Question: Consumer Reports has hired you to study gas mileage obtained by the new Federico Fellini Fuel-Efficient Ferrari. Specifically, you are asked to test the manufacturer's claim that, on aver-age, the car travels 30 miles per gallon. To conduct the test, you fill the tank of the test car 10 times, and you record the car's mileage per gallon on each tank. The results of the ten trials are 29.0, 29.5, 28.8, 28.7, 31.0, 26.0, 28.0, 30.5, 30.2, 28.3. Suppose you have good reason to believe that miles per gallon on each tank of gas is an independent normal variable with a standard deviation of 3.
a. If the manufacturer's claim is true, what is the probability of a sample mean less than or equal to the one you observed?
b. After you turn in your report, you regret not having been able to make more confident statements about the validity of the manufacturer's claim_ You decide that, in future tests of other cars, if the sample mean is at least 1 mile per gallon less than the manufacturer's claim, you would like to reject the claim. How many tankfuls must you sample in a future test to make the probability less than 0.05 that you reject a claim when in fact it is true? Continue to assume that miles per gallon on a tank of gas is independently, normally distributed with a standard deviation of 3.