Problem: In 1848, a gold sovereign was supposed to weigh 123 grains, so that a sample of 5000 coins should have weighed about 615000 (=123*5000) grains. From historical records we know that the allowable tolerance in those days for a sample of 5000 coins was 1280 grains. Thus, if the actual total weight of the 5000 coins in the sample differed from 615000 grains by more than 1280 grains, then the Master of the Mint had failed the Trial.

Suppose the Master of the Mint in 1848 was honest and was manufacturing gold coins which weighed 123 grains on average with a standard deviation of 1 grain. It can then be asserted, using the Central Limit Theorem, that the distribution of the total weight of a random sample of 5000 coins would be normal, with a mean of 615000 grains and a standard deviation of 70.7 grains.

With this information, what is the probability that the Master would not fail the Trial (i.e. the probability that the total weight of the 5000 coins would not differ from 615000 grains

by more than 1280 grains)? (Note: your answer must include a pic of a normal curve with the relevant region shaded, along with the numerical answer)