Gasoline. Many drivers of cars that can run on regular gas actually buy premium in the belief that they will get better gas mileage. To test that belief, we use 10 cars from a company fleet in which all the cars run on regular gas. Each car is filled first with either regular or premium gasoline, decided by a coin toss, and the mileage for that tankful is recorded. Then the mileage is recorded again for the same cars for a tankful of the other kind of gasoline. We don't let the drivers know about this experiment. Here are the results (miles per gallon):

a) Is there evidence that cars get significantly better fuel economy with premium gasoline?

b) How big might that difference be? Check a 90% confidence interval.

c) Even if the difference is significant, why might the company choose to stick with regular gasoline?

d) Suppose you had done a "bad thing." (We're sure you didn't.) Suppose you had mistakenly treated these data as two independent samples instead of matched pairs. What would the significance test have found? Carefully explain why the results are so different.

e) Use a matched-pairs sign test to test the appropriate hypothesis. Do your conclusions change from those in part a?