(Solved): The manager of a restaurant believes that waiters and waitresses who introduce themselves by tellin ...

The manager of a restaurant believes that waiters and waitresses who introduce themselves by telling customers their names will get larger tips than those who don't. In fact, she claims that the average tip for the former group is \( 18 \% \), whereas that of the latter is only \( 15 \% \). If tips are normally distributed with a standard deviation of \( 3 \% \), what is the probability that in a random sample of 10 tips recorded from waiters and waitresses who introduce themselves and 10 tips from waiters and waitresses who don't, the mean of the former will exceed that of the latter? The average South African loses an average of 15 days per year to colds and flu. The natural remedy echinacea reputedly boosts the immune system. One manufacturer of echinacea pills claims that consumers of its product will reduce the number of days lost to colds and flu by one-third. To test the claim, a random sample of 50 people was drawn. Half took echinacea, and the other half took placebos. If we assume that the standard deviation of the number of days lost to colds and flu with and without echinacea is 3 days, find the probability that the mean number of days lost for echinacea users is less than that for nonusers.