giving on Saturday.  The talk was given at the Association of Math Teachers of Philadelphia and Vicinity (ATMoPAV) fall conference.  The title of my talk was "The German Tank Problem: Simulating a Statistic".  For those of you who teach AP Statistics, this is an activity you likely do to introduce sampling distributions.  When I first did this activity, it bombed, because I wasn't comfortable with using Fathom to create a simulation of a sampling distribution.

Then in 2009, I took an online course to learn how to use Fathom.  I still wasn't comfortable with using it entirely and spent about a year screencasting demos to show Fathom in class.  Then in 2010, I started to use Fathom with my students.  It is a great visualization tool and it is great for helping students understand the difference between the distribution of a population, the distribution of a sample and the distribution of a sampling statistic. (Note: Fathom has a free trial version and a 1-year subscription is $5.95.)

For the German Tank problem, students are given a bag with N "tanks" numbered in their bag consecutively from 1 to N.  They randomly select 7 tanks and use the numbers on the tanks to create a statistic to estimate the total number of tanks. The eight participants at my workshop on Saturday worked in pairs and spent a good fifteen minutes to create their statistics.  At the right you can see the common statistics students often create.  The Partition Method is the one that was used by the mathematicians in WW II to estimate the number of tanks.  At the workshop on Saturday, participants created several of these common statistics.

I had been doing this problem for several years and last year I had a pair of students create a statistic that was good as (or perhaps better than) the Partition statistic.  I named the statistic after the two students, Cecily Redfern and Neelam Ferrari.  It's called the Redfern-Ferrari statistic.  Two hundred trials of the simulation can be seen on the screenshot at the left.  Note that 344 is the true number of tanks in the bag and that both Partition and Redfern-Ferrari are centered near 344 with similar variability.

What actually is the Redfern-Ferrari statistic?  The formula for the statistic is Max. + Range/6.  The students found the average difference between two consecutive numbers by calculating Range/6. Then, they added it to the maximum number they drew to get the approximation for the value of N.

If you have tried this activity and abandoned it, I would encourage you try it again. If you are interested in the Fathom files to do this activity and/or the handout I give to my students to do the activity, send an email to me at mathteacher@ptd.net or leighnataro@gmail.com.