49. The trial of the pyx. In 1150, it was recognized in England that coins should have a standard weight of precious metal as the basis for their value. A guinea, for example, was supposed to contain 128 grains of gold. (There are 360 grains in an ounce.) In the “trial of the pyx,” coins minted under contract to the crown were weighed and compared to standard coins (which were kept in a wooden box called the pyx). Coins were allowed to deviate by no more than 0.28 grains—roughly equivalent to specifying that the standard deviation should be no greater than 0.09 grains (although they didn’t know what a standard deviation was in 1150). In fact, the trial was performed by weighing 100 coins at a time and requiring the sum to deviate by no more than or 28 grains—

equivalent to the sum having a standard deviation of about 9 grains.

a) In effect, the trial of the pyx required that the mean weight of the sample of 100 coins have a standard deviation of 0.09 grains. Explain what was wrong with performing the trial in this manner.

b) What should the limit have been on the standard deviation of the mean?

Note: Because of this error, the crown was exposed to being cheated by private mints that could mint coins with greater variation and then, after their coins passed the trial, select out the heaviest ones and recast them at the proper weight, retaining the excess gold for themselves. The error persisted for over 600 years, until sampling distributions became better understood.

50. Safe cities. Allstate Insurance Company identified the 10 safest and 10 least-safe U.S. cities from among the 200 largest cities in the United States, based on the mean number of years drivers went between automobile accidents.

The cities on both lists were all smaller than the 10 largest cities. Using facts about the sampling distribution model of the mean, explain why this is not surprising.