riddles and brain teasers

Two hospitals

A certain town is served by two hospitals. In the larger hospital, about 45 babies are born each day. In the smaller one, about 15 babies are born each day. Although the overall proportion of girls is about 50%, the actual proportion at either hospital may be greater or less on any day. At the end of a year, which hospital will have the greater number of days on which more than 60% of the babies born were girls?

The answer is 2: the smaller hospital. The Law of Large Numbers says:

The more people we survey, the more accurate the result.

Larger samples are better.

In this case, the number of children born each day is a random sample of the number of children born in a year. There is much more deviation in a small (daily, in this case) sample than over an entire year. Similarly, there will be more deviation in a sample of size 15 than in one of size 45. Even though both hospitals will have about 50% girls over the year, the smaller hospital will have more days in which the number is higher than 60% (and more days when it is less than 40%). (Consider the extreme case of a hospital that has one baby a day. Every day will be either more than 60% girls (one) or less than 40% girls (zero).

People commonly and mistakenly believe that even very small samples will have the same composition as large ones. Remember this next time someone uses a small survey sample to "prove" something -- the results may be invalid!
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