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A certain town is served by two hospitals. In the larger hospital about 45 babies are born each day, and in the smaller hospital about 15 babies are born each day. About 50% of all babies are boys. However, the exact percentage varies from day to day. Sometimes it may be higher than 50%, sometimes lower.

For a period of 1 year, each hospital recorded the days on which more than 60% of the babies born were boys. Which hospital do you think recorded more such days?

Three possible answers:

A. the larger hospital

B. the smaller hospital

C. about the same ( within 5% of each other).

Would it be the smaller hospital, since there's less babies are born each year?

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    $\begingroup$ I don't get the title: did that doctor won a Nobel Prize (in what?) for solving that problem? It seems to be a rather simple problem to win aNobel, even if those prizes are heavily over-rated. $\endgroup$ – DonAntonio Feb 26 '14 at 4:50
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    $\begingroup$ It is since the variance is bigger. The variance of the sample proportion is $\frac{p(1-p)}{n}$, where $p$ is the proability of a boy birth and $n$ is the sample size. Big $n$ (hospital), small variance. $\endgroup$ – André Nicolas Feb 26 '14 at 4:51
  • $\begingroup$ @DonAntonio : Kahneman's Nobel Prize was for prospect theory: en.wikipedia.org/wiki/Prospect_theory, en.wikipedia.org/wiki/Daniel_Kahneman $\endgroup$ – Michael Hardy Feb 26 '14 at 5:03
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    $\begingroup$ The title is ridiculous. $\endgroup$ – Did Feb 26 '14 at 6:00
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B. The smaller hospital. The intuitive Law of Large Numbers says that with more observations the experimental average tends to get closer to the true proportion. A big hospital has more observations, so their number will be less likely to stray from 50%, or have less variance.

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