I'm probably unfit to solve this fitness question
A resident facility has a fitness room available 10 hours a day. A session starts at the hour or half-hour. A resident will do fitness for exactly 30 minutes from 1 to 3 times a week. Hero uses the room 3 times a week (1.5 hours) at random start times. Residents who do fitness also use random start times.
Over a large number of weeks, hero saw only X (e.g., 10) of the 100 other residents when he is doing fitness, some more than once.
Question: Can one use this data to estimate the proportion of residents who do fitness knowing it is at least X%?
Make reasonable simplifying assumptions as necessary. I think this problem can be viewed as one of choosing numbered balls out of a number of urns to see if they match hero's choices. Urns are available only to residents that do fitness.
3 Replies
With no responses in a week, I guess I ought to attempt a solution, so here is one.
I used simulation and regression to find a prediction equation for the total number of fitness users in a resident facility given hero’s observed number of fitness users over a number of weeks. The following assumes hero does fitness 3 times a week:
R = total number of residents
F = total number of fitness users (the dependent variable)
W3 = weeks of observation at 3 hero fitness sessions per week.
The number of hero fitness sessions/week can be varied to yield a different equation.
N = number of different residents hero observed doing fitness over W3 weeks
For the specified number of simulations, I determined the number of users hero will observe given the total number of fitness users. The following prediction equation was then developed using a power series regression of the simulation results with N as the independent variable. The R^2 was > 99%.
F = Min(R, N * 18 * W3^(-0.786)) [May become known as STatmanhal’s Fitness User equation (STFU)]
The equation will differ for hero fitness frequency but it will be of the same form and have high R^2. Of course, multiple regression could also be done to provide a single equation but I found that separate equations for hero fitness frequency did better.
Example: With 3 hero fitness sessions/week, over an 8 week period in a facility with 100 residents, hero observed 10 different users. His estimate for the total number of residents using the fitness room is
F = 10 * 18 *8^(-0.786) = 35
I would think there is an analytical solution. Excluding the cases where a user was seen more than once was more than I wanted to tackle. Anybody?
sounds like a variation of the german tank problem
Thanks for the reference.
With a quick read, the German Tank Problem refers to the Allies using the sequential serial numbers of captured tanks to estimate the total number produced. The similarity, then, is using a sample of the number of “successes: (tanks or fitness users) to estimate the total number. But numbering the residents, say by apartment numbers, provides no useful data IMO. Yet, I believe the analytic approaches described in the article may, with appropriate revision, be applicable
