|A WWII-era German Tank|
The German forces had an advantage that the Allies had to overcome: more, and technologically superior, tanks. Knowing how many tanks the Germans were producing was the first step to figuring out how to take care of this threat. So, the Allies tried conventional methods of gathering intelligence: interrogation, spies, and decoding messages.
Using the methods of Alan Turing only caused frustration for the Allied forces (PS- to decode a message (kinda) like Turing, check out this past Problem of the Week). Intercepted messages and espionage led to the belief that the Germans were churning out 1,400 tanks per month! And this just didn't seem feasible.
So, the Allies turned to their mathletes and another important clue: the Germans carefully marked each tank with its own unique serial number. By carefully observing the numbers on captured tanks, the Allies created a mathematical model to determine the rate of production. And their results were highly accurate.The Allies used the model and estimated 255 tanks per month; after the war, German notes and data described production at 256 per month.
Above is the mathematical model, written out. They made a point estimation by giving first the minimum-variance unbiased estimator in the top equation, and then calculating the variance.
After the mathematicians determined how many tanks were being produced, the Allies knew how much manpower was needed to stop them.