AP Statstistics Exploration: The German Tank Problem
Overview
The German tank problem for AP Statistics.
Intro to the German Tank Problem
- Requirements of the Activity
• Level of students: Introductory statistics, probability, or mathematical statistics students
• Classroom size: Works well with 25-30 students; students work in small groups of sizes 3 or 4
• Time to do activity in class: 60 minutes
Teaching materials: Paper sheets with numbers 1 through N printed on them, Brown lunch bags for each group for holding the cut out slips of paper 1 through N(312) You could just have one bag and have each group come up and draw out 5 , record and then put back. Shake and have next group come up.
Today’s German Tank Problem activity is based on this real-world problem, You will play the role of the Mathematicians in WWII.
First, Read pages 448 and 449 in TPS 6e and then page 452 starting at CHOOSING AN ESTIMATOR to the bottom ofpage 453. This way you are familiar with what an unbiased estimator is.
The German Tank Problem: Capturing the Tanks
Learning Goals of the German Tank Problem Activity
Bring up the topic of estimation before starting statistical inference
What is a parameter? What is an estimator, or a statistic?
What is a good estimator? What qualities does a good estimator have?
Biased versus unbiased estimators
Minimum variance estimators
Simulation is a powerful tool for studying distributions and their properties
Instructions for Students
0. Form Allied Statistician Units of size 3 or 4
1. Your unit will obtain (through non-violent military action) a bag filled with the serial numbers of the entire fleet of tanks. Please do not look at the numbers in the bag.
Randomly draw five slips of paper out of the bag without replacement. DO NOT LOOK IN THE BAG. Record your sample:
Sample:
________,__________,_________,__________,_________
Have someone from your unit write your sample results on the board for your military unit.
The German Tank Problem: The Mathematicians at work. Decide on a formula for an unbiased estimator
Give students about 15 mintues to work through some options. I do not let students use the example options.
2. Discuss in your group how you could you use the data above (and only this data) to estimate the total number of “tanks” (slips of paper) in the bag. Allow yourself to think “outside the box.”
Here are some ideas (not necessarily correct or incorrect) to get you started:
(a). Use the largest of the five numbers in your sample.
(b). Add the smallest and largest numbers of your sample.
(c). Double the mean of the five numbers obtained in your sample.
3. Come up with an estimator for determining the total number of “tanks” (slips of paper) N in the bag. That is, develop a rule or formula to plug the 5 serial numbers into for estimating N.
Write down your military unit’s formula for estimating N:
The German Tank Problem: Check out your formula on other samples
4. Plug in your sample of 5 serial numbers from #1 to get an estimate of N using the formula your unit constructed.
5. Apply your rule to each of the samples drawn by the other groups (on the board) to come up with estimates for N. Construct a dot plot of these estimates below.
<----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o----o---->
Estimates for N using each group’s sample values
The German Tank Problem: How did you do?
N=312 for
8. In your group, decide on what you think the true value of N is. Record it.
9. I will give you the correct value of N after the majority of the units are done. It is:
N =
Did you make a “good” estimate in #8? Why or why not? Did you have a good estimation formula?
Is any unit’s dotplot or histogram centered about the value N = ______ approximately? In other words, do any of the estimators (formulas) appear to be unbiased?
The German Tank Problem: Conclusion
10. The records of the Speer Ministry, which was in charge of Germany's war production, were recovered after the war. The table below gives the actual tank production for three different months, the estimate by statisticians from serial number analysis, and the number obtained by traditional American/British “intelligence” gathering.
Alternatives to German Tanks
In 2008 a Londoner started asking for people to post the serial number of their phone and the date they bought it. From the posted information and using estimation formulas, he was able to calculate that Apple had sold 9.1 million iPhones by the end of September 2008