# Solutions to Coin Questions 1

The population of interest is all coins minted in 1799, with mean 128 gr. and a standard deviation (SD) of 1 gr. From this population a simple random sample of 100 coins was selected for the Trial of the Pyx.

Let S be the sum of the weights of the 100 coins in the sample. The expected value of the sum, E(S), is 100 times the population mean: 12800 gr. The standard error (SE) is given by the SD times the square-root of the sample size: 1*10 = 10 gr. Due to the Central Limit Theorem, the long run histogram of S is normal, centered at 12800 gr with variance equal to 100 (the square of the SE). The Master of the Mint survives the Trial of the Pyx only if the total weight S of the 100 coins in the sample weighs between (12800-32 and 12800+32) gr. We need to calculate P(survive)=P(12768 < S < 12832). Normalize by subtracting the mean (12800) from all three quantities inside the probability and by dividing by the standard error. Using Z = (S - E(S))/SE makes this probability equal to P((12768-12800)/10 < Z < (12832-12800)/10) = P(-3.2 < Z < 3.2) which is approximately 99.9 %.

Therefore, if the Master of the Mint is honest he is virtually certain to survive the Trial.