While three watchmen were guarding an orchard, a thief slipped in and stole some apples. On his way out, he met the three watchmen one after another, and to each in turn he gave half the apples he had and two besides. In this way he managed to escape with one apple. How many apples had he stolen originally?

An automobile assembly line operation has a scheduled mean completion time, μ, of 12 minutes. the standard deviation of completion times is 1.6 minutes. it is claimed that, under new management, the mean completion time has decreased. to test this claim, a random sample of 33 completion times under new management was taken. the sample had a mean of 11.2 minutes. assume that the population is normally distributed. can we support, at the 0.05 level of significance, the claim that the mean completion time has decreased under new management? assume that the standard deviation of completion times has not changed.