Now that we covered combinations with factorials and a bit about sampling without replacement, I’d thought we go on to sample with replacement and bootstrapping.  Yeah, cuz maybe if we’ve been to Kokomo once, say, we want to go there again — or we want to go to Key Largo three times, assuming we can go to a place three times, or Montego twice or … you get the picture.  Anyway, so we could do this many, many times and then come up with a distribution of values and summarize the general characteristic of those values, say with a mean.  Like, for example, in case you didn’t know the major American Physical Society March (APS) meeting in 2018 will be happening in Los Angeles.  Eh? APS physics, meeting … LA … would be perfect for an Order of The Dimensions movie premiere!  Oh, just humor me and pretend it’s possible.  So say, the APS and the execs at Universal fly me in for the meeting/premiere and want to put me at a hotel by the convention center where the meeting will be held.  Now, I checked the hotels around that area and saw that they range in ratings from 2.5 to 4.0.  I would hope that they (read: they better) put me in a hotel closer to a 4.0-rating, but for argument’s sake, I generated a set of 10 values between 2.5 and 4.0 using a uniform distribution so I got a fairly equal chance of getting a rating between those two values.  Now, I sampled with replacement 100 values from those 10 numbers and got a distribution that looked like this:

where the mean of all 100 sampled values was 3.22.  Which I guess is okay for a hotel rating, although in reality, you future planners out there — think Ritz-Carlton.  Thanks.  But anyway, that is one application of the bootstrap.  But how can we assess if the parameters obtained from this particular simulation or any random number generation or imputation  technique ?  Well,we can look at stuff like the standardized bias, root mean squared error, and the coverage rate.  But we can get into that next time.  Until then … APS … Universal … March meeting 2018 –think about it 😉