A gasoline manufacturer believes a new additive will result in more miles per gallon. A large number of mileage measurements on the gasoline without the additive showed a mean of 24.7 miles per gallon and a standard deviation of 4.8. An experiment was performed in which 75 cars were tested using the gasoline plus the additive. The results showed a sample mean of 26.5 miles per gallon. To evaluate these data, a directional test with α = 0.051 tail was used. Suppose that before doing the experiment, the manufacturer wants to determine the probability that he will be able to detect a real mean increase of 2.0 miles per gallon with the additive if the additive is at least that effective.

(a) If he tests 20 cars, what is the power to detect a mean increase of 2.0 miles per gallon?

(c) How many cars should he use if he wants to have a 99% chance of detecting a mean increase of 2.0 miles per gallon?