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All the workers at a certain company drive to work and park in the company’s lot. The company is interested in estimating the average number of workers in a car. Which of the following methods will enable the company to estimate this quantity?

  1. Randomly choose $n$ workers, find out how many were in the cars in which they were driven, and take the average of the $n$ values.
  2. Randomly choose $n$ cars in the lot, find out how many were driven in those cars, and take the average of the $n$ values.

My intuition goes for number 2, but I'm not able to justify it formally.

srodriguex
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  • Say half the cars had two people in them, and half of them had one person. What would you expect the outcome of each of the two methods to be? – Arthur Oct 10 '13 at 14:07

1 Answers1

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In the first case, you introduce "inflation" in the sense of over-counting: any two randomly chosen workers may have very well driven in the same car. The more passengers in a car, say k of them in car $C_i$, ($1 \leq i\leq n$) the greater the chance that two or more workers will each, separately, report "k" passengers for the same car, overcounting the number of passengers in car $C_i$. That is, two or more of the selected workers may belong to the same "set."

In the second case, since the passengers in each car belong to disjoint sets, randomly selecting the cars, and then determining the number of passengers in the selected cars, ensures that the number of passengers in one car, if counted once, will not be counted again.

amWhy
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