Introduction: During World War II, the Germans tried to make a complete tabulation of how much they were producing, but reports from individual factories were often late and not always reliable.  British and U.S. statisticians (also known as the "Allied Statisticians") were keenly interested in estimating German war production, too, but they could hardly ask the German factories to send them reports.  Instead, they based their estimates on the manufacturing serial numbers of captured equipment (specifically the tire molds and tank gearboxes).  These serial numbers provided a sample that was very small, but reliable.  The records of the Speer Ministry, which was in charge of Germany's war production, were recovered after the war.  Special studies made after the war discovered that the British and U.S. statisticians' estimates of German production were more accurate and timely than Germany's own estimates.

1.  DO NOT LOOK IN THE CUP.  Draw three numbers out of the cup.  Record your sample.  Return the chips, mix well, and repeat.

Sample 1: ________,__________,_________ Sample 2: ________,__________,_________
Sample 3: ________,__________,_________ Sample 4: ________,__________,_________

2.  Discuss in your group how could you use the data above (and only this data) to estimate the number of chips.

Our estimate of N is : __________
Come up with an estimator (rule or formula), and write it here.
 
 
 
 
 

3.  Apply your rule to each of the samples drawn by the other groups (minimum of 20) to come up with estimates for N.  Construct a dot plot of these estimates on the sheet provided.

4. Calculate the mean and standard deviation of the estimates you produced in #3.

                                     mean = ___________      standard deviation = _________
    Write the mean and standard deviation on the sheet provided next to the dot plot.  Label your paper with your group number, but not your names.  Take your paper to the front of the room and tape it to the white board in the designated area.

5. Study the dot plots, means, and standard deviations from all the estimators produced by the class.  Vote for the one which appears to be the best estimator.

6.  The instructor will now give you the correct value of N.
                             N = __________
     Did you make a good estimate in #2?  Why or why not?

     Obtain from the instructor the explanation of the method used by Allied statisticians in finding the estimate for number of tanks produced.

7.  Write the estimator used by Allied statisticians here:
 

8. Use this estimator with your four original samples to make four estimates of the number of chips in the cup.

                                              ______, _______, _______, _______

How close are your estimates to the actual number?  (record the differences).
 

9.  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.

Month	Actual # of Tanks Produced	Allied Statisticians Estimate	Estimate by Intelligence Agencies
June 1940	122	169	1000
June 1941	271	244	1550
Sept 1942	342	327	1550

Interpret the results of the table in a paragraph (how good or bad were the estimates?).
 

10. What is the difference between an estimator and an estimate?
 

11.  During World War II, Allied statisticians also conducted a serial number study of tires on several German
Mark V tanks to determine the production of one tire manufacturer.  Each tire was stamped with the number of
the mold in which it was made; captured tires had 20 different mold numbers from this manufacturer.  The
largest mold number captured was 77.
a. What is the best estimate of the total number of molds?

b. What additional piece of information do you need to estimate this manufacturer's daily tire production?
 

12. Suppose you are standing on a corner watching taxis go by.  You see that the numbers of the first five taxis
are 284, 570, 321, 319, and 35.  What is the best estimate for the total number of taxis?  List the assumptions
you are making to get this estimate.
 
 

13.  You are collecting tickets at the Battle of the Bands concert.  How could you use the technique developed
here to estimate the number of tickets sold?  What assumptions do you need to make?